# Min-max construction of two capillary embedded geodesics on Riemannian
$2$-disks

Dongyeong Ko  Department of Mathematics, Rutgers University - New Brunswick,
Piscataway, NJ 08854<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove the existence of two capillary embedded geodesics with
a contact angle $\theta\in(0,\pi/2)$ on Riemannian $2$-disks with strictly
convex boundary, where the absence of a simple closed geodesic loop based on a
point of boundary is given. In particular, our condition contains the cases of
Riemannian $2$-disks with strictly convex boundary, nonnegative Gaussian
curvature and total geodesic curvature lower bound $\pi$ of the boundary.
Moreover, by providing examples, we prove that our total geodesic curvature
condition is sharp to admit a capillary embedded geodesic with a contact angle
$\theta\in(0,\pi/2)$ under the nonnegative interior Gaussian curvature
condition. We also prove the existence of Morse Index $1$ and $2$ capillary
embedded geodesics for generic metric under the assumptions above.

##  1\. Introduction

Capillary minimal submanifolds in a manifold with boundary $(M,\partial M,g)$
are submanifolds with boundary in $\partial M$ which are minimal which
intersect $\partial M$ with a constant angle $\theta\in(0,\pi/2)$, which
describe the model of the interfaces between incompressible immiscible fluids
physically. We refer to the angle $\theta$ as the contact angle. There is a
beautiful survey of capillary surfaces in the book of Finn [20]. Recently, the
analysis of capillary minimal surface on Riemannian polyhedra played a key
role to prove Gromov’s geometric comparison theorems in Li [32] (see also
[33]).

In the context of variational methods in geometry, a capillary minimal surface
$\Sigma^{2}=\partial_{rel}\Omega$ on a closed manifold with boundary
$(M^{3},\partial M^{3},g)$ with a contact angle $\theta$ is defined as a
critical point of the capilarity energy functional defined as follows:

$A^{\theta}(\Omega)=|\partial\Omega\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathring{M}|+\cos\theta|\partial\Omega\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\partial M|.$

In particular, we consider an existence problem of capillary embedded
geodesics on Riemannian $2$-disks $(D^{2},\partial D^{2},g)$.

A search of geodesics by the min-max theory dates back to Birkhoff [7] on the
existence of a closed geodesic on Riemannian $2$-spheres. Lusternik-
Schnirelmann [35] obtained three closed geodesics by finding non-trivial
higher parameter families of simple closed curves in the space of embedded
curves. Grayson [22] proved the existence of three simple closed geodesics on
any Riemannian $2$-spheres by the curve shortening flow.

In the free boundary and capillary setting, there are relatively fewer
results. Lusternik-Schnirelmann [35] proved that any bounded domain in
$\mathbb{R}^{n}$ with a smooth convex boundary contains at least $n$ distinct
orthogonal geodesic chords. Bos [8] extended the existence results to
Riemannian $n$-balls with locally convex boundary. Moreover, he provided an
example of a non-convex domain in $\mathbb{R}^{n}$ which does not admit a free
boundary embedded geodesic, which shows the necessity of the boundary
convexity condition (See Figure 1 of [8]). The discrete curve shortening
process in the free boundary setting has been developed by Gluck-Ziller [21]
and Zhou [55] which produces free boundary immersed geodesics. Moreover,
Donato-Montezuma [18] recently produced a free boundary embedded geodesic or a
geodesic loop on nonnegatively curved surfaces with a convex boundary in the
one-parameter Almgren-Pitts min-max setting (see also [3] and [31] for a few
examples of constructions).

In the forthcoming paper of the author [29], we prove the existence of two
free boundary embedded geodesics with Morse Index bound on surfaces with
convex boundary.

There has been progress in obtaining capillary surface via Almgren-Pitts min-
max theory. De Masi-De Philippis [15] constructed a min-max capillary minimal
surface on the convex domain in $\mathbb{R}^{3}$. Li-Zhou-Zhu [34] developed
min-max theory of constant mean curvature capillary surfaces in $3$-manifolds
with smooth boundary. However, Almgren-Pitts min-max procedure only gives rise
to geodesic networks rather than the embedded curves on surfaces (see [42] and
[1]). We are forced to apply additional techniques such as curve shortening
flow to obtain an embedded geodesic.

Regarding the Morse theory of minimal submanifolds obtained by min-max
methods, Marques-Neves ([37], [38]) derived upper and lower bound of Morse
Index of min-max minimal hypersurfaces on closed manifolds with dimension
$3\leq n+1\leq 7$ by developing deformation techniques. The author [28]
obtained Morse Index bounds of the simple closed geodesics on Riemannian
$2$-spheres (see also [16]). In contrast to the hypersurface case, in the
curve setting, the deformation relies on the interpolation, arising from the
curve shortening flow, between two families of simple closed curves with
length control.

For critical points of perturbed functionals of the length functional on
closed surfaces, a few results on the existence of curves with constant
geodesic curvature have been established through degree theory and min-max
methods. It has been a long-standing conjecture of Novikov [41] that for any
metric on a Riemannian $2$-sphere, there exists a closed embedded curve of
constant curvature $c$ for every $c>0$ (whose immersed version is referred as
Arnold’s conjecture [4]). Schneider [45] constructed two simple closed
magnetic curves on quarter-pinched Riemannian $2$-spheres using degree theory.
Recently, Sarnataro-Stryker [44] improved the existence result by constructing
a closed embedded curve with any prescribed constant curvature on
$1/8$-pinched Riemannian $2$-spheres, based on the Almgren-Pitts frameworks by
Zhou-Zhu [57] and Ketover-Liokumovich [27] (For the existence results of
immersed constant curvature curves, see [5] and [10]).

Additionally, while it is motivating to expect the existence of multiple
critical points of perturbed functionals associated with open domains on
manifolds, for instance, constant curvature curves and CMC hypersurfaces,
there are very few results. The difficulty in constructing multiple critical
points of weighted function mainly arises from the topological simplicity of
the space of domains compared to the space of curves (or hypersurfaces,
cycles) in many cases. Dey [17] and Mazurowski [39] have made some progress in
the multiple existence of CMC hypersurfaces using variational methods.
Mazurowski-Zhou [40] defined a half-volume spectrum and proved it is achieved
by CMC hypersurfaces together with (potentially) minimal hypersurfaces via
Allen-Cahn approach, where the construction of such hypersurfaces by Almgren-
Pitts min-max remains open.

To our knowledge, our min-max construction is the first of the multi-parameter
min-max construction of domains associated to the perturbed functionals. For
critical points of unperturbed area functionals, there have been successful
programs to construct infinitely many minimal hypersurfaces on closed
manifolds based on Almgren-Pitts min-max theory; Marques-Neves [36] proved the
existence of infinitely many closed embedded minimal hypersurfaces in closed
manifolds $(M^{n+1},g)$ with $3\leq n+1\leq 7$ satisfying the Frankel
property: any two closed embedded minimal hypersurfaces intersect each other,
and Song [49] proved the infinite existence in the general metric. Zhou [56]
proved the infinite existence for generic metric by settling the multiplicity
conjecture (See [51] and [52] for the free boundary setting).

We consider a multiple existence problem and Morse theory of embedded
geodesics with a contact angle $\theta\in(0,\pi/2)$ at boundaries on
Riemannian $2$-disks $(D^{2},\partial D^{2},g)$ with a strictly convex
boundary whose boundary total (signed) curvature $\int_{\partial D^{2}}\kappa$
is bounded below by $\pi$. A capillary geodesic $\gamma=\partial_{rel}\Omega$
with a contact angle $\theta\in(0,\pi/2)$ is a critical point of the following
functional $L^{\theta}$:

$L^{\theta}(\Omega)=|\partial\Omega\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathring{D}^{2}|+\cos\theta|\partial\Omega\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\partial D^{2}|.$

One version of our main result is as following:

######  Theorem 1.1.

For $\theta\in(0,\pi/2)$, there exists at least 2 capillary embedded geodesics
with contact angle $\theta$ on Riemannian disks $(D^{2},\partial D^{2},g)$
with a strictly convex boundary and nonnegative interior Gaussian curvature in
$D^{2}$ and a total signed geodesic curvature on boundary satisfies
$\int_{\partial D^{2}}\kappa\geq\pi$.

We show that the total curvature value $\pi$ is threshold to achieve two
capillary embedded geodesics in a general angle $\theta\in(0,\pi/2)$ in the
following sense:

######  Proposition 1.2 (Sharpness of the curvature condition).

For a given $k\in(0,\pi)$, there exists a Riemannian $2$-disk $(D^{2},\partial
D^{2},g)$ with a strictly convex boundary, interior nonnegative Gaussian
curvature and a total signed curvature $\int_{\partial D^{2}}\kappa=k$ such
that there exists $\theta\in(0,\pi/2)$ satisfying the following: The
Riemannian disk $(D^{2},\partial D^{2},g)$ does not admit a capillary embedded
geodesic with a contact angle $\theta$.

######  Remark 1.3.

We can also consider Almgren-Pitts min-max construction for the
$L^{\theta}$-functional on non-negatively curved surfaces with a convex
boundary as in the free boundary version in Donato-Montezuma [18]. In the
capillary case, we may obtain even worse boundary regularity than that of free
boundary cases. By analyzing the possible singularity formulation relying on
the cut-and-paste arguments originating from Calabi-Cao [9], in the one-
parameter min-max, we obtain a capillary geodesic network whose density
$\theta^{1}(p)$ at a single boundary singular point $p$ is bounded above by
$2$, consisting of $4$ half-geodesics locally.

######  Remark 1.4.

It would be interesting to know whether there are two capillary immersed
geodesics. We expect that the discrete curve shortening process to obtain a
free boundary geodesic in Gluck-Ziller [21] and Zhou [55] can be generalized
to the capillary setting.

We denote a _critical geodesic lasso_ as a simple geodesic loop which is a
critical point of the length functional (see Section 2.3 for the precise
definition). Our min-max argument to derive Theorem 1.1 works if the min-max
construction does not achieve a critical geodesic lasso as a min-max limit. In
this context, we formulate a more general version of Theorem 1.1 after
assuming the following hypothesis on the nonexistence of geodesic lassos with
bounded length (See Section 3.2 for the more precise formulation of the
hypothesis).

($\star$) $\text{Min-max widths are not realized by critical geodesic lassos
on }(D^{2},\partial D^{2},g).$

######  Theorem 1.5.

Suppose $(D^{2},\partial D^{2},g)$ has a strictly convex boundary and
satisfies the hypothesis $(\star)$. Then there exist at least two smooth
embedded capillary geodesics on $(D^{2},\partial D^{2},g)$.

Theorem 1.1 is obtained from Theorem 1.5 by the simple analysis based on
Gauss-Bonnet theorem. Moreover, by the same reasoning, Theorem 1.5 covers the
cases of Riemannian $2$-disks with a nonpositive Gaussian curvature or with a
total curvature of positive curvature part does not exceed $\pi$. A Riemannian
metric $g$ is called $\theta$-_bumpy_ if every capillary embedded geodesic
with a fixed contact angle $\theta\in(0,\pi/2)$ is nondegenerate i.e. there is
no capillary embedded geodesic with a fixed contact angle $\theta\in(0,\pi/2)$
that admits a non-trivial Jacobi field.

######  Corollary 1.6.

There are at least distinct two capillary embedded geodesics on Riemannian
disks $(D^{2},\partial D^{2},g)$ with a strictly convex boundary and endowed a
$\theta$-bumpy metric satisfying the hypothesis $(\star)$. Then for $k=1,2$,
there exists a capillary embedded geodesic $\gamma_{k}$ and its corresponding
domains $\Omega_{k}$ such that $\gamma_{k}=\partial_{rel}(\Omega_{k})$
satisfying $L^{\theta}(\Omega_{1})<L^{\theta}(\Omega_{2})$.

As in the Morse theoretic characterization in [28] by the author, we obtain
the following Morse Index bound of capillary embedded geodesics:

######  Theorem 1.7.

Suppose $(D^{2},\partial D^{2},g)$ is a Riemannian $2$-disk whose boundary is
strictly convex and satisfies the hypothesis $(\star)$ endowed with a
$\theta$-bumpy metric. Then for each $k=1,2$, there exists a capillary
embedded geodesic $\gamma_{k}=\partial_{rel}\Omega_{k}$ with

$index(\gamma_{k})=k$

and the $L^{\theta}$-functional of two geodesics satisfy
$L^{\theta}(\Omega_{1})<L^{\theta}(\Omega_{2})$.

We also obtain an Index bound for general metrics as:

######  Corollary 1.8.

For a $2$-Riemannian disk $(D^{2},\partial D^{2},g)$ with a strictly convex
boundary satisfying the hypothesis $(\star)$, for $k=1,2$, there exists a
capillary embedded geodesic $\gamma_{k}$ with

$index(\gamma_{k})\leq k\leq index(\gamma_{k})+nullity(\gamma_{k}).$

We outline the proof of our main results. We discuss the parametrization of
one and two parameter family of domains first. Roughly speaking, motivated by
the fact that the space of (unparametrized) embedded intervals on $D^{2}$
whose endpoints are on $\partial D^{2}$ relative to the space of point curves
retracts onto $(M^{2},\partial M^{2})$, where $M^{2}$ is a Möbius strip, which
is proven by Smale [47] (see also Appendix of [23]). Then the space of
connected domains whose relative boundary is an interval can be retracted to
an annulus $S^{1}\times I$, which is a double cover of the space of curves.
The boundary of this annulus consists of the empty set and the whole disk
which corresponds to point curves in the space of embedded curves. Moreover,
in the viewpoint of Morse theory, the empty set and the whole disk can be
regarded as Morse Index zero critical points (see Proposition 3.3). Then by
the strong Morse inequality, we can expect the existence of Morse Index $1$
and $2$ critical points. Based on this intuition, it is natural to parametrize
domains by an annulus $S^{1}\times I$ to perform $2$-parameter min-max
construction.

To find these critical points we expect and to ensure they are distinct, we
consider a $2$-parameter family of domains parametrized by the annulus
$S^{1}\times I$ whose two boundary components $S^{1}\times\\{0\\}$ and
$S^{1}\times\\{1\\}$ correspond to the empty set and the whole disk,
respectively. Then we parametrize the endpoints of relative boundaries of
domains in the family and add a condition that the endpoints of relative
boundaries to have a topological degree one. By applying Lusternik-
Schnirelmann type arguments, unless a geodesic lasso is achieved as a min-max
limit, we obtain two distinct capillary embedded geodesics with different
$L^{\theta}$-functional values or a $S^{1}$-cycle of capillary embedded
geodesics.

For the tightening process to accumulate domains whose $L^{\theta}$-functional
is close to a critical value toward capillary embedded geodesics, we apply the
curve shortening flow with fixed boundary established by Huisken [25] and
Edelen [19], and the standard tightening process by vector fields as in
Colding-De Lellis [12]. This procedure relying on the flow with a Dirichlet
boundary condition forces us to obtain a critical geodesic lasso with an
arbitary contact angle instead of capillary embedded geodesics with the
prescribed contact angle $\theta$. While Langford-Zhu [30] recently obtained
results on the long-time behavior of the free boundary curve shortening flow
on domains in $\mathbb{R}^{2}$, there is no such analysis in general Neumann
boundary condition and even it is expected to behave badly compared to the
free boundary curve shortening flow.

Furthermore, we obtain a generic Morse Index bound of capillary embedded
geodesics produced by the min-max procedure. We refine the interpolation
arguments in [28] by the author in the closed geodesics case using curve
shortening flow with fixed boundary. Together with the bumpy metric analysis
in Appendix C, we obtain capillary embedded geodesics with Morse Index $1$ and
$2$ for generic metrics satisfying the hypothesis $(\star)$.

We then construct examples showing that our boundary condition is sharp given
a nonnegatively curved surface. We consider an Euclidean cone as a disk with
an extremal metric in the sense that the interior curvature concentrates to a
vertex of the cone. If a cone angle is smaller than $\pi/6$, we can find a
simple geodesic loop based on a vertex on the boundary. This corresponds to a
cone whose boundary total curvature is smaller than $\pi$. Then if we consider
a geodesic emanating from any boundary point with a contact angle slightly
larger than the contact angle of the simple geodesic loop, then we have an
self-intersection point. Then by deforming the “tip part” of this cone, we
obtain example surfaces which do not admit a capillary embedded geodesic for
some contact angles. Hence we deduce that our boundary curvature condition in
Theorem 1.1 is optimal. This constructive proof is inspired by the example
surface illustrated in Donato-Montezuma (See Figure 1 in [18]). We expect the
absence of a capillary geodesic with a certain contact angle in case of the
existence of simple geodesic loop and the rotationally symmetric disk by the
idea above.

For $k\in(0,\pi)$, we conjecture that on Riemannian disks $(D^{2},\partial
D^{2},g)$ with a strictly convex boundary, nonnegative interior Gaussian
curvature in $D^{2}$ and a total signed geodesic curvature on boundary
satisfies $\int_{\partial D^{2}}\kappa\geq k$, there exists a capillary
embedded geodesic with a contact angle $\theta\in(0,\min(k/2,\pi/2))$ based on
cone examples in the proof of Proposition 1.2. However, some geodesics do not
appear as a min-max limit of our smooth min-max construction since they
achieve smaller $L^{\theta}$-functional than the critical value (see Theorem
1.1 of [18] and the following discussion therein). Moreover, even by the
consideration of Almgren-Pitts min-max to construct such capillary geodesics,
it seems to be delicate to rule out singular stationary varifolds of
$L^{\theta}$-functional discussed in Remark 1.3. The adaption of idea of
Mazurowski-Zhou [40] to consider boundary length preserving sweepouts may give
rise to obtain short capillary embedded geodesics which are not obtained by
our method, however we lose a control of a contact angle so there is a trade-
off.

The organization of this paper is as follows. In Section $2$, we introduce
basic notions on capillary geodesics and their properties. In Section $3$, we
construct families and domains, explain the hypothesis, and establish
tightening arguments toward capillary geodesics. In Section $4$, we show the
existence of two capillary geodesics on convex disks with a certain condition.
In Section $5$, we prove the Morse Index estimate. In Section $6$, we prove
that our boundary curvature condition is sharp.

In Appendix A, we prove the compactness theorem of capillary geodesics. In
Appendix B, we recall the curve shortening flow with fixed boundary points. In
Appendix C, we prove bumpy metric theorems of capillary embedded geodesics.

## Acknowledgments

The author is grateful to his advisor Daniel Ketover for suggesting this topic
and valuable discussions, and giving valuable comments on the draft. The
author thanks Otis Chodosh and Tom Ilmanen for providing insights on curve
shortening flow with boundary. The author is also thankful to Yevgeny
Liokumovich and Xin Zhou for valuable discussions relating to this work. The
author was partially supported by NSF grant DMS-1906385.

##  2\. Capillary geodesics and basic properties

Our ambient surface with boundary $(D^{2},\partial D^{2},g)$ is a Riemannian
$2$-disk with strictly convex boundary $\partial D^{2}$. For a compact subset
$K\subset D^{2}$, we define the relative interior $int_{D^{2}}(K)$ as an
interior with respect to the relative topology on $D^{2}$, and denote a
topological interior of $D^{2}$ as $\mathring{D^{2}}$. We denote the _relative
boundary_ $\partial_{rel}(\Omega)$ as a set of points in $D^{2}$ which are
neither in $int_{D^{2}}(\Omega)$ nor $int_{D^{2}}(D^{2}\setminus\Omega)$. We
define the space of simple domain $\mathcal{D}(D^{2})$ as

$\mathcal{D}(D^{2})=\\{\Omega|\partial_{rel}(\Omega)\text{ is a simple curve
segment with endpoints on }\partial D^{2}\\}.$

We allow $\partial_{rel}(\Omega)$ to be empty. i.e. $\mathcal{D}(D^{2})$
contains $\emptyset$ and $D^{2}$.

Also we denote that $\mathfrak{X}(D^{2})$ as a set of smooth vector fields on
$D^{2}$, and $\mathfrak{X}_{tan}(D^{2})$ as a set of smooth vector fields on
$D^{2}$ that are tangent to $\partial D^{2}$ on $\partial D^{2}$. We always
assume that $D^{2}$ is contained in some closed surface $M$ such that
$D^{2}\subset M$. Let us denote that $B_{r}(p)$ as a geodesic open ball with
radius $r$ centered at $p\in D^{2}$, and $\tilde{B}_{r}^{+}(p)$ as a half
geodesic open ball with radius $r$ centered at $p\in\partial D^{2}$.

Let $\gamma$ be a smooth, embedded curve which can be presented as a relative
boundary of a connected domain $\Omega$ in $\mathcal{D}(D^{2})$. We define
$\nu$ as a unit normal vector field of $\gamma$ pointing outward $\Omega$, and
$\overline{\eta}$ as a unit normal vector field of $\partial D^{2}$ which
points outward of $D^{2}$. Moreover, we denote $\eta$ as outward pointing unit
normal vectors of $\partial\gamma$ in $\gamma$, and $\overline{\nu}$ as
outward pointing unit vectors on $\partial\gamma$ in $\partial D^{2}$.

We recall the definition of Hausdorff distance between two nonempty subsets
$A$ and $B$ which is given by

$d_{\mathcal{H}}(A,B)=\max\Big{\\{}\sup_{a\in A}d(a,B),\sup_{b\in
B}d(b,A)\Big{\\}},$

where $d(a,B)=\inf_{b\in B}d(a,b)$ and $d(a,b)$ is an (intrinsic) distance
between two points on $(D^{2},\partial D^{2},g)$. We also denote that
$d_{\partial D^{2}}(\cdot,\cdot)$ as an intrinsic distance between two points
on $\partial D^{2}$, and $N_{h,\partial D^{2}}(A)$ as a $h$-neighborhood in
$\partial D^{2}$ of $A\subset\partial D^{2}$ with the distance function
$d_{\partial D^{2}}$.

Let $V_{1}(D^{2})$ be the space of rectifiable $1$-varifolds whose support is
in $D^{2}$, and $IV_{1}(D^{2})$ be a space of integer rectifiable
$1$-varifolds in $D^{2}$; $G_{1}(D^{2})$ be the Grassmannian line bundle over
$D^{2}$. We endow the varifold $F$-metric on $V_{1}(D^{2})$ as of 2.1(19)(20)
in [43].

###  2.1. Capillary geodesics

Capillary geodesics are critical points of weighted length functional defined
on $\mathcal{D}(D^{2})$. For $\Omega\in\mathcal{D}(D^{2})$, we denote
$\partial^{b}\Omega$ and $\partial^{i}\Omega$ by varifolds
$\partial\Omega\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\partial D^{2}$ and
$\partial\Omega\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathring{D^{2}}$, respectively.
For a fixed capillary contact angle $\theta\in(0,\pi/2)$, we define a
_capillary varifold_ $\partial^{\theta}\Omega\in V_{1}(D^{2})$ as

$\partial^{\theta}\Omega=\partial^{i}\Omega+\cos\theta\cdot\partial^{b}\Omega.$

Let us call $\partial^{i}\Omega$ by _interior varifold_ and
$\partial^{b}\Omega$ by _boundary varifold_. Then we define a capillary length
functional $L^{\theta}(\Omega)$ on $\mathcal{D}(D^{2})$ as

(1)
$L^{\theta}(\Omega)=|\partial^{\theta}\Omega|=|\partial^{i}\Omega|+\cos\theta\cdot|\partial^{b}\Omega|.$

Let $p_{1}$ and $p_{2}$ be two endpoints of $\gamma=\partial_{rel}(\Omega)$ on
$\partial D^{2}$. Now we define $\gamma$ as a _capillary embedded geodesic
with a contact angle_ $\theta\in(0,\pi/2)$ if it is an embedded geodesic and
$\langle\eta,\overline{\nu}\rangle=-\cos\theta$ at $p_{1}$ and $p_{2}$. By
calculations of the first variation (see 16.2 of [46]), the first variation
formula of $L^{\theta}$ for $X\in\mathfrak{X}_{tan}(D^{2})$ holds:

(2) $\partial^{\theta}\Omega(X)=-\int_{\gamma}X\cdot Hds+\langle
X,(\eta+\cos\theta\overline{\nu})\rangle|_{p_{1}}+\langle
X,(\eta+\cos\theta\overline{\nu})\rangle|_{p_{2}},$

where $H$ is a mean curvature vector of $\gamma$. We call a varifold $V\in
V_{1}(D^{2})$ as a stationary varifold if $\partial V(X)=0$ for any
$X\in\mathfrak{X}_{tan}(D^{2})$. (2) gives the relation between stationary
varifold and capillary geodesics:

######  Proposition 2.1.

For $\Omega\in\mathcal{D}(D^{2})$, a capillary varifold
$\partial^{\theta}\Omega$ is stationary if and only if
$\gamma:=\partial_{rel}(\Omega)$ is a capillary geodesic with contact angle
$\theta$.

Moreover, we prove the following lemma which will be useful later (cf.
Corollary 1.3 in [34]).

######  Proposition 2.2.

Let $\mathcal{S}$ be a compact subset of $\mathcal{D}(D^{2})$ in the
$F$-distance metric sense of their associated capillary varifolds. Then for
every $\epsilon>0$, there exists $\delta>0$ such that for every
$\Omega^{\prime}\in\mathcal{D}(D^{2})$ and $\Omega\in\mathcal{S}$

$F(\partial^{\theta}\Omega^{\prime},\partial^{\theta}\Omega)<\delta\Rightarrow\,\,F(\partial^{i}\Omega^{\prime},\partial^{i}\Omega)<\epsilon.$

###### Proof.

For a fixed $\Omega\in\mathcal{S}$, it holds since for a sequence
$\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ of $\Omega_{i}\in\mathcal{D}(D^{2})$,
$\lim_{i\rightarrow\infty}F(\partial^{\theta}\Omega_{i},\partial^{\theta}\Omega)=0$
holds if and only if
$\lim_{i\rightarrow\infty}F(\partial^{i}\Omega_{i},\partial^{i}\Omega)=0$. The
compactness property completes the proof. ∎

###  2.2. Second variation formula and Morse Index

For $X\in\mathfrak{X}_{tan}(D^{2})$, let us denote $f=\langle X,\nu\rangle$.
For a capillary embedded geodesic $\gamma$ which is induced by
$\Omega\in\mathcal{D}(D^{2})$ and $f\in C^{\infty}(\gamma)$, by Section 2.1 of
[24], we have a second variation formula of $L^{\theta}$ as following:

(3)
$Q(f,f):=\partial^{2}L^{\theta}|_{\Omega}(X,X)=\int_{\gamma}(|\nabla_{\gamma}f|^{2}-Kf^{2})ds-\frac{1}{\sin\theta}(\kappa(p_{1})f^{2}(p_{1})+\kappa(p_{2})f^{2}(p_{2})),$

where $\kappa$ is a geodesic curvature of $\partial D^{2}$. We call a
capillary geodesic $\gamma$ is stable if $Q(f,f)$ is nonnegative for all $f\in
C^{\infty}(\gamma)$, and we define Morse index of $\gamma$ to be a maximal
dimension of a subspace of $C^{\infty}(\gamma)$ where $Q$ is negative
definite. i.e. the number of negative eigenvalues of the stability operator
associated with $Q$.

###  2.3. Critical simple geodesic lassos

On $(D^{2},\partial D^{2},g)$, we define a (smooth) closed curve
$\gamma:[0,L]\rightarrow D^{2}$ to be a _simple geodesic lasso_ if
$\gamma(0)=\gamma(L)\in\partial D^{2}$ and $\gamma((0,L))$ is an (unit-speed
parametrized) embedded geodesic, where $L$ is a length of $|\gamma|$. In
particular, we call a simple geodesic lasso $\gamma$ to be a _critical
geodesic lasso_ if

(4) $\langle\gamma^{\prime}(0)-\gamma^{\prime}(L),v\rangle=0$

for any $v\in T_{\gamma(0)}\partial D^{2}$. Note that (4) provides a geometric
description of a critical geodesic lasso that $\gamma$ is a geodesic without
self-intersection point but at endpoints intersecting with the same angle with
tangent vector of $\partial D^{2}$ at the endpoint.

Let us denote (nonparametrized) a domain $\Omega\subset D^{2}$ such that
$\partial_{rel}(\Omega)=\gamma$ for some critical geodesic lasso $\gamma$ by a
_crtical lasso domain_. Even though $\Omega$ may not be in the set of simple
domains $\mathcal{D}(D^{2})$, we define capillary varifolds of $\Omega$
correspondingly in such cases. Note that
$supp(\partial^{b}\Omega)=\emptyset\text{ or }\partial M$ in this case. We
also denote $\Lambda_{las,a}$ as a set of critical lasso domains $\Omega$ such
that $L^{\theta}(\Omega)\leq a$. Note that the first variation formula (2)
still can be applied to $\partial^{\theta}\Omega$ after idntifying endpoints.
By combining the definition and the first variation formula (2), we obtain the
following:

######  Proposition 2.3.

If $\Omega$ is a critial lasso domain, then a corresponding capillary varifold
$\partial^{\theta}\Omega$ is a stationary varifold.

###### Proof.

Let us denote a critical geodesic lasso and its vertex by $\gamma$ and $p$,
respectively. We parametrize $\gamma$ by its length $L$. Then by considering
boundary terms of the first variation of $\partial^{\theta}\Omega$ in (2), we
obtain the first variation formula of $\partial^{\theta}\Omega$ as

$\partial^{\theta}\Omega(X)=-\int_{\gamma}X\cdot Hds+\langle
X,-\gamma^{\prime}(0)+\gamma^{\prime}(L)\rangle|_{p}.$

Since $\gamma$ is a geodesic, the interior term vanishes and the boundary term
vanishes by (4). Hence $\partial^{\theta}\Omega(X)=0$ for any
$X\in\mathcal{X}_{tan}(D^{2})$ and $\partial^{\theta}\Omega$ is a stationary
varifold. ∎

##  3\. Smooth min-max construction of capillary geodesics

In this section, we introduce one and two parameter smooth min-max
construction to construct two distinct capillary geodesics (see [12] for
minimal surface version, [13] for multi-parameter version for minimal surface,
and [14] for boundary version).

We also discuss the hypothesis we impose to, and technical parts such as
nontriviality of sweepouts and the pull-tight procedure via curve shortening
flow with boundary.

###  3.1. One and two parameter families

We take a smooth topology of capillary varifolds we introduced in the last
section on $\mathcal{D}(D^{2})$. First we describe a $1$-parameter family of
domains. Let us denote $I=[0,1]$ as a unit interval. We call the family of
domains $\\{\Phi_{t}\\}_{t\in I}$ a $1$-_sweepout_ if

  1. (1)

$\Phi_{0}=\emptyset$ and $\Phi_{1}=D^{2}$.

  2. (2)

$\Phi_{t}$ converges to $\Phi_{t_{0}}$ in the Hausdorff topology as
$t\rightarrow t_{0}$.

  3. (3)

$\Phi_{t}\in\mathcal{D}(D^{2})$ for $t\in[0,1]$.

Now we construct a $2$-parameter family of domains. Note that $\partial D^{2}$
is diffeomorphic to a unit circle $S^{1}$ and we fix a counterclockwise
orientation on $\partial D^{2}$. Let us define _a pair of endpoints function_
$\partial:\mathcal{D}(D^{2})\setminus\\{\emptyset,D^{2}\\}\rightarrow\partial
D^{2}\times\partial D^{2}$ as an ordered pair of points which represents the
endpoints of $\gamma=\partial_{rel}(\Omega)$ for
$\Omega\in\mathcal{D}(D^{2})\setminus\\{\emptyset,D^{2}\\}$, where we take the
first and second endpoints of $\partial^{b}{\Omega}$ with the order on the
counterclockwise orientation on $\partial D^{2}$ as convention. We define
$\pi_{1},\pi_{2}:\partial D^{2}\times\partial D^{2}\rightarrow\partial D^{2}$
as a projection map of $\partial$ to the first and second coordinate,
respectively. We define an _endpoint function_
$e:\mathcal{D}(D^{2})\setminus\\{\emptyset,D^{2}\\}\rightarrow\partial D^{2}$
as

(5) $e(\Omega)=\pi_{1}\partial(\Omega),$

for $\Omega\in\mathcal{D}(D^{2})\setminus\\{\emptyset,D^{2}\\}$.

We define the family of domains $\\{\Phi_{s,t}\\}_{(s,t)\in S^{1}\times I}$ as
a $2$-_sweepout_ if

  1. (1)

$\Phi_{s,0}=\emptyset$ and $\Phi_{s,1}=D^{2}$ for any $s\in S^{1}$.

  2. (2)

$\Phi_{s,t}$ converges to $\Phi_{s_{0},t_{0}}$ in the Hausdorff topology as
$(s,t)\rightarrow(s_{0},t_{0})$.

  3. (3)

$\Phi_{s,t}\in\mathcal{D}(D^{2})$ for $(s,t)\in S^{1}\times[0,1]$.

  4. (4)

The map $f_{t}:S^{1}\rightarrow\partial D^{2}$ defined by
$f_{t}(s):=e(\Phi_{s,t})$ has a mapping degree $1$ for every $t\in(0,1)$.

We denote $\mathcal{P}_{i}$ to be the set of $i$-sweepouts for $i=1,2$.

######  Remark 3.1.

  1. (1)

By the convention to take an order of endpoints and the definition of mapping
degree, we obtain that a corresponding second endpoint function in condition
(4), namely $\pi_{2}\partial$, also has a mapping degree $1$ for every
$t\in(0,1)$.

  2. (2)

Since the continuity of $\Phi_{s,t}$ in Hausdorff topology implies the
continuity of the endpoint function $e$, the mapping degree $1$ property at
one point $t\in(0,1)$ in the condition (4) indeed ensures the mapping degree
$1$ for whole $t\in(0,1)$.

  3. (3)

We use the nonzero mapping degree property to prove the nontriviality of
$2$-sweepouts, hence fixing a nontrivial mapping degree to define a
$2$-sweepout does not affect the two parameter min-max construction.

######  Remark 3.2.

While our $2$-sweepouts of domains do not directly correspond to $2$-parameter
families of embedded curves whose boundary points are on $\partial D^{2}$,
there is an analogy between two classes. More specifically, certain
$2$-sweepouts of domains appear as a double cover of $2$-parameter family of
embedded curves with endpoints on $\partial D^{2}$ detecting a second
nontrivial (relative) homology in the space of such embedded curves. The
covering map is a projection map sending domains to its relative boundary
here.

For instance, on the flat disk
$D^{2}=\\{(x,y)\,|\,x^{2}+y^{2}=1\\}\subset\mathbb{R}^{2}$, we denote the
lines $\mathcal{L}(a,b,c)$ to be

$\mathcal{L}(a,b,c):=\\{ax+by=c\\}\cap D^{2}$

and

$\mathcal{L}:=\bigcup_{(a,b,c)\in\mathbb{R}^{3}}\mathcal{L}(a,b,c).$

Then the $2$-parameter family

$\mathbb{R}P^{2}\rightarrow\mathcal{L},\,[a,b,c]\mapsto\mathcal{L}(a,b,c)$

is a $2$-sweepout of curves on $(D^{2},\partial D^{2})$ and its double cover
in $\mathcal{D}(D^{2})$ is a $2$-sweepout of domains after proper
reparametrization.

For $i=1,2$, we define the $i$-_width_ $w_{i}(D^{2})$ of sweepouts
$\mathcal{P}_{i}$ by

(6) $w_{i}(D^{2}):=\inf_{\Phi\in\mathcal{P}_{i}}\sup_{x\in
X_{i}}L^{\theta}(\Phi_{x}),$

where $X_{1}=I$ and $X_{2}=S^{1}\times I$. By the definition of $k$-sweepouts,
we have $w_{1}(D^{2})\leq w_{2}(D^{2})$.

We denote the sequence of $k$-sweepout $\\{\Phi^{j}\\}$ to be a _minimizing
sequence_ if $\lim_{j\rightarrow\infty}\sup_{x\in
X_{i}}L^{\theta}(\Phi^{j}_{x})=w_{i}$. For a minimizing sequence
$\\{\Phi^{j}\\}$ and for some sequence of parameters $\\{x_{j}\\}$, if
$\lim_{j\rightarrow\infty}L^{\theta}(\Phi^{j}_{x_{j}})=w_{i}$ then we call
$\\{\Phi^{j}_{x_{j}}\\}$ as a _min-max sequence_. Moreover, we define
$\Lambda_{e,w_{k}}$ to be the set of capillary varifolds
$\partial^{\theta}\Omega$ such that $L^{\theta}(\Omega)=w_{k}$, and
$\partial_{rel}(\Omega)$ is a multiplicity one embedded geodesic whose both
endpoints are on $\partial D^{2}$.

We define $\Lambda_{\theta,\infty,a}$ to be the set of stationary capillary
varifolds $V$ such that $|V|\leq a$ with respect to
$\mathfrak{X}_{tan}(D^{2})$. Moreover, we define $\Lambda_{\theta,w_{k}}$ to
be a set of capillary varifolds $\partial^{\theta}\Omega$ whose
$L^{\theta}$-functional achieves value $w_{k}$ and relative boundary is a
multiplicity one embedded geodesic whose both endpoints are on $\partial
D^{2}$, and both contact angles at endpoints are $\theta$. Note that
$\Lambda_{e,w_{k}}\cap\Lambda_{\theta,\infty,w_{k}}=\Lambda_{\theta,w_{k}}$ if
$\Lambda_{las,w_{k}}=\emptyset$. We denote $\Lambda_{\theta,w_{k},j}$ and
$\Lambda^{j}_{\theta,w_{k}}$ by subsets of $\Lambda_{\theta,w_{k}}$ consisting
of capillary varifolds whose Morse index less than or equal to $j$ or larger
than or equal to $j$, respectively. We denote $\mathcal{D}_{\theta,w_{k}}$ as
the corresponding set of domains of the set of capillary varifolds
$\Lambda_{\theta,w_{k}}$. Let the _critical set_ $\Lambda(\\{\Phi_{j}\\})$ be
a set of stationary capillary varifolds can be obtained by the limit of min-
max sequence induced by $\\{\Phi_{j}\\}$.

We can also deduce that $w_{k}>\cos\theta|\partial D^{2}|$ so that both widths
are achieved by nontrivial stationary varifolds. Here we follow Section 2.3 of
[34] by taking constant curvature $c=0$ there.

######  Proposition 3.3 (Theorem 2.10 in [34]).

For a fixed $\theta\in(0,\pi/2)$ and $k=1,2$, $\cos\theta|\partial
D^{2}|<w_{k}$.

###  3.2. Hypothesis

Let $(D^{2},\partial D^{2},g)$ be a Riemannian disk with a strictly convex
boundary. By Proposition 2.2, capillary varifolds of critical lasso domains
are critical varifolds so that
$\Lambda_{las,a}\subseteq\Lambda_{\theta,\infty,a}$ for any
$a\in\mathbb{R}_{+}$. Now we assume the following to avoid detecting a
critical geodesic lasso appearing as min-max limit capillary varifolds:

($\star$) $\Lambda_{las,2w_{2}}=\emptyset.$

Saying in more detail, there is no criticial lasso whose length is bounded by
$2w_{2}$.

###  3.3. Pulling-tight toward embedded capillary geodesics

In this section, we discuss the pull-tight process to accumulate capillary
varifolds whose $L^{\theta}$-functional is close enough to the widths $w_{k}$
toward capillary embedded geodesics. Our pull-tight procedure is two fold. The
first procedure is to accumulate toward embedded geodesics, and the second one
is to accumulate towards stationary capillary varifolds so that toward
capillary embedded geodesics. Moreover, we obtain the Morse index upper bound
of limit capillary embedded geodesics.

We sketch the first procedure by applying convergence and regularity of curve
shortening flow with fixed endpoints in surfaces with strictly convex boundary
(in Appendix B) à la Edelen [19]. This is a boundary analog of Grayson’s
theorem [22]. We obtain the boundary version of Theorem 2.1 in [28] as
follows.

######  Theorem 3.4.

Suppose $(D^{2},\partial D^{2},g)$ to be a smooth Riemannian $2$-disk with
strictly convex boundary satisfying the hypothesis $(\star)$. For $k=1,2$, for
any minimizing sequence $\\{\Phi^{j}\\}$ of $k$-sweepouts, there is a deformed
minimizing sequence $\\{\hat{\Phi}^{j}\\}$ of $\\{\Phi^{j}\\}$ satisfying the
following property. For any $s>0$, there is some $0<a<w_{k}$ satisfying

(7) $\\{\partial^{\theta}\hat{\Phi}^{j}_{x}\in
V_{1}(D^{2}):L^{\theta}(\hat{\Phi}^{j}_{x})\geq
w_{k}-a\\}\subset\bigcup_{\gamma\in\Lambda_{e,w_{k}}}B^{F}_{s}(\gamma)$

for all sufficiently large $j$, where $B^{F}_{s}(\gamma)$ is a $F$-metric ball
with the center $\gamma$.

###### Proof.

By the compactness theorem of embedded geodesics with endpoints on boundary
(Theorem A.1), we have compactness of $\Lambda_{e,w_{k}}$ and $F$-distance
function $F(\cdot,\Lambda_{e,w_{k}})$ is well-defined.

Due to Theorem B.1, the regularity and smoothness of curve shortening flow
with fixed boundary condition, curves converges to geodesic segments as time
goes to infinity by the curve shortening flow with convex boundary. Note that
the length of curves monotonically decrease by flow, and so
$L^{\theta}$-functional also monotonically decreases.

For a given relative boundary of domain $\alpha$ whose both endpoints are on
$\partial D^{2}$, let us denote $H(t,\alpha)$ as a curve deformed by curve
shortening flow with boundary by time $t$. For any time $t\in[0,\infty)$,
$H(t,\alpha)$ is an embedded curve if $\alpha$ is embedded. We define the
tightened domain $\hat{\Phi}^{j}_{x}\in\mathcal{D}(D^{2})$ by flow by taking a
sequence $t_{j}\rightarrow\infty$ and domain $\hat{\Phi}^{j}_{x}$ such that

$\displaystyle\partial^{i}\hat{\Phi}^{j}_{x}$
$\displaystyle=H(t_{j},\partial^{i}\Phi^{j}_{x})$
$\displaystyle\partial^{b}\hat{\Phi}^{j}_{x}$
$\displaystyle=\partial^{b}\Phi^{j}_{x},$

for $x\in X_{k}$. Note that $\hat{\Phi}^{j}$ is well-defined by the continuity
and regularity property of curve shortening flow with boundary of curves with
finite length. Let us argue by contradiction.

Suppose there is a sequence of
$\\{\hat{\Phi}^{j_{k}}_{x_{j_{k}}}\\}_{k\in\mathbb{N}}$ such that
$L^{\theta}(\hat{\Phi}^{j_{k}}_{x_{j_{k}}})\geq w_{k}-a_{j}$ and
$F(\partial^{\theta}\hat{\Phi}^{j_{k}}_{x_{j_{k}}},\Lambda_{e,w_{k}})\geq s$
and $a_{j}\rightarrow 0$. Notice that
$L^{\theta}(\hat{\Phi}^{j_{k}}_{x_{j_{k}}})-L^{\theta}(\Phi^{j_{k}}_{x_{j_{k}}})\rightarrow
0$ as $k\rightarrow\infty$ by the convergence of the length. By monotone
decreasing property of $L^{\theta}$-functional through curve shortening flow
with boundary, $\\{\partial^{\theta}\Phi^{j_{k}}_{x_{j_{k}}}\\}$ converges to
elements of $\Lambda_{e,w_{k}}$ and
$\\{\partial^{\theta}\hat{\Phi}^{j_{k}}_{x_{j_{k}}}\\}$ as well. This
contradicts to our assumption
$F(\partial^{\theta}\hat{\Phi}^{j_{k}}_{x_{j_{k}}},\Lambda_{e,w_{k}})\geq s$.
Also note that a crticial geodesic lasso does not appear as a limit geodesic
by the hypothesis $(\star)$. ∎

Now we can adopt standard pull-tight arguments of Proposition 4.1 in [12] by
modifying the functional from area functional to $L^{\theta}$-functional, and
from $\mathfrak{X}(D^{2})$ to $\mathfrak{X}_{tan}(D^{2})$. We have the pull-
tight type theorem as follows:

######  Theorem 3.5.

Suppose $(D^{2},\partial D^{2},g)$ is a smooth Riemannian $2$-disk with
strictly convex boundary satisfying the hypothesis $(\star)$. For $k=1,2$, for
any minimizing sequence $\\{\Phi^{j}\\}$ of $k$-sweepouts, there is a deformed
minimizing sequence $\\{\hat{\Phi}^{j}\\}$ of $\\{\Phi^{j}\\}$ satisfying the
following property. For any $s>0$, there is some $0<a<w_{k}$ satisfying

$\\{\partial^{\theta}\hat{\Phi}^{j}_{x}\in
V_{1}(D^{2}):L^{\theta}(\hat{\Phi}^{j}_{x})\geq
w_{k}-a\\}\subset\bigcup_{\gamma\in\Lambda_{\theta,w_{k}}}B^{F}_{s}(\gamma)$

for all sufficiently large $j$, where $B^{F}_{s}(\gamma)$ is a $F$-metric ball
with the center $\gamma$.

###### Proof.

By Theorem 3.4, for given minimizing sequence $\\{\Phi^{j}\\}$, there is a
deformed minimizing sequence $\\{\Psi^{j}\\}$ such that for any $s>0$ there is
some $0<a^{\prime}<w_{k}$ satisfying (7):

$\\{\partial^{\theta}\Psi^{j}_{x}\in
IV_{1}(D^{2}):L^{\theta}(\Psi^{j}_{x})\geq
w_{k}-a^{\prime}\\}\subset\bigcup_{\gamma\in\Lambda_{e,w_{k}}}B^{F}_{s}(\gamma).$

Now we follow Section 4 in [12], modify the functional and change
$\mathfrak{X}(M)$ to $\mathfrak{X}_{tan}(D^{2})$. We have a compactness of
$\Lambda_{\theta,2w_{2}}$ by Corollary A.2. We construct a tightening map as

$H:[0,1]\times\\{\Omega\in\mathcal{D}(D^{2})\,|\,L^{\theta}(\Omega)\leq
2w_{2}\\}\rightarrow\\{\Omega\in\mathcal{D}(D^{2})\,|\,L^{\theta}(\Omega)\leq
2w_{2}\\},$

satisfying

  1. (i)

$H(0,\Omega)=\Omega$ for all $\Omega$;

  2. (ii)

$H(t,\Omega)=\Omega$ if
$\partial^{\theta}\Omega\in\Lambda_{\theta,\infty,2w_{2}}$

  3. (iii)

if $\partial^{\theta}\Omega\notin\Lambda_{\theta,\infty,2w_{2}}$, then

$L^{\theta}(H(1,\Omega))-L^{\theta}(\Omega)\leq-L(F(\partial^{\theta}\Omega,\Lambda_{\theta,\infty,2w_{2}}))<0,$

where $L:[0,\infty)\rightarrow[0,\infty)$ is a continuous function with
$L(0)=0$ and $L(t)>0$ when $t>0$.

Let us define $\hat{\Phi}^{j}_{x}:=H(1,\Psi^{j}_{x})$. Then the proof of
Proposition 4.1 in [12] gives that a capillary varifold of every min-max
sequence $\\{\hat{\Phi}^{j}_{x_{j}}\\}$ of $\\{\hat{\Phi}^{j}\\}$ converges to
a stationary capillary varifold $V$. Moreover, by the continuity of $H$ and
the fact that a stationary varifold $V$ is a fixed point of $H$, and the fact
that an interior varifold of every min-max sequence of $\\{\Psi^{j}\\}$ indeed
converges to an embedded geodesic, we have $V\in\Lambda_{\theta,w_{k}}$ and
$|V|=w_{k}$. By the hypothesis $(\star)$, a capillary varifold of critical
lasso domain does not appear as a min-max limit capillary varifold, and the
proof is completed. ∎

Combining with the finiteness theorem on $\theta$-bumpy metric, we can apply
Deformation Theorem A in [37] to rule out capillary embedded geodesics with
large Morse index by the applying same arguments in Theorem 2.3 in [28]:

######  Theorem 3.6.

Suppose $(D^{2},\partial D^{2},g)$ to be a smooth Riemannian $2$-disk with
strictly convex boundary satisfying the hypothesis $(\star)$. For $k=1,2$, for
any minimizing sequence $\\{\Phi^{j}\\}$ of $k$-sweepouts, there is a deformed
minimizing sequence $\\{\hat{\Phi}^{j}\\}$ of $\\{\Phi^{j}\\}$ satisfying the
following property. For any $s>0$, there is some $0<a<w_{k}$ satisfying

$\\{\partial^{\theta}\hat{\Phi}^{j}_{x}\in
V_{1}(D^{2}):L^{\theta}(\hat{\Phi}^{j}_{x})\geq
w_{k}-a\\}\subset\bigcup_{\gamma\in\Lambda_{\theta,w_{k},k}}B^{F}_{s}(\gamma)$

for all sufficiently large $j$, where $B^{F}_{s}(\gamma)$ is a $F$-metric ball
with the center $\gamma$.

##  4\. Existence of multiple capillary geodesics

In this section, we prove the existence of multiple capillary embedded
geodesics on $(D^{2},\partial D^{2},g)$ satisfying the hypothesis $(\star)$
based on our min-max constructions in Section 3. For $k=1,2$, we proved that
each $k$-sweepout produces at least one capillary embedded geodesic in Theorem
3.5 as a limit of min-max sequence (cf. Section 6 of [36] and Section 3 of
[26]). We now show that either $k$-widths $w_{k}(D^{2})$ are different each
other or there is an $S^{1}$-cycle of capillary geodesics by Lusternik-
Schnirelmann type arguments. Hence there are at least two capillary embedded
geodesics on $(D^{2},\partial D^{2},g)$.

First of all, we prove that if a capillary varifold of
$\Omega^{\prime}\in\mathcal{D}(D^{2})$ is sufficiently close to a capillary
varifold of a given domain $\Omega\in\mathcal{D}(D^{2})$ in an $F$-metric
topology then the endpoint function $e(\Omega^{\prime})$ should be contained
in the small neighborhood $N_{h,\partial D^{2}}(supp(\partial^{b}\Omega))$
which is a proper open subset of $\partial D^{2}$. We apply the arguments in
the proof of Lemma 3.1 in [28] with the modification of constants on the
relation between $F$-distance and Hausdorff distance.

######  Lemma 4.1.

Let $\Omega\in\mathcal{D}(D^{2})$ be a given domain. For every $h>0$, there
exists $\delta=\delta(h,\Omega)>0$ if

(8) $F(\partial^{\theta}\Omega,\partial^{\theta}\Omega^{\prime})<\delta$

then

(9) $e(\Omega^{\prime})\in N_{h,\partial D^{2}}(supp(\partial^{b}\Omega)).$

###### Proof.

For a given $h>0$, we can find $0<h^{\prime}<h$ depending on $\Omega$ such
that

(10) $\partial D^{2}\cap N_{h^{\prime}}(\partial\Omega)\subset N_{h,\partial
D^{2}}(supp(\partial^{b}\Omega)).$

For $p\in\partial\Omega$, note that the density $\theta(p)\geq\cos\theta>0$.
We can apply the arguments of the proof to bound Hausdorff distance of Lemma
3.1 in [28] after changing constants depending on the density lower bound. Let
us take $\delta=h^{\prime 2}\cos\theta/10$. Then by (8) and the application of
the arguments in the proof of Lemma 3.1 in [28] directly, we have

(11) $d_{\mathcal{H}}(\partial\Omega,\partial\Omega^{\prime})<h^{\prime}.$

(11) implies that $e(\Omega^{\prime})$ is contained in $\partial D^{2}\cap
N_{h^{\prime}}(\partial\Omega)$, and we obtain (9) from (10). ∎

We now prove the multiple existence of capillary embedded geodesics.

######  Theorem 4.2.

Suppose $(D^{2},\partial D^{2},g)$ satisfies the hypothesis $(\star)$. If
$w_{1}(D^{2})=w_{2}(D^{2})$, then there exist infinitely many distinct smooth
embedded capillary geodesics in $(D^{2},\partial D^{2},g)$.

###### Proof.

Let us argue by contradiction. Suppose that there are only finitely many
capillary embedded geodesics in $(D^{2},\partial D^{2},g)$. Denote the set of
domains $\Omega_{i}\in\mathcal{D}(D^{2})$ such that
$L^{\theta}(\Omega_{i})=w_{2}(D^{2})$ and $\partial^{i}\Omega_{i}$ is a
capillary embedded geodesic for each $1\leq i\leq k$ by
$\mathcal{S}=\\{\Omega_{1},...,\Omega_{k}\\}$, and the corresponding set of
capillary varifolds by $\mathcal{V}_{\mathcal{S}}$, namely
$\mathcal{V}_{\mathcal{S}}=\\{\partial^{\theta}\Omega_{1},...,\partial^{\theta}\Omega_{k}\\}$.
There exists $s_{0}>0$ such that $B^{F}_{s_{0}}(\partial^{\theta}\Omega_{i})$
are disjoint each other for $1\leq i\leq k$. We denote that
$\gamma_{i}=\partial^{\theta}\Omega_{i}$.

Also there exists $h_{0}>0$ such that $N_{h_{0},\partial
D^{2}}(supp(\partial^{b}\Omega_{i}))$ are proper open subsets of $\partial
D^{2}$ for all $1\leq i\leq k$. Then we take
$\delta_{i}:=\delta(h_{0},\Omega_{i})$ in Lemma 4.1 for $1\leq i\leq k$.

We consider a minimizing sequence of $2$-sweepouts
$\\{\Phi^{j}\\}_{j\in\mathbb{N}}$ and its tightened minimizing sequence
$\\{\hat{\Phi}^{j}\\}_{j\in\mathbb{N}}$ by applying Theorem 3.5 so that it
satisfies the following property: For any $s>0$, there exists $a>0$ such that

(12) $\\{\partial^{\theta}\hat{\Phi}^{j}_{x}\in
V_{1}(D^{2}):L^{\theta}(\hat{\Phi}^{j}_{x})\geq
w_{2}-a\\}\subset\bigcup_{\gamma\in\mathcal{V}_{\mathcal{S}}}B^{F}_{s}(\gamma)$

for sufficiently large $j$. We take $0<s<\min(s_{0}/2,\min_{1\leq i\leq
k}\delta_{i}/2)$. For each $j$, we define open sets $\mathcal{Y}_{j}$ and
$\mathcal{Z}_{j}$ to be

$\mathcal{Y}_{j}:=\\{x\in S^{1}\times
I\,|\,F(\partial^{\theta}\hat{\Phi}^{j}_{x},\mathcal{V}_{\mathcal{S}})<2s\\}$

and

(13) $\mathcal{Z}_{j}:=\\{x\in S^{1}\times
I\,|\,F(\partial^{\theta}\hat{\Phi}^{j}_{x},\mathcal{V}_{\mathcal{S}})>s\\}.$

Let us divide $\mathcal{Y}_{j}$ into

$\mathcal{Y}_{j,i}=\\{x\in S^{1}\times
I\,|\,F(\partial^{\theta}\hat{\Phi}^{j}_{x},\partial^{\theta}\Omega_{i})<2s\\}.$

By the choice of $s$, $\mathcal{Y}_{j}=\cup_{1\leq i\leq k}\mathcal{Y}_{j,i}$
and open sets $\mathcal{Y}_{j,i}$ are disjoint each other.

For a fixed $j$, we first claim that there exists a closed curve
$\alpha_{j}:S^{1}\rightarrow\mathcal{Y}_{j}$ which is not nullhomotopic in
$S^{1}\times I$. Suppose not. Then since all the curves in $\mathcal{Y}_{j}$
are contractible, we can find a continuous path
$\beta_{j}:[0,1]\rightarrow(S^{1}\times
I)\setminus\mathcal{Y}_{j}\subset\mathcal{Z}_{j}$ such that $\beta_{j}(0)\in
S^{1}\times\\{0\\}$ and $\beta_{j}(1)\in S^{1}\times\\{1\\}$ (See the proof of
Theorem 1.6 in [26] for a similar Lusternik-Schnirelmann type argument).

We have $\hat{\Phi}^{j}_{\beta_{j}(0)}=\emptyset$ and
$\hat{\Phi}^{j}_{\beta_{j}(1)}=D^{2}$ and
$\Psi(t):=\hat{\Phi}^{j}_{\beta_{j}(t)}$ is a $1$-sweepout. On the other hand,
(12) and (13) gives

$L^{\theta}(\hat{\Phi}^{j}_{\beta_{j}(t)})<w_{2}(D^{2})-a=w_{1}(D^{2})-a$

for $t\in I$. This contradicts to the definition of $1$-sweepout and we have
an existence of a closed curve $\alpha_{j}$ which is not nullhomotopic in
$S^{1}\times I$.

Then by the fact of $\alpha_{j}$ being not nullhomotopic in $S^{1}\times I$
and the definition (4) of $2$-sweepout,
$f_{j}:=e\circ\alpha_{j}:S^{1}\rightarrow\partial D^{2}$, the endpoint
function of $\alpha_{j}$, is a continuous map and has a nonzero mapping
degree. By the degree theory argument, $f_{j}$ is a surjective map. Since
$\alpha_{j}$ is a continuous loop and $\mathcal{Y}_{j,i}$ are disjoint open
sets, $\alpha_{j}(S^{1})$ is contained in a single $\mathcal{Y}_{j,i}$ for
some $1\leq i\leq k$ and
$F(\partial^{\theta}\hat{\Phi}^{j}_{\alpha_{j}(t)},\gamma_{i})<2s$ for $t\in
S^{1}$.

By our choice of $s$, we have

$F(\partial^{\theta}\hat{\Phi}^{j}_{\alpha_{j}(t)},\gamma_{i})<2s<\delta_{i}$

and we apply Lemma 4.1 and obtain the following:

$f_{j}(t)\in N_{h_{0},\partial D^{2}}(supp(\partial^{b}\Omega_{i}))$

for $t\in S^{1}$. Since $N_{h_{0},\partial
D^{2}}(supp(\partial^{b}\Omega_{i}))$ is a proper open subset of $\partial M$,
we have

$f_{j}(S^{1})\subseteq N_{h_{0},\partial
D^{2}}(supp(\partial^{b}\Omega_{i}))\subsetneq\partial D^{2}.$

This contradicts to the surjectivity of $f_{j}$. Thus if
$w_{1}(D^{2})=w_{2}(D^{2})$, then there are infinitely many distinct smooth
embedded capillary geodesics in $(D^{2},\partial D^{2},g)$. ∎

For $(D^{2},\partial D^{2},g)$ endowed with a $\theta$-bumpy metric, there is
no capillary embedded geodesic with a nontrivial Jacobi field. This implies a
finiteness of capillary embedded geodesics with an $L^{\theta}$-functional
bound $w_{2}(D^{2})$ and we obtain two distinct capillary geodesics with
different $L^{\theta}$-functional values. We discuss the genericness of
$\theta$-bumpy metrics of capillary embedded geodesics in Appendix C.

######  Corollary 4.3.

There are at least distinct two capillary embedded geodesics on Riemannian
disks $(D^{2},\partial D^{2},g)$ with a $\theta$-bumpy metric satisfying the
hypothesis $(\star)$. In particular, if $(D^{2},\partial D^{2},g)$ is endowed
with a $\theta$-bumpy metric, then for $k=1,2$, there exists a capillary
embedded geodesic $\gamma_{k}$ and its corresponding domains $\Omega_{k}$ such
that $\gamma_{k}=\partial_{rel}(\Omega_{k})$ and

$w_{k}(D^{2})=L^{\theta}(\Omega_{k}),$

their widths satisfy $w_{1}(D^{2})<w_{2}(D^{2})$.

###### Proof.

By Theorem 3.5, for $k=1,2$, there exists a capillary embedded geodesic
$\gamma_{k}$ which is obtained as a min-max limit of a tightened minimizing
sequence $\\{\hat{\Phi}^{j}\\}$. If $w_{1}(D^{2})<w_{2}(D^{2})$, then
$\gamma_{1}$ and $\gamma_{2}$ are distinct. Suppose
$w_{1}(D^{2})=w_{2}(D^{2})$. Then by Theorem 4.2, there are infinitely many
capillary embedded geodesics whose $L^{\theta}$-functional value is
$w_{1}(D^{2})=w_{2}(D^{2})$.

Now we consider the bumpy metric case. Suppose $w_{1}(D^{2})=w_{2}(D^{2})$.
Then there is an infinite sequence of capillary embedded geodesics
$\\{\gamma_{k}\\}$ where their corresponding domains are $\\{\Omega_{k}\\}$.
Denote the limit capillary geodesic and limit domain by $\gamma$ and $\Omega$,
respectively. By compactness theorem of capillary embedded geodesics (Theorem
A.1), this induces a nontrivial Jacobi field of the limit capillary varifold
$\partial^{\theta}\Omega$ and this contradicts to the $\theta$-bumpy
assumption. Hence $w_{1}(D^{2})<w_{2}(D^{2})$ and two capillary geodesics are
distinct each other. ∎

We now prove Theorem 1.1 which shows a possible curvature condition of
Riemannian disks containing two min-max capillary embedded geodesics.

######  Theorem 4.4.

There are at least 2 capillary embedded geodesics on Riemannian disks
$(D^{2},\partial D^{2},g)$ with nonnegative Gaussian curvature in $D^{2}$ and
total signed geodesic curvature on boundary satisfies $\int_{\partial
D^{2}}\kappa\geq\pi$.

###### Proof.

It suffices to show that if a Riemmanian disk $(D^{2},\partial D^{2},g)$
satisfies the conditions on Gaussian curvature and total signed geodesic
curvature then it satisfies the hypothesis $(\star)$. We prove that there is
no simple geodesic lasso on $(D^{2},\partial D^{2},g)$.

Suppose there exists a simple geodesic lasso $\gamma:[0,L]\rightarrow D^{2}$
with a vertex $\gamma(0)=\gamma(L)=p\in\partial D^{2}$. We define two angles
$\alpha_{0},\alpha_{L}\in[0,\pi]$ by angles between tangent vectors at $p$
satisfying

(14) $\displaystyle\langle\gamma^{\prime}(0),v_{p}\rangle$
$\displaystyle=\cos\alpha_{0}$ (15)
$\displaystyle\langle\gamma^{\prime}(L),v_{p}\rangle$
$\displaystyle=\cos\alpha_{L},$

where $v_{p}$ is a unit tangent vector at $p$ of $\partial D^{2}$. Here we
choose the direction of $v_{p}$ to satisfy $\alpha_{0}+\alpha_{L}<\pi$. Also
let us denote the region bounded by $\gamma$ by $\Omega$. Note that
$\alpha_{0}$ and $\alpha_{L}$ are exterior angles of $\Omega$.

By Gauss-Bonnet theorem, we know that $\int_{D^{2}}K\leq\pi$ from
$\int_{\partial D^{2}}\kappa\geq\pi$. Then we have

(16) $\displaystyle 2\pi$
$\displaystyle=\int_{\Omega}K+\int_{\gamma}\kappa+\alpha_{0}+\alpha_{L}$ (17)
$\displaystyle\leq\int_{D^{2}}K+\int_{\gamma}\kappa+\alpha_{0}+\alpha_{L}$
$\displaystyle\leq\pi+\alpha_{0}+\alpha_{L}<2\pi$

and this shows the contradiction. (16) holds since $\Omega$ is topological
disk and the angle condition (14) and (15) hold and (17) comes from the
nonnegativity of Gaussian curvature on $D^{2}$. Hence there is no simple
geodesic lasso on $(D^{2},\partial D^{2},g)$ and this nonexistence implies the
hypothesis $(\star)$. ∎

##  5\. Morse index bound of min-max capillary and free boundary embedded
geodesics

In this section, we prove the generic Morse index bound of min-max capillary
embedded geodesics which we constructed in previous sections. Throughout this
section, we assume the hypothesis $(\star)$ and the $\theta$-bumpiness of the
metric.

By the compactness theorem of capillary varifolds (Corollary A.2) and the
bumpy metric theorem of capillary geodesics (Theorem C.1), we have the generic
finiteness of capillary embedded geodesics with bounded length.

We mainly follow the ideas to prove Morse index bound in [28] and explain the
necessary modifications in capillary case. We apply curve shortening flow with
fixed boundary instead of curve shortening flow in simple closed geodesic
cases.

###  5.1. Local min-max theorem

We state the boundary version of local min-max theorem in [28] (See also [38]
and [54]). For a domain $\Omega$ and curve $\gamma$ with
$\partial_{rel}\Omega=\gamma$, denote $\Omega_{v}$ as a domain induced by
$(F_{v})_{\sharp}(\gamma)$ where $\\{F_{v}\\}_{v\in\bar{B}^{k}}$ is a local
min-max diffeomorphism. By taking the vector fields with normal negative
eigensections, the following directly holds by the proof of Theorem 4.1 in
[28]:

######  Theorem 5.1 ([28], Theorem 4.1).

Let $(D^{2},\partial D^{2},g)$ be a Riemannian disk with $\theta$-bumpy metric
and strictly convex boundary. Let $\gamma$ be an embedded capillary geodesic
with contact angle $\theta$ with corresponding domain $\Omega$, Morse index
$k$ and multiplicity one. For every $\beta>0$, there is $\epsilon>0$ and a
smooth family $\\{F_{v}\\}_{v\in\bar{B}^{k}}\subset Diff(D^{2})$ such that

  1. (i)

$F_{0}=Id,\,F_{-v}=F_{v}^{-1}$ for all $v\in\bar{B}^{k}$;

  2. (ii)

the function

$L^{\theta}_{\gamma}:\bar{B}^{k}\rightarrow[0,\infty),\,\,\,\,L^{\theta}_{\gamma}(v)=L^{\theta}(\Omega_{v}),$

is strictly concave;

  3. (iii)

$||F_{v}-Id||_{C^{1}}<\beta$ for all $v\in\bar{B}^{k}$;

and such that for every $V\in IV_{1}(D^{2})$ induced by a related boundary of
a domain $\Omega^{\prime}$ whose endpoints are on $\partial D^{2}$ with
$F(V,\gamma)<\epsilon$, we have

$\max_{v\in\bar{B}^{k}}L^{\theta}(\Omega^{\prime}_{v})\geq L^{\theta}(\Omega)$

with equality only if $\gamma=(F_{v})_{\sharp}V$ for some $v\in\bar{B}^{k}$.

###  5.2. Metric perturbation

We elaborate the metric perturbation in [28] to settle the boundary version of
interpolation lemmas. Suppose we have a capillary embedded geodesic $\gamma$
and adopt the Fermi coordinate (12) on the tubular neighborhood
$N_{h}(\gamma)$ (on the extended closed surface $M$) in [28]:

(18) $ds^{2}=J(x,y)^{2}dx^{2}+dy^{2}$

such that

  1. (1)

$J_{yy}=-KJ$

  2. (2)

$J(x,0)=1$

  3. (3)

$J_{x}(x,0)=\kappa=0$

  4. (4)

$J_{y}(x,0)=0$

where $K$ is a Gaussian curvature and $\kappa$ is a geodesic curvature of
$\gamma$.

We perturb the metric to be foliated by mean convex curves to obtain a
suitable squeezing lemma. We take $M>\max K_{(D^{2},g)}$ and small $h>0$
satisfying $h<\max(M^{-1},1/10)$. We use the bump function
$\phi_{\beta}:D^{2}\rightarrow\mathbb{R}$ constructed in [28].

######  Proposition 5.2 ([28], Proposition 5.2).

Given a capillary embedded geodesic $\gamma$ with contact angle $\theta$, for
any $\beta>0$, there exists a smooth bump function
$\phi_{\beta}:D^{2}\rightarrow\mathbb{R}$ such that

  1. (i)

$\phi_{\beta}(x)=0$ when $x\notin N_{h}(\gamma)$

  2. (ii)

$\phi_{\beta}(x)<0$ for $x\in\gamma$

  3. (iii)

$||\exp(2\phi_{\beta})-1||_{C^{0}}<\beta$

  4. (iv)

$\nabla_{\dot{\gamma}}\phi_{\beta}=0$ on $\gamma$ [$\phi_{\beta}$ is constant
on $\gamma$]

  5. (v)

The Hessian $\partial^{2}\phi_{\beta}(\nu,\nu)=M$ on $\gamma$.

For a fixed $\beta>0$, let us consider the metric
$g_{\beta}=\exp(2\phi_{\beta})g$ whose Fermi coordinate representation is
given by

(19)
$ds_{\beta}^{2}=J(x,y)^{2}\exp(2\phi_{\beta})dx^{2}+\exp(2\phi_{\beta})dy^{2}$

and we consider a canonical diffeomorphism
$G_{\beta}:(D^{2},g)\rightarrow(D^{2},g_{\beta})$ given by $G_{\beta}(z)\equiv
z$ for all $z\in D^{2}$. Let $\gamma_{g_{\beta}}:=G_{\beta}(\gamma)$ and
$N_{h,g_{\beta}}(\gamma_{g_{\beta}}):=G_{\beta}(N_{h}(\gamma_{g_{\beta}}))$ in
$(D^{2},g_{\beta})$. The following is a boundary version of Lemma 5.3 in [28]
and it also ensures the conservation of the convexity of the boundary along
diffeomorphism $G$.

######  Lemma 5.3.

For any $\beta>0$, $\gamma_{g_{\beta}}$ is a capillary embedded geodesic with
contact angle $\theta$. The Gaussian curvature satisfies $K_{g_{\beta}}(z)<0$
for $z\in\gamma_{g_{\beta}}$ on $(D^{2},g_{\beta})$. Moreover, for two
endpoints $z_{1},z_{2}$ of $\gamma_{g_{\beta}}$ on $\partial D^{2}$, the
geodesic curvature $\kappa_{\partial D^{2}}$ of $\partial D^{2}$ satisfies
$\kappa_{\partial D^{2}}(z_{1}),\kappa_{\partial D^{2}}(z_{2})>0$.

###### Proof.

Since $G$ is a conformal deformation and by the arguments in Lemma 5.3 in
[28], $\gamma_{g_{\beta}}$ is a capillary embedded geodesic with contact angle
$\theta$ and Gaussian curvature is negative on $\gamma_{g_{\beta}}$. Consider
$\partial D^{2}$ in the extended surface $M$. By the calculation of the change
of the geodesic curvature by conformal deformation in Besse [6], we have

(20)
$\kappa_{g_{\beta}}(z_{i})=e^{-\phi_{\beta}}\Big{(}\kappa_{g}(z_{i})-\frac{\partial\phi_{\beta}}{\partial\nu}\Big{)}$

for $i=1,2$. Note that $\nabla_{\nu}\phi_{\beta}=0$ by the construction of
$\phi_{\beta}$ in Proposition 5.2. Hence $\kappa_{g_{\beta}}(z_{i})>0$ for
$i=1,2$ by the convexity of the boundary. ∎

###  5.3. Identifying the endpoints by sliding the curves

We introduce the deformation of the family of domains $\\{\Phi^{j}\\}$ to
identify all endpoints of relative boundaries to be $\partial\gamma$ with an
$F$-distance upper bound along the deformation. We prove this sliding
deformation to apply the curve shortening flow with fixed boundary in the
procedure to construct squeezing deformation in the next subsection.

######  Lemma 5.4.

Let $(D^{2},\partial D^{2},g)$ be a Riemannian disk with $\theta$-bumpy metric
and a strictly convex boundary. Let $\gamma=\partial^{i}\Omega$ be a fixed
embedded capillary geodesic with contact angle $\theta$ and corresponding
region $\Omega$, and $X$ be a finite dimensional simplicial complex. Then
there is $\delta_{0}>0$ satisfying the following property: For
$0<\delta<\delta_{0}$, if $\Phi:X\rightarrow\mathcal{D}(D^{2})$ satisfying

(21) $F(\partial^{i}\Phi(x),\gamma)<\delta,$

then there is a homotopy $H:[0,1]\times X\rightarrow\mathcal{D}(D^{2})$ such
that

(22) $F(\partial^{i}H(t,x),\gamma)<C(\theta)\sqrt{\delta}$

and two endpoints satisfy $\pi_{i}\partial(H(1,x))=\pi_{i}\partial(\Omega)$
for any $x\in X$ and $i=1,2$.

###### Proof.

Consider the tubular neighborhood $N_{h}(\gamma)$ for small $h$. Then since
$\gamma$ and $\partial M$ intersect with an angle $\theta$, there exists
$h_{0}>0$ such that for $0<h<h_{0}$, the Hausdorff distance in $\partial
D^{2}$ satisfies

(23) $d_{\mathcal{H},\partial D^{2}}(\partial\gamma,\partial
N_{h}(\gamma)\cap\partial D^{2})<\frac{2h}{\sin\theta}.$

Let us set $\delta_{0}=h_{0}^{2}/10$. By (23) and Lemma 3.1 in [28],
$\partial^{i}\Phi(x)$ is supported in $N_{\sqrt{10\delta}}(\gamma)\subset
N_{h_{0}}(\gamma)$.

We add two small segments connecting between endpoints of
$\partial^{i}\Phi(x)$ and those of $\gamma$ along $\partial D^{2}$ and push
this curve slightly to the interior of $D^{2}$, and consider the corresponding
domain. Then we connect the family of initial domains $\Phi(x)$ and the new
family of domains explained above with small additional segments via homotopy.

Now we consider the exponential map $\exp_{p}:T_{p}\partial
D^{2}\rightarrow\partial D^{2}$ on $\partial D^{2}$. For $x\in X$, $i=1,2$ and
$t\in[0,1]$, we define two segments $\gamma_{i,t}$ by segments connecting
$\pi_{i}\partial\Phi(x)$ and
$\exp_{\pi_{i}\partial\Omega}[(1-t)(\exp_{\pi_{i}\partial\Omega})^{-1}\pi_{i}\partial\Phi(x)]$
along $\partial D^{2}$. We define a homotopy $H^{\prime}(t,x)$ by domains
satisfying

(24)
$\partial^{i}H^{\prime}(t,x)=\gamma_{1,t}*\partial^{i}\Phi(x)*\gamma_{2,t}$

for $i=1,2$. We regard $\gamma_{i,t}$ as curves in $\mathring{D}^{2}$ here.
Then now we prove the $F$-distance estimate for this homotopy. For
$f:G_{1}(D^{2})\rightarrow\mathbb{R}$ satisfying $|f|\leq 1$ and $Lip(f)\leq
1$ (we regard $f$ on $G_{1}(D^{2}\setminus int(D^{2}))$ as $f$ on the extended
surface $M$ here),

$\displaystyle\Big{|}\int fd(\partial^{i}H^{\prime}(t,x))-\int
fd\gamma\Big{|}$ $\displaystyle\leq\Big{|}\int
fd(\partial^{i}H^{\prime}(t,x))-\int
fd(\partial^{i}\Phi(x))\Big{|}+\Big{|}\int fd(\partial^{i}\Phi(x))-\int
fd\gamma\Big{|}$ (25) $\displaystyle\leq\Big{|}\int fd\gamma_{1,t}+\int
fd\gamma_{2,t}\Big{|}+\delta$
$\displaystyle\leq|\gamma_{1,t}|+|\gamma_{2,t}|+\delta$ (26)
$\displaystyle<\frac{4\sqrt{10\delta}}{\sin\theta}+\delta$
$\displaystyle<C(\theta)\sqrt{\delta},$

where (25) follows from (21), (24), and (26) comes from (23) and the fact that
$\partial^{i}\Phi(x)$ is supported in $N_{\sqrt{10\delta}}(\gamma)$. Then by
the definition of the $F$-distance, we have the following $F$-distance
estimate

(27) $F(\partial^{i}H^{\prime}(t,x),\gamma)<C(\theta)\sqrt{\delta}.$

At last, we push the curves along the homotopy $\partial^{i}H^{\prime}(t,x)$
slightly into the interior of $D^{2}$, and define a corresponding homotopy of
domains by $H(t,x)$. By (27), the $F$-distance estimate (22) holds and
$H(t,x)$ is a desired homotopy. ∎

###  5.4. Squeezing lemma

Now we prove the capillary version of squeezing lemma, whose simple closed
curve version is Lemma 5.4 in [28]. The main difference with simple closed
curve version is the curve shortening flow part. We apply the curve shortening
flow with fixed boundary in Appendix B instead of curve shortening flow for
closed curves. We apply Lemma 5.4 to identify the endpoints to squeeze whole
families to a single capillary curve, provided a capillary embedded geodesic
whose ambient Gaussian curvature is negative on the geodesic.

The following technical lemma shows the existence of mean convex foliation of
$N_{h}(\gamma)$ where fixed points are two endpoints of $\gamma$, in case of a
geodesic is strictly stable and achieves a negative ambient Gaussian
curvature.

######  Lemma 5.5.

There exists $h_{1}>0$ such that for $0<h<h_{1}$, there exists a foliation by
curves with fixed endpoints $\\{S_{t}\\}_{t\in[0,h]}$ of $N_{h}(\gamma)$
satisfying the following

  1. (1)

$S_{0}=\gamma$,

  2. (2)

For $t\in(0,h]$, each leave $S_{t}$ consists of two embedded curves whose
endpoints are two endpoints of $\gamma$,

  3. (3)

For $t\in(0,h]$, the curvature vector of $S_{t}$ points toward $\gamma$.

###### Proof.

We consider $h<\min(inj(D^{2}),h_{0})$ and set
$\partial\gamma=\\{p_{1},p_{2}\\}$, where $h_{0}$ is a constant in the proof
of Lemma 5.4 satisfying (23). Note that $\gamma$ has a (rounded) trapezoidal
tubular neighborhood $N_{h}(\gamma)$. For small $h>0$, $\partial
N_{h}(\gamma)\cap\partial D^{2}$ consists of four points, where two points are
on the neighborhood of $p_{1}$ and two points are on the neighborhood of
$p_{2}$. We denote $\partial N_{h}(\gamma)\cap\partial
D^{2}=\\{q_{1,1,h},q_{1,2,h},q_{2,1,h},q_{2,2,h}\\}$, where
$q_{1,1,h},q_{1,2,h}\in N_{2h/\sin\theta,\partial M}(p_{1})$ and
$q_{2,1,h},q_{2,2,h}\in N_{2h/\sin\theta,\partial M}(p_{2})$. We label points
$q_{i,j,h}$ to be $q_{1,j,h}$ and $q_{2,j,h}$ are on the same side of $\gamma$
for $j=1,2$. Without loss of generality, the angle between a unit tangent
vector $v_{i}$ of $\gamma$ at $p_{i}$ toward $int(D^{2})$ and a unit tangent
vector $w_{i}$ of $\partial D^{2}$ at $p_{i}$ toward $q_{i,1,h}$ is an acute
contact angle $\theta\in(0,\pi/2)$. Then the angle between $v_{i}$ and
$-w_{i}$ which is a unit tangent vector of $\partial M$ at $p_{i}$ toward
$q_{i,2,h}$ is $\pi-\theta$.

For some $\delta>0$, we adopt the Fermi coordinate
$c:[-\delta,|\gamma|+\delta]\times(-h,h)\rightarrow M$ on $N_{h}(\gamma)$ in
(18), where $c(0,0)=p_{1}$ and $c(|\gamma|,0)=p_{2}$. We set $\delta>0$ here
to satisfy $N_{h}(\gamma)\subset c([-\delta,|\gamma|+\delta]\times(-h,h))$.
Also we define
$\pi_{y}:[-\delta,|\gamma|+\delta]\times(-h,h)\rightarrow(-h,h)$ by
$y$-coordinate function of Fermi coordinate $c$. Without loss of generality,
we take Fermi coordinate $c$ to satisfy $\pi_{y}c^{-1}q_{i,1,h}>0$ and
$\pi_{y}c^{-1}q_{i,2,h}<0$ for $i=1,2$.

We divide $N_{h}(\gamma)$ into four disjoint small convex regions
$\\{R_{i,j,h}\\}_{i,j\in\\{1,2\\}}$ where each $R_{i,j,h}$ contains
$q_{i,j,h}$, respectively, and the interior region
$N_{h}(\gamma)\setminus\cup_{i,j=1,2}R_{i,j,h}$. We construct the foliations
on $N_{h}(\gamma)\setminus\cup_{i,j=1,2}R_{i,j,h}$ and $R_{i,j,h}$’s
separately and join those to a combined foliation.

For $i=1,2$, let us connect $p_{i}$ and $q_{i,1,h}$ by geodesics
$\alpha_{i,1,h}$. We define the small regions enclosed by $\partial D^{2}$ and
$\alpha_{i,1,h}$ by $R_{i,1,h}$. Also we define two vertical geodesics
$\alpha_{1,2,h}:=c(\\{0\\}\times[-h,0])$ and
$\alpha_{2,2,h}:=c(\\{|\gamma|\\}\times[-h,0])$. We denote the small regions
enclosed by $\partial D^{2}$, $\alpha_{i,2,h}$ and $\partial N_{h}(\gamma)$ by
$R_{i,2,h}$.

Let us construct the foliation in the interior part first. As in [28], for
$t\in[-h,h]$ we take

(28)
$\gamma_{int,t,h}:=c(\\{y=t\\})\cap(N_{h}\setminus\cup_{i,j=1,2}R_{i,j,h}).$

There exists $h^{\prime}>0$ such that for $0<h<h^{\prime}$, by the arguments
in Section 5.3 of [28], the curvature vector of each leave
$\\{\gamma_{int,t,h}\\}_{t\in[-h,h]}$ points toward into $\gamma$.

Now we consider each small boundary region $R_{i,j,h}$ for $i,j\in\\{1,2\\}$.
Note that $R_{i,j,h}$ is a convex region so that is star-shaped. We foliate
$R_{i,j,h}$ by scaled curves of parts of $\partial
R_{i,j,h}\setminus\alpha_{i,j,h}$. For $s\in[0,1]$, let us define

(29) $\gamma_{i,j,s,h}=\exp_{p_{i}}(s(\exp_{p_{i}})^{-1}(\partial
R_{i,j,h}\setminus\alpha_{i,j,h})),$

which foliates $R_{i,j,h}$ with a fixed point $p_{i}$ by star-shapedness of
$R_{i,j,h}$ and $h<inj(D^{2})$. Since $\partial\tilde{B}_{r}(p_{i})$ is convex
for small $r>0$, $\partial D^{2}$ is convex, and each $\gamma_{i,j,s,h}$
consists of the scaled curve of these based on $p_{i}$, there exists
$h^{\prime\prime}>0$ such that each leave $\gamma_{i,j,s,h}$ has a curvature
vector points toward $\gamma$ for $0<h<h^{\prime\prime}$.

Now we connect the foliations we constructed in (28) and (29). Let us
parametrize $\alpha_{i,1,h}$ by the regular curve by
$\alpha_{i,1,h}:[0,1]\rightarrow D^{2}$ where
$\alpha_{i,1,h}(t)=\exp_{p_{i}}(t(\exp_{p_{i}})^{-1}(q_{i,1,h}))$ for $i=1,2$.
Since $\gamma$ and $\partial D^{2}$ meets with capillary angle $\theta$
transversely and $\partial D^{2}$ is convex, there exists
$h^{\prime\prime\prime}>0$ such that for $0<h<h^{\prime\prime\prime}$,
$y_{\alpha_{i,1,h}}:=\pi_{y}c^{-1}\alpha_{i,1,h}:[0,1]\rightarrow[0,h]$ is a
strictly increasing function. We define the foliation
$\\{\gamma_{t,h}\\}_{t\in[0,h]}$ by

(30)
$\gamma_{t,h}:=(\gamma_{1,1,y_{\alpha_{1,1,h}}^{-1}(t),h}*\gamma_{int,t,h}*\gamma_{2,1,y_{\alpha_{2,1,h}}^{-1}(t),h})\cup(\gamma_{1,2,t/h,h}*\gamma_{int,-t,h}*\gamma_{2,2,t/h,h}).$

Then we take
$h_{1}=\min\\{inj(D^{2}),h_{0},h^{\prime},h^{\prime\prime},h^{\prime\prime\prime}\\}$.
By taking $S_{t}=\gamma_{t,h}$ in (30), by constructions in (28) and (29), the
foliation by curves with fixed endpoints $\\{S_{t}\\}_{t\in[0,h]}$ satisfies
conditions (1), (2) and (3) for $0<h<h_{1}$. ∎

Let us call $N_{h}(\gamma)$ having a mean convex foliation with fixed point
$\\{S_{t}\\}_{t\in[0,h]}$ as a _mean convex neighborhood of $\gamma$_.
Following is a squeezing lemma which is a capillary analog of Lemma 5.4 in
[28].

######  Lemma 5.6.

Let $(D^{2},\partial D^{2},g)$ be a Riemannian disk with a $\theta$-bumpy
metric and strictly convex boundary, $\gamma$ be a capillary embedded
geodesic, $\Omega$ be a corresponding domain with
$\gamma=\partial_{rel}\Omega$, and $X$ be a simplicial complex with finite
dimension $k$. There exists $\delta_{1}=\delta_{1}((M,g),\gamma)>0$ with the
following property:

For $0<\delta<\delta_{1}$, if $\Phi:X\rightarrow\mathcal{D}(D^{2})$ is a
continuous map in the smooth topology such that

$\sup\\{F(\partial^{i}\Phi(x),\gamma):x\in X\\}<\delta,$

then there is a homotopy $H:[0,1]\times X\rightarrow\mathcal{D}(D^{2})$ such
that $H(0,x)=\Phi(x)$ and $H(1,x)=\Omega$ so that $\Phi$ is nullhomotopic.

###### Proof.

We mainly discuss the difference with the proof of Lemma 5.4 in [28]. We
consider a capillary embedded geodesic whose ambient negative Gaussian
curvature is negative case first. We can assume that the only capillary
embedded geodesic in the tubular neighborhood $N_{h}(\gamma)$ is $\gamma$
since the metric is $\theta$-bumpy.

We take $\delta_{1}=h_{1}^{2}/10$ where $h_{1}$ is a constant in Lemma 5.5.
Note that $0<\delta_{1}<\delta_{0}$ by the construction of mean convex
neighborhood in the proof of Lemma 5.5. We apply Lemma 5.4 for
$0<\delta<\delta_{1}$ and obtain a homotopy $H_{1}:[0,1]\times
X\rightarrow\mathcal{D}(D^{2})$ such that

$F(\partial^{i}H_{1}(t,x),\gamma)<C(\theta)\sqrt{\delta}$

and endpoints are fixed as $\pi_{i}\partial H_{1}(1,x)=\pi_{i}\partial\Omega$
for $i=1,2$. The entire support of $\partial^{i}H_{1}(1,x)$ are in
$N_{\sqrt{10\delta}}(\gamma)$ by Lemma 3.1 in [28].

Now we apply the curve shortening flow with boundary (see Appendix B) to
curves $\\{\partial^{i}H_{1}(1,x)\\}_{x\in X}$. $N_{h_{0}}(\gamma)$ is a mean
convex neighborhood by Lemma 5.5. We define
$Y(\cdot,t)|_{\partial^{i}H_{1}(1,x)}$ to be the curve shortening flow with
fixed boundary, where
$Y(\cdot,0)|_{\partial^{i}H_{1}(1,x)}=\partial^{i}H_{1}(1,x)$ and
$Y_{t}(y,t)|_{\partial^{i}H_{1}(1,x)}=\kappa N(y)$ where $\kappa$ is a
geodesic curvature and $N$ is a normal vector field on
$Y_{t}(\cdot,t)|_{\partial^{i}H_{1}(1,x)}$ for $t>0$. Then by the regularity
theorem of curve shortening flow with boundary (Theorem B.1) and mean-
convexity of $N_{h_{0}}(\gamma)$, for any small $s>0$ there is a large time
$t_{0}$ such that for $t>t_{0}$ and any $x\in X$, the total curvature
$\int_{Y(\cdot,t)}|\kappa|<s$ since curves uniformly smoothly converge to
$\gamma$ (See also Theorem 2.6 of [25]). Moreover, the curves are graphical
over $\gamma$. Now we construct the homotopy $H_{2}:[0,1]\times
X\rightarrow\mathcal{D}(D^{2})$ induced by the flow by $H_{2}(t,x)$ satisfying

$\partial^{i}H_{2}(t,x)=Y(\cdot,2t_{0}t)|_{\partial^{i}H_{1}(1,x)},$

for all $x\in X$ and $t\in[0,1]$. We apply the squeezing homotopy as in [28]
and define this homotopy as $H_{3}:[0,1]\times
X\rightarrow\mathcal{D}(D^{2})$. By combining three homotopies, we obtain a
desired homotopy $H:[0,1]\times X\rightarrow\mathcal{D}(D^{2})$ as

(31) $H=H_{1}\cdot H_{2}\cdot H_{3}.$

Now we consider the general cases. We fix some $\beta>0$ and consider the
perturbed metric $(S^{2},g_{\beta})$ in (19). By Lemma 5.3 and Lemma 5.5,
there exists a mean convex neighborhood $N_{h^{\prime}}(\gamma_{g_{\beta}})$
where $N_{h^{\prime}}(\gamma_{g_{\beta}})\cap\partial M$ is strictly convex.
Let us take $\delta_{1}=h^{\prime 2}/20$. We complete the proof by following
remaining arguments in the proof of Lemma 5.4 in [28] and taking the pullback
map of (31). ∎

###  5.5. $F$-distance and $L^{\theta}$-functional estimate

In this section, we prove the $F$-distance and $L^{\theta}$-functional
estimate along the homotopy we constructed in Lemma 5.6. This is an analog of
Section 6 of [28]. We will focus on the modification we need in the capillary
case.

Let $\gamma$ be a capillary embedded geodesic with a contact angle
$\theta\in(0,\pi/2)$ in $(D^{2},\partial D^{2},g)$ and Gaussian curvature on
$\gamma$ is negative. Take $h_{1}>0$ in Lemma 5.5. Suppose $0<h<h_{1}$ in the
following Lemma 5.7 and Lemma 5.8, then $N_{h}(\gamma)$ is mean convex
neighborhood by Lemma 5.5. We take the Fermi coordinate
$c:[-\delta,|\gamma|+\delta]\times(-h,h)\rightarrow M$ of (19). The following
lemma is a capillary analog of Lemma 6.1 in [28]. The proof of the following
lemma directly follows from the proof of Lemma 6.1 in [28].

######  Lemma 5.7.

Let $(D^{2},\partial D^{2},g)$, $\gamma$, $c$ be as above and $K(z)<0$ for
$z\in N_{h}(\gamma)$. There exists $C=C(|\gamma|)>0$ satisfying the following
property: For $0<\epsilon<h^{2}$, if an embedded curve $\alpha$ has its
endpoints on different components of $\partial D^{2}\cap N_{h}(\gamma)$
satisfies $|\alpha|<|\gamma|+\epsilon$, then

(32) $F(\alpha,\gamma)<C(|\gamma|)(h+\sqrt{\epsilon}).$

Then we obtain the $F$-distance estimate along the squeezing map in Lemma 5.6
in negative ambient Gaussian curvature case. This is an analog of Lemma 6.2 in
[28]. The sliding procedure in Lemma 5.4 causes the change of order on the
$F$-distance bound.

######  Lemma 5.8.

Let $\gamma=\partial\Omega$ be a capillary embedded geodesic with contact
angle $\theta$ on $(D^{2},\partial D^{2},g)$, Gaussian curvature $K(z)<0$ for
$z\in N_{h}(\gamma)$, and $X$ be a $k$-dimensional simplicial complex. There
exists $\epsilon_{0}=\epsilon_{0}(h,\theta)$ such that there exists
$C=C(|\gamma|,\theta)>0$ satisfying the following property: For
$0<\epsilon<\epsilon_{0}$, if $\Phi:X\rightarrow\mathcal{D}(D^{2})$ is a
continuous map in the smooth topology such that

$\sup\\{F(\partial^{i}\Phi(x),\gamma):x\in X\\}<\epsilon,$

then there is a homotopy $H:[0,1]\times X\rightarrow\mathcal{D}(D^{2})$ such
that $H(0,x)=\Phi(x)$, $H(1,x)=\Omega$ and the following $F$-distance estimate
holds along $H$:

$\sup\\{F(\partial^{i}H(t,x),\gamma):x\in X\text{ and
}t\in[0,1]\\}<C(|\gamma|,\theta)\sqrt[4]{\epsilon}.$

###### Proof.

Take $\epsilon_{0}=\min\\{1,C(\theta)^{-2}h^{2}/10\\}h^{2}/10$ where
$C(\theta)$ is the constant in (22). We consider the homotopy $H=H_{1}\cdot
H_{2}\cdot H_{3}$ in (31). We apply Lemma 5.4 and obtain the following
$F$-distance estimate:

(33) $\sup\\{F(\partial^{i}H_{1}(t,x),\gamma):x\in X\text{ and
}t\in[0,1]\\}<C(\theta)\sqrt{\epsilon}.$

Note that $C(\theta)\sqrt{\epsilon}<h^{2}/10$ and
$supp(\partial^{i}H_{1}(t,x))\subset N_{\sqrt{10\epsilon}}(\gamma)$ for
$t\in[0,1]$ and $x\in X$. Since $0<\sqrt{10\epsilon}<h$,
$N_{\sqrt{10\epsilon}}(\gamma)$ is a mean convex neighborhood of $\gamma$ by
Lemma 5.5.

Since the length functional monotonically decreases by the curve shortening
flow with fixed boundary, the arguments in Lemma 6.2 in [28] can be directly
applied to the homotopy $H_{2}\cdot H_{3}$ by applying Lemma 5.7 in place of
Lemma 6.1 in [28]. We have following for $t\in[0,1]$ and $x\in X$:

(34) $\displaystyle F(\partial^{i}H(t,x),\gamma)$
$\displaystyle<C(|\gamma|)(\sqrt{10\epsilon}+(\sup\\{F(\partial^{i}H_{1}(1,x),\gamma):x\in
X\\})^{\frac{1}{2}})$ (35)
$\displaystyle<C(|\gamma|)(\sqrt{10\epsilon}+C(\theta)^{\frac{1}{2}}\sqrt[4]{\epsilon})$
(36) $\displaystyle<2C(|\gamma|)C(\theta)^{\frac{1}{2}}\sqrt[4]{\epsilon}.$

(34) follows from (32) in Lemma 5.7 and (35) is obtained by (33). By taking
$C(|\gamma|,\theta)=2C(|\gamma|)C(\theta)^{\frac{1}{2}}$ in (36), we obtain a
desired estimate. ∎

Then with Lemma 5.3 and Lemma 5.6, we can apply the arguments in Lemma 6.3 in
[28], and we obtain the following $F$-distance estimate in general cases.

######  Lemma 5.9.

Let $\gamma=\partial\Omega$ be a capillary embedded geodesic on
$(D^{2},\partial D^{2},g)$, and $X$ be a $k$-dimensional simplicial complex.
There exists $C=C(|\gamma|)>0$ and $\epsilon_{0}=\epsilon_{0}(\gamma)>0$ such
that satisfying the following property: For $0<\epsilon<\epsilon_{0}$, if
$\Phi:X\rightarrow\mathcal{D}(D^{2})$ is a continuous map in the smooth
topology such that

$\sup\\{F(\partial^{i}\Phi(x),\gamma):x\in X\\}<\epsilon,$

then there is a homotopy $H:[0,1]\times X\rightarrow\mathcal{D}(D^{2})$ such
that $H(0,x)=\Phi(x)$, $H(1,x)=\Omega$ and the following $F$-distance estimate
holds along $H$:

$\sup\\{F(\partial^{i}H(t,x),\gamma):x\in X\text{ and
}t\in[0,1]\\}<C(|\gamma|,\theta)\sqrt[4]{\epsilon}.$

The following interpolation lemma associated with a local min-max theorem
(Theorem 5.1) is an analog of Lemma 6.4 in [28].

######  Lemma 5.10.

Let $\gamma=\partial\Omega$ be a capillary embedded geodesic with contact
angle $\theta$ and $X$ be a simplicial complex with finite dimension $k$.
Suppose $\Phi:X\rightarrow\mathcal{D}(D^{2})$ be a $k$-parameter family of
domains and $\\{F_{v}\\}_{v\in\overline{B}^{j}}$ be a local min-max family of
diffeomorphisms in Theorem 5.1. There exists
$\epsilon_{1}=\epsilon_{1}(\gamma)>0$ such that satisfying the following
property : If there is a continuous function $w:X\rightarrow\partial B^{j}$
satisfying

$F(\partial^{i}\Phi(x),(F_{w(x)})_{\sharp}\gamma)<\epsilon$

with $0<\epsilon<\epsilon_{1}$, then there exists a homotopy $H:[0,1]\times
X\rightarrow\mathcal{D}(D^{2})$ such that $H(0,x)=\Phi(x)$,
$H(1,x)=(F_{w(x)})_{\sharp}\Omega$ satisfying

(37) $\sup\\{F(\partial^{i}H(t,x),(F_{w(x)})_{\sharp}\gamma):x\in X\text{ and
}t\in[0,1]\\}<C(|\gamma|,\theta)\sqrt[4]{\epsilon}.$

Finally we obtain the following $L^{\theta}$-functional estimate along the
squeezing homotopy.

######  Corollary 5.11.

Suppose $\gamma$, $\Omega$, $\Phi$, $\\{F_{v}\\}_{v\in\overline{B}^{j}}$ are
as in Lemma 5.10, and $\epsilon_{1}=\epsilon_{1}(\gamma)$ is obtained by Lemma
5.10. Then for $0<\epsilon<\epsilon_{1}$, the following holds: If there is a
continuous function $w:X\rightarrow\partial B^{j}$ satisfying

$F(\partial^{i}\Phi(x),(F_{w(x)})_{\sharp}\gamma)<\epsilon$

with $0<\epsilon<\epsilon_{1}$, then there exists a homotopy $H:[0,1]\times
X\rightarrow\mathcal{D}(D^{2})$ such that $H(0,x)=\Phi(x)$,
$H(1,x)=(F_{w(x)})_{\sharp}\Omega$ satisfying

(38)
$L^{\theta}(H(t,x))<L^{\theta}((F_{w(x)})_{\sharp}\Omega)+C(|\gamma|,\theta)\sqrt[4]{\epsilon}$

for $x\in X$ and $t\in[0,1]$.

###### Proof.

Let us recall the definition (1) of $L^{\theta}$-functional. Note that the
homotopy satisfying (37) in Lemma 5.10 is constructed by the pullback map of
(31) in Lemma 5.6. Moreover, by tracking the homotopies consisting (31) of, we
have the pullback map of $H_{1}$ is the only homotopy involves the boundary
part of $L^{\theta}$-functional.

By (iii) of Theorem 5.1, let us take local min-max ball to satisfy

(39) $|(F_{v})_{\sharp}(\alpha)|<2|\alpha|$

for $v\in\overline{B}^{j}$ and $\alpha\in V_{1}(M)$. Moreover, by (43) and the
choice of $\beta$ in the proof of Lemma 6.3 in [28], we take $G_{\beta}$ such
that

(40) $|(G_{\beta})^{-1}_{\sharp}(\alpha)|<3|\alpha|$

for $\alpha\in V_{1}(M)$. Then we obtain

$\displaystyle L^{\theta}(H(t,x))$
$\displaystyle=|\partial^{i}H(t,x)|+\cos\theta|\partial^{b}H(t,x)|$ (41)
$\displaystyle<|(F_{w(x)})_{\sharp}\gamma|+\cos\theta|\partial^{b}H(t,x)|+C(|\gamma|,\theta)\sqrt[4]{\epsilon}$
$\displaystyle\leq|(F_{w(x)})_{\sharp}\gamma|+\cos\theta(|\partial^{b}(F_{w(x)})_{\sharp}\Omega|+|\partial^{b}(F_{w(x)})_{\sharp}\Omega\triangle\partial^{b}H(t,x)|)+C(|\gamma|,\theta)\sqrt[4]{\epsilon}$
(42)
$\displaystyle\leq|(F_{w(x)})_{\sharp}\gamma|+\cos\theta|\partial^{b}(F_{w(x)})_{\sharp}\Omega|+6\cos\theta\cdot\frac{4\sqrt{10\epsilon}}{\sin\theta}+C(|\gamma|,\theta)\sqrt[4]{\epsilon}$
$\displaystyle<L^{\theta}((F_{w(x)})_{\sharp}\Omega)+C(|\gamma|,\theta)\sqrt[4]{\epsilon}.$

We obtain (41) by applying (37), and (42) by applying (26), (39), and (40). ∎

###  5.6. Morse Index bound

Now we prove the Morse index bound of two capillary embedded geodesics on
Riemannian disk with a convex boundary $(D^{2},\partial D^{2})$ endowed with a
$\theta$-bumpy metric $g$ and the hypothesis $(\star)$. $\theta$-Bumpy metrics
are generic in Baire sense by Theorem C.1.

######  Theorem 5.12.

Suppose $(D^{2},\partial D^{2},g)$ is a Riemannian $2$-disk whose boundary is
strictly convex and satisfy the hypothesis $(\star)$ with a $\theta$-bumpy
metric. Then for each $k=1,2$, there exists a capillary embedded geodesic
$\gamma_{k}=\partial_{rel}\Omega_{k}$ with

$index(\gamma_{k})=k$

and the $L^{\theta}$-functional of two geodesics satisfy
$L^{\theta}(\Omega_{1})<L^{\theta}(\Omega_{2})$.

###### Proof.

We explain the necessary modifications from Theorem 7.1 in [28]. We apply
Corollary A.3 for the finiteness of capillary embedded geodesics with bounded
$L^{\theta}$-functional, and Theorem 5.1 for the capillary version of local
min-max theorem.

Then for each capillary geodesic $\gamma=\partial_{rel}\Omega$ with an index
$j<k$, we take the local min-max ball $\\{F_{v}\\}_{v\in\overline{B}^{j}}$ to
satisfy

(43) $F((F_{v})_{\sharp}(V_{1}),(F_{v})_{\sharp}(V_{2}))\leq 2F(V_{1},V_{2})$

for $V_{1},V_{2}\in V_{1}(M)$. If $j=1$ and $k=2$, then there exists some
$b(\gamma)$ such that

(44) $L^{\theta}((F_{v})_{\sharp}(\Omega))<L^{\theta}(\Omega)-b(\gamma),$

where $\partial_{rel}(\Omega)=\gamma$ and for $v\in\partial B^{1}$ by the
concaveness of $L^{\theta}$-functional in the local min-max ball. We pick
small $0<s<\min_{\Omega\in
D_{w_{k}}}\min(\epsilon(\gamma),b(\gamma)^{4}C(L_{k},\theta)^{-4}/3^{5},\epsilon_{1}(\gamma)/3)$,
where $\epsilon(\gamma)$ is a constant in Theorem 5.1, $b(\gamma)$ is a
constant in (44), $\epsilon_{1}(\gamma)$ is a constant in Corollary 5.11 for
$\gamma$ and $C(w_{k},\theta)$ is a constant in (38). By tracking the proof of
Lemma 6.1 in [28] and Lemma 5.8, we notice that $C_{\theta}(t):=C(t,\theta)$
can be chosen to be a monotone increasing function. We take $C(w_{k},\theta)$
here since $w_{k}\geq|\gamma|$ for every capillary embedded geodesic
$\gamma=\partial_{rel}\Omega$ with $L^{\theta}(\Omega)=w_{k}$ by the
definition of $L^{\theta}$-functional.

We apply Proposition 2.2 for the compact set
$\mathcal{S}=\mathcal{D}_{\theta,w_{k}}$ which is compact in the varifold
$F$-topology of its corresponding capillary varifolds: There exists
$\delta_{s}$ such that

$F(\partial^{\theta}\Omega^{\prime},\partial^{\theta}\Omega)<\delta_{s}\Rightarrow\,\,F(\partial^{i}\Omega^{\prime},\partial^{i}\Omega)<s.$

We apply Theorem 3.6 to obtain a tightened sequence
$\\{\hat{\Phi}^{j}(x)\\}_{x\in dom(\Phi)}$ with the Morse index bound:

(45) $\\{\partial^{\theta}\hat{\Phi}^{j}\in
V_{1}(D^{2}):L^{\theta}(\hat{\Phi}^{j})\geq
w_{k}-a\\}\subset\bigcup_{\gamma\in\Lambda_{i,w_{k},k}}B^{F}_{\delta_{s}}(\gamma).$

Then
$\Lambda(\\{\hat{\Phi}_{i}\\})=\\{\tilde{\Omega}_{1},...,\tilde{\Omega}_{q}\\}$.
Denote corresponding capillary geodesics $\tilde{\gamma}_{j}$ to be
$\tilde{\gamma}_{j}=\partial^{i}\tilde{\Omega}_{j}$. We proceed by the
contradiction argument. Let us assume that $index(\tilde{\gamma}_{j})<k$ for
every $i=1,...,q$.

Consider the $m_{i}$’th barycentric subdivision $Y_{i}$ of $Dom(X)$ so that
$|L^{\theta}(\hat{\Phi}_{i}(x))-L^{\theta}(\hat{\Phi}_{i}(y))|<a/2$ whenever
$x,y$ is in a same simplex in $Dom(X)$. Note that $Dom(X)=I$ for $k=1$ and
$Dom(X)=S^{1}\times I$ for $k=2$. We define $W_{i}$ to be a union of all
$k$-dimensional simplices $t\in Y_{i}$ such that
$L^{\theta}(\hat{\Phi}_{i}(x))\geq w_{k}-a/2$ for some $x\in t$. Then
$L^{\theta}(\hat{\Phi}_{i}(y))\geq w_{k}-a$ for every $y\in W_{i}$. Let
connected components be $W_{i,1},...,W_{i,r}$. Then by (45), for each $1\leq
p\leq r$ there exists $1\leq q_{p}\leq q$ with

$F(\partial^{i}\hat{\Phi}_{i}(y),\tilde{\gamma}_{q_{p}})<\delta_{s}.$

Note that

$w_{k}-a\leq L^{\theta}(\hat{\Phi}_{i}(y))\leq w_{k}-a/2$

since $y\in W_{i,p}$ since it belongs to both $W_{i}$ and $Y_{i}\setminus
W_{i}$.

Let us assume $j=0$. Then arguments in Theorem 7.1 in [28] apply and conclude
to $\partial W_{i,p}=0$. However, since $Dom(X)$ is connected, this cannot
happen and $j\geq 1$.

Now we consider the case of $k=2$ and $j=1$. The boundary of each $W_{i,p}$
can be separated to simple closed curves. Let us take an outermost curve
$\alpha_{i,p}$ among this curves hence $supp(\alpha_{i,p})\subseteq\partial
W_{i,p}$. Let us define a disk $D_{i,p}\in S^{1}\times I$ to be a disk
separated by $\alpha_{i,p}$ containing $W_{i,p}$. Moreover, for some
$\\{p_{1},p_{2}\\}\subseteq\\{1,...,q\\}$, if some $W_{i,p_{1}}$ is contained
in the region $D_{i,p_{2}}$, we do not perform the deformation for
$\alpha_{i,p_{1}}$. From now, let us only consider curves $\alpha_{i,p}$ which
we will apply the deformations, and renumber these curves to
$\alpha_{i,1},...,\alpha_{i,p^{\prime}}$. We apply the rest of deformation
arguments of Theorem 7.1 in [28] to curves
$\\{\alpha_{i,l}\\}_{l=1}^{p^{\prime}}$. Note that interpolation homotopy
$H_{2}$ contracts family of domains to a single domain
($(F_{1})_{\sharp}\Omega_{q_{p}}$ or $(F_{-1})_{\sharp}\Omega_{q_{p}}$) if
$j=1$. Then now consider the family of domains $\Psi_{i}$ defined by

$\Psi_{i}=\hat{\Phi}_{i}(S^{1}\times
I\setminus\cup_{j=1}^{p^{\prime}}D_{i,j})\cup_{\cup_{j=1}^{p^{\prime}}\alpha_{i,j}}(H_{1}\cdot
H_{2})_{\sharp}([0,1]\times\cup_{j=1}^{p^{\prime}}\alpha_{i,j}).$

Notice that $\Psi_{i}$ is a family of domains constructed by the replacement
of $\hat{\Phi}_{i}(D_{i,j})$’s by the family of domains $(H_{1}\cdot
H_{2})_{\sharp}([0,1]\times\alpha_{i,j})$. Hence we can reparametrize
$\Psi_{i}$ to be a $2$-sweepout $\hat{\Psi}_{i}$ for large $i$ and this
satisfies $\sup_{x\in S^{1}\times I}L^{\theta}(x)<w_{k}-\min(b/6,a/2)$ and
this gives the contradiction. ∎

Also we obtain a general Morse index characterization by bumpy approximation
as in Corollary 7.3 in [28]:

######  Corollary 5.13.

For a $2$-Riemannian disk $(D^{2},\partial D^{2},g)$ with a strictly convex
boundary satisfying the hypothesis $(\star)$, for $k=1,2$, there exists a
capillary embedded geodesic $\gamma_{k}$ with

$index(\gamma_{k})\leq k\leq index(\gamma_{k})+nullity(\gamma_{k}).$

##  6\. The optimality of the condition

In this section, we prove that the boundary total curvature condition in
Theorem 4.4 is sharp. Our proof of the following Proposition 6.1 relies on the
elementary construction of “a rounded Euclidean cone”.

######  Proposition 6.1.

For a given $k\in(0,\pi)$, there exists a Riemannian $2$-disk $(D^{2},\partial
D^{2},g)$ with a strictly convex boundary, interior nonnegative Gaussian
curvature and a total signed curvature $\int_{\partial D^{2}}\kappa=k$ such
that there exists $\theta\in(0,\pi/2)$ satisfying the following: The
Riemannian disk $(D^{2},\partial D^{2},g)$ does not admit a capillary embedded
geodesic with a contact angle $\theta$.

###### Proof.

Consider a Riemannian disk $M$ of revolution explicitly given by $(u,r(u)\cos
t,r(u)\sin t)$ for $u\in[s,1]$ and $t\in[0,2\pi]$ in $\mathbb{R}^{3}$ endowed
with Euclidean metric such that

  1. (1)

$0<s<1/2\cos(k/2)$ and $r(s)=0$,

  2. (2)

$r(u)\geq 0$ for $u\in[s,1]$,

  3. (3)

$r^{\prime}_{+}(s)=+\infty$ and $r^{\prime}(u)>0$,

  4. (4)

$r^{\prime\prime}(u)\leq 0$,

  5. (5)

$r(u)=(k/\sqrt{4\pi^{2}-k^{2}})u$ for $u\geq 1/2\cos(k/2)$.

By a simple calculation, we can obtain total signed curvature $\int_{\partial
D^{2}}\kappa=k$. Moreover, the condition (4) gives $K\geq 0$ on $M$. Then
since a flat plane and a cone are isometric, by straightforward trigonometry,
we obtain that a geodesic emanating from a boundary point
$(1,k/\sqrt{4\pi^{2}-k^{2}},0)$ with a contact angle $k/2$ is a critical
lasso. Note that the geodesic only passes through the cone part ($x\geq
1/2\cos(k/2)$). Notice that a small addition of the contact angle of a
critical lasso on an Euclidean cone induces an interior self-intersection
point. Hence there exists small $\epsilon>0$ such that the geodesic emanating
from $(1,k/\sqrt{4\pi^{2}-k^{2}},0)$ with a contact angle
$k/2+\epsilon\in(0,\pi/2)$ which develops an interior self-intersection point.
Since $M$ is rotationally symmetric, we see that $M$ does not admit a
capillary embedded geodesic with a contact angle $k/2+\epsilon$. ∎

##  Appendix A Compactness theorem of embedded geodesics

The following theorem asserts the compactness of the set of embedded geodesics
on surfaces.

######  Theorem A.1.

Let $(M,\partial M,g)$ be a closed surface with strictly convex boundary. For
a fixed contact angle $\theta$, if $\\{\gamma_{k}\\}$ is a sequence of
connected embedded geodesics whose endpoints are on $\partial M$ with length
bound

$\mathcal{H}^{1}(\gamma_{k})\leq L<\infty$

for some fixed constant $L\in\mathbb{R}$ independent of $k$. Then up to
subsequence, there exists a connected capillary embedded geodesic $\gamma$
with a contact angle $\theta$ where $\gamma_{k}\rightarrow\gamma$ in the
varifold sense with

$\mathcal{H}^{1}(\gamma)\leq L<\infty$

and convergence is smooth and graphical for all $x\in M$. Moreover, if for
every $k$, $\gamma_{k}$’s are capillary embedded geodesics with a contact
angle $\theta$, then $\gamma$ is also a capillary embedded geodesic with a
contact angle $\theta$. The multiplicity of convergence $m$ is 1. Furthermore,
if $\gamma_{k}\cap\gamma=\emptyset$ eventually, then $\gamma$ is stable, and
$index(\gamma)\geq 1$ otherwise.

###### Proof.

We can directly adopt the compactness theorem of embedded geodesics without
boundary of Theorem A.1 in [28]. For the multiplicity part, a small tubular
neighborhood of capillary embedded geodesic $\gamma$ is diffeomorphic to
rectangular strips and this gives multiplicity $1$. Moreover, by standard
arguments of elliptic equations, we can ensure the fact that the first
eigenfunction $f$ associated with the second variation formula (1) is a signed
function and this gives a stability analysis as in Claim 4 of Theorem A.1 in
[28]. The contact angle condition is preserved along the convergence. ∎

Following corollary shows the compactness property of associated capillary
varifolds of capillary embedded geodesics.

######  Corollary A.2.

Let $(M,\partial M,g)$ be a closed surface with strictly convex boundary.
Suppose that a sequence of capillary geodesics $\\{\gamma_{k}\\}$ with contact
angle $\theta$ satisfies $\partial^{i}\Omega_{k}=\gamma_{k}$ where
$\Omega_{k}\in\mathcal{D}(M)$ and the $L^{\theta}$-functional bound as

$L^{\theta}(\Omega_{k})\leq L<\infty.$

Then up to subsequence, there exists a domain $\Omega\in\mathcal{D}(M)$ such
that $\partial^{i}\Omega=\gamma$ and an associated capillary varifold
$\partial^{\theta}\Omega$ with

$L^{\theta}(\Omega)\leq L<\infty.$

Hence, we have a varifold convergence of capillary varifolds
$\partial^{\theta}\Omega_{k}\rightarrow\partial^{\theta}\Omega$.

###### Proof.

By Theorem A.1 above, we have a smooth convergence
$\gamma_{k}\rightarrow\gamma$ with the conservation of contact angles. This
gives a convergence of endpoints of $\gamma_{k}$ to endpoints of $\gamma$, and
convexity of boundary implies the existence of domain $\Omega\in\mathcal{M}$
such that $\gamma=\partial\Omega$ and associated capillary varifolds
$\partial^{\theta}\Omega$. Then we obtain the convergence of varifolds
$\partial^{i}\Omega_{k}\rightarrow\partial^{i}\Omega$ and
$\partial^{b}\Omega_{k}\rightarrow\partial^{b}\Omega$, and convergence of
capilary varifolds
$\partial^{\theta}\Omega_{k}\rightarrow\partial^{\theta}\Omega$ follows. ∎

We have the following finiteness theorem on $\theta$-bumpy metric.

######  Corollary A.3.

On Riemannian $2$-disk $(M,\partial M,g)$ with strictly convex boundary and a
$\theta$-bumpy metric, the set of all domains $D_{L}$ whose relative boundary
is a capillary embedded geodesic whose boundary points are on $\partial M$ and
$L^{\theta}$-functional achieves the value less than or equal to $L$ is
finite.

###### Proof.

Suppose $D_{L}$ is an infinite set. By Corollary A.2, there exists an infinite
sequence of associated capillary varifolds $\\{\partial^{\theta}\Omega_{k}\\}$
converging to $\partial^{\theta}\Omega$ in $\Lambda_{\theta,L}$. This induces
a nontrivial Jacobi field on $\partial^{\theta}\Omega$ and this contradicts to
the bumpiness of the metric. ∎

##  Appendix B Regularity theorem of curve shortening flow with fixed boundary
points

In this section, we include the existence of regularity theorem of curve
shortening flow with fixed boundary points. While it is stated and proven in
Edelen [19], we include it here without the proof for completeness. The
arguments of the proof relies on the distance comparison arguments which is
initiated by Huisken [25] (See also Theorem 2.6 in [25]).

######  Theorem B.1 ([19], Corollary 3.2).

Let $(M,\partial M,g)$ be a compact Riemannian surface with a strictly convex
boundary. For $p,q\in\partial M$, let $F_{0}:[a,b]\rightarrow M$ be a smooth
embedded curve where $F_{0}(a)=p$ and $F_{0}(b)=q$. Then the curve shortening
flow for the initial curve $F_{0}$ with boundary conditions
$F_{t}(a)=F_{0}(a),\,F_{t}(b)=F_{0}(b)$ has a smooth solution for all times
$t>0$, which for $t\rightarrow\infty$ converges to a geodesic segment
connecting $p$ and $q$.

##  Appendix C Bumpy metric theorem of capillary embedded geodesics

In this section, we prove the bumpy metric theorem of capillary embedded
geodesics. The bumpy metric theorem asserts that capillary embedded geodesics
do not have nontrivial Jacobi fields in a generic metric. We will mostly
modify the bumpy metric theorem in embedded minimal hypersurface in [53]. Our
modifications mainly come from the free boundary version in [2] and the
stationary geodesic network version of [11] (See also [50]). We outline the
proof and mainly explain about the necessary modifications.

We state the bumpy metric theorem first:

######  Theorem C.1.

Let $(M,\partial M,g)$ be a compact Riemannian surface with a strictly convex
boundary. For a $C^{\infty}$-generic set of a metric on $M$, no capillary
embedded geodesic with contact angle $\theta$ on $M$ (whose boundary points
are on $\partial M$) has a nontrivial Jacobi fields.

For $q\geq 3$ a positive integer or $\infty$, denote $\Gamma^{q}$ to be the
set of $C^{q}$ metrics on $M$ endowed with the smooth topology. For $3\leq
j+1\leq q$, let us consider the space of embeddings $Emb(I,M)$ of $I=[0,1]$ to
$M$. For an embedding $w\in Emb(I,M)$ we can take an equivalence class
$[w_{\Omega}]$ associated with a connected domain $\Omega\subseteq M$ by

$[w_{\Omega}]:=\\{(w\cdot\phi,\Omega)|\phi\in\text{Diff}(I)\text{ and
}w=\partial^{i}\Omega\\},$

We consider the set $\mathcal{PE}(I,M)$ of equivalence classes $[w_{\Omega}]$
for an embedding $w$ and corresponding domain $\Omega$. We take a subset
$S^{q}$ of $\mathcal{PE}=\mathcal{PE}(I,M)$ defined by

(46)
$\mathcal{S}^{q}:=\\{(\gamma,[w_{\Omega}])\in\Gamma^{q}\times\mathcal{PE}|w\text{
is a }\gamma-\text{embedded geodesic with a contact angle }\theta\\},$

where $\theta\in(0,\pi/2)$ is a fixed angle and defined by the outer angle
between $T_{z}\partial M$ and $T_{z}w(I)$ for $z\in w(I)\cap\partial M$ from
$\Omega$. Let us endow the topology in $\mathcal{PE}$ induced by the Hausdorff
distance between two associated domains of the embeddings, and endow the
topology in $\mathcal{S}^{q}$ as an inherited topology from the product
topology in $\Gamma^{q}\times\mathcal{PE}$. Let us denote
$\pi^{q}_{\mathcal{S}}:\mathcal{S}^{q}\rightarrow\Gamma^{q}$ to be the
projection map onto the first factor as

(47) $\pi^{q}_{\mathcal{S}}(\gamma,[w_{\Omega}])=\gamma.$

Given a Sard-Smale Theorem (Theorem C.2, Theorem 1.2 in [48]) and a capillary
version of the structure theorem (Theorem C.3, cf. Theorem 35 of [2]) below,
we can modify arguments in Section 7.1 of [2] and apply Theorem C.2 to prove
Theorem C.1. We prove Theorem C.3 in Subsection C.1.

######  Theorem C.2 ([48], Theorem 1.2).

Let $\mathcal{M}$ and $\Gamma$ be a separable $C^{q}$-Banach manifolds.
Suppose $\pi:\mathcal{M}\rightarrow\Gamma$ is a $C^{q}$ Fredholm map of
Fredholm index $0$. Then the set of regular values for $\pi$ is a Baire
generic set.

######  Theorem C.3.

[cf. [2], Theorem 35] Let $(M,\partial M,g)$ be a compact Riemannian surface
with a strictly convex boundary. Then $\mathcal{S}^{q}$ is a separable Banach
manifold of class $C^{q-1}$ and $\pi^{q}_{\mathcal{S}}$ is a $C^{q-1}$
Fredholm map of Fredholm index $0$. Furthermore, given any
$(\gamma,[w_{\Omega}])\in\mathcal{S}^{q}$, the nullity of $w_{\Omega}$ equals
the dimension of the kernel of the linear map
$D\pi^{q}_{\mathcal{S}}(\gamma,[w_{\Omega}])$. In particular, $w$ admits a
non-trivial Jacobi field associated with $L^{\theta}$-functional if and only
if the point $(\gamma,[w_{\Omega}])$ is critical for $\pi^{q}_{\mathcal{S}}$.

###  C.1. The structure theorem

In this subsection, we prove Theorem C.3. We will outline the proof of Theorem
C.3 and mostly emphasize the necessary change we need here. Let
$V:=I\times\mathbb{R}$ be a normal bundle over $I$ and $V_{I}:=T^{*}I\otimes
V$. We take the convention that the positive direction of the normal bundle is
chosen toward outside of an enveloping domain for each embedding of curves.
Also denote $E:V\rightarrow M$ to be an embedding of $V$ to $M$ such that
$E(\partial V)\subseteq\partial M$ with contact angle $\theta$ and each
section of $V$ corresponds to the normal coordinate of the embedding in $M$.
We consider a smooth map $l:\Gamma\rightarrow C^{q}(I\times V\times
V_{I};\mathbb{R})$ where $\Gamma$ is an open subset of a Banach space
$\mathcal{S}^{q}$. For $u\in C^{2}(I,V)$, we define a functional $L$ to be

(48)
$L(\gamma,u)=\int_{0}^{1}l(\gamma)(x,u,u^{\prime}(x))dx+\bigg{[}b(\gamma)(x,u(x))\bigg{]}^{1}_{0}.$

Let us write $l_{\gamma}$ and $b_{\gamma}$ in place of $l(\gamma)$ and
$b(\gamma)$. The first variational formula (see Proposition 41 of [2]) gives
the following:

(49) $\displaystyle\frac{d}{dt}\Big{|}_{t=0}L(\gamma,u+tv)$
$\displaystyle=\int_{0}^{1}(D_{2}l_{\gamma}(x,u,u^{\prime})-(D_{3}l_{\gamma}(x,u,u^{\prime}))_{x})v(x)dx$
$\displaystyle+\bigg{[}(D_{3}l_{\gamma}(x,u,u^{\prime})(x)+D_{2}b_{\gamma}(x,u))v(x)\bigg{]}_{0}^{1}.$

We define the first variation map $(H,\Theta):\Gamma\times
C^{2}(I,V)\rightarrow C^{0}(I,V)\times C^{0}(\partial I,V)$ by

(50) $\displaystyle H(\gamma,u)$
$\displaystyle=-D_{2}l_{\gamma}(x,u,u^{\prime})-(D_{3}l_{\gamma}(x,u,u^{\prime}))_{x}$
(51) $\displaystyle\Theta(\gamma,u)$
$\displaystyle=D_{3}l_{\gamma}(x,u,u^{\prime})+D_{2}b_{\gamma}(x,u).$

The first variation (49) gives the following directly.

######  Proposition C.4 (cf. [2], Proposition 41 (1)).

Suppose $L,H,\Theta$ is defined as in (48), (50), (51). Then $H$ and $\Theta$
are $C^{q-2}$ banach maps.

Now we consider the linearization of $(H,\Theta)$. The linearization
$L_{H}:C^{2}(I,V)\rightarrow C^{0}(I,V)$ and $L_{\Theta}:C^{2}(I,V)\rightarrow
C^{0}(\partial I,V)$ are defined by

$L_{H}(\gamma,u)[v]=\frac{d}{dt}\bigg{|}_{t=0}H(\gamma,u+tv),\,\,L_{\Theta}(\gamma,u)[v]=\frac{d}{dt}\bigg{|}_{t=0}\Theta(\gamma,u+tv)$

and we define a Jacobi operator $L(\gamma,u):C^{2}(I,V)\rightarrow
C^{0}(I,V)\times C^{0}(\partial I,V)$ by

(52) $L(\gamma,u)=(L_{H}(\gamma,u),L_{\Theta}(\gamma,u)).$

Note that the formula of $L_{H}$ and $L_{\theta}$ is explicitly written by

(53) $\displaystyle L_{H}(\gamma,u)[v](\cdot)$
$\displaystyle=-(D_{22}l_{\gamma}(x,u,u^{\prime})v+D_{23}l_{\gamma}(x,u,u^{\prime})v^{\prime})(\cdot)$
$\displaystyle+(D_{33}l_{\gamma}(x,u,u^{\prime})v^{\prime}(\cdot)+D_{23}l_{\gamma}(x,u,u^{\prime})v(\cdot))_{x}$
$\displaystyle L_{\Theta}(\gamma,u)[v](\cdot)$
$\displaystyle=((D_{23}l_{\gamma}(x,u,u^{\prime})+D_{22}b_{\gamma}(x,u))v+D_{33}l_{\gamma}(x,u,u^{\prime})v^{\prime})(\cdot)$

######  Proposition C.5 (cf. [2], Proposition 45 (2),(3)).

Suppose the Jacobi operator $L$ is given by (52) and $L_{H}$ is uniformly
elliptic in the sense of

(54) $D_{33}l_{\gamma}(x,u,u^{\prime})>0$

for $x\in[0,1]$. Then

  1. (1)

Let us denote $\epsilon:\partial I\rightarrow I$ as the canonical inclusion.
The operator $L$ is self-adjoint in the sense of

$(L(\gamma,u)[v],(w,w\circ\epsilon))=(L(\gamma,u)[w],(v,v\circ\epsilon))$

for any $v,w\in C^{2}(I,V)$.

  2. (2)

The operator $L$ is Fredholm of index $0$.

###### Proof.

(1) follows from integration by parts. $L$ is a Fredholm operator by ODE
uniqueness theorem and boundary conditions $L^{\Theta}$ on $\partial I$. Since
$L$ is a self-adjoint operator and $L_{H}$ is uniformly ellptic in the sense
of (54), the equation $(L_{H},L_{\Theta})(\phi,\psi)=(\xi,\eta)$ becomes a
nondegenerate 2nd order ODE with Robin boundary condition by (53). Hence, the
simpler ODE version of arguments in the proof of Proposition 45 in [2] applies
and we obtain that the Fredholm index is $0$. ∎

The arguments in the proof of Proposition 46 in [2] and we obtain the
following structural result on the neighborhood of a fixed embedding. For a
point $(\gamma_{0},u_{0})$ with

$H(\gamma_{0},u_{0})=\Theta(\gamma_{0},u_{0})=0,$

we set $K=kerL(\gamma,u)$.

######  Proposition C.6 ([2], Proposition 46).

Suppose $L$ is elliptic in the sense of (54). Assume that for every nonzero
$\kappa\in K$ there exists a differentiable curve
$\gamma:(-1,1)\rightarrow\Gamma$ with $\gamma(0)=\gamma_{0}$ and such that

$\frac{\partial}{\partial
s}\bigg{|}_{s=0}((H(\gamma(s),u_{0}),\Theta(\gamma(s),u_{0})),(\kappa,\kappa\circ\epsilon))\neq
0.$

Then the map $(H,\Theta):\Gamma\times C^{2}(I,V)$ is a submersion near
$(\gamma_{0},u_{0})$, so there exists a neighborhood $U_{\Gamma}\times
U_{C^{2}(I,V)}$ of such point such that

$\mathcal{M}=\\{(\gamma,u)\in U_{\Gamma}\times
U_{C^{2}(I,V)}|H(\gamma,u)=0,\Theta(\gamma,u)=0\\}$

is a $C^{q-2}$-Banach submanifold with tangent space
$kerD(H,\Theta)(\gamma,u)$ at each $(\gamma,u)\in\mathcal{M}$. Furthermore,
the restricted projection operator
$\Pi_{|\mathcal{M}}:\mathcal{M}\rightarrow\Gamma$, where $\Pi:\Gamma\times
C^{2}(I,V)\rightarrow\Gamma$ is given by $\Pi(\gamma,u)=\gamma$, is a
$C^{q-2}$-Fredholm map of index $0$.

We denote a domain associated to the variation $u$ by $\Omega_{u}$. Let a
signed boundary length function
$b_{\gamma}:I\times\mathbb{R}\rightarrow\mathbb{R}$ which only depends on
boundary values $u(0)$ and $u(1)$ to be

(55) $b_{\gamma}(x,u(x))=\cos\theta|\partial^{b}\Omega_{u}|.$

###### Proof of Theorem C.3.

We take $l_{\gamma}$ to be the nonparametric length functional and a boundary
length function in (55). Then $(\gamma,[\omega_{\Omega}])\in\mathcal{S}^{q}$
if and only if a corresponding pair of an embedding $(\gamma,u)$ is stationary
for the functional $L$.

We apply proposition C.5 and arguments in the proof of Theorem 35 in [2]
directly works and we obtain the structural theorem. ∎

## References

  * [1] N. S. Aiex, The width of ellipsoids, Commun. Anal. Geom., 27 (2019), 251–285
  * [2] L. Ambrozio, A. Carlotto, and B. Sharp, Compactness analysis for free boundary minimal hypersurfaces, Calc. Var. Partial Differential Equations 57 (2018), no. 1, 57:22. MR3740402
  * [3] L. Ambrozio, R. Montezuma, R. Santos, The Width of curves in Riemannian manifolds, arXiv: 2307.12939 (2023).
  * [4] V. I. Arnold, Arnold’s problems, Springer-Verlag, Berlin; PHASIS, Moscow, 2004, Translated and revised edition of the 2000 Russian original, With a preface by V. Philippov, A. Yakivchik and M. Peters.
  * [5] L. Asselle and G. Benedetti, The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles, J. Topol. Anal. 8 (2016), no. 3, 545–570.
  * [6] A. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag (1987).
  * [7] G. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199-300.
  * [8] W. Bos, Kritische Sehnen auf Riemannschen Elementarraumstücken, Math. Ann. 151 (1963), 431–451.
  * [9] E. Calabi and J. Cao, Simple closed geodesics on convex surfaces. J. Differential Geom. 36 (1992), no. 3, 517–549.
  * [10] D. Cheng and X. Zhou. Existence of curves with constant geodesic curvature in a Riemannian 2-sphere. Trans. Amer. Math. Soc., 374:9007–9028, 2021.
  * [11] O. Chodosh and C. Mantoulidis, The p-widths of a surface, Publications mathématiques de l’IHÉS volume 137, 245–342, 2023.
  * [12] T. Colding and C. De Lellis, The min-max construction of minimal surfaces, Surveys in Differential Geometry VIII, International Press, (2003), 75–107.
  * [13] T.H. Colding, D. Gabai, and D. Ketover, On the classification of Heegaard splittings, Duke Math. J. 167 (2018), no. 15, 2833–2856.
  * [14] C. De Lellis and D. Tasnady. The existence of embedded minimal hypersurfaces. J. Differential Geom., 95(3):355–388, 2013.
  * [15] L. De Masi and G. De Philippis. Min-max construction of minimal surfaces with a fixed angle at the boundary. 2021. arXiv: 2111.09913 (2021).
  * [16] G. De Philippis, M. Marini, M. Mazzucchelli, and S. Suhr, Closed geodesics on re- versible Finsler 2-spheres, J. Fixed Point Theory Appl. 24 (2022), no. 2, Paper No. 19.
  * [17] A. Dey, Existence of multiple closed cmc hypersurfaces with small mean curva- ture. J. Differential Geom. to appear, arXiv:1910.00989 (2019).
  * [18] S. Donato and R. Montezuma, The first width of non-negatively curved surfaces with convex boundary, preprint arXiv:2211.14815 (2022).
  * [19] N. Edelen, Noncollapsing of curve-Shortening flow in surfaces, Int. Math. Res. Not., 2015, No. 20, pp. 10143-10153.
  * [20] R. Finn, Equilibrium capillary surfaces, Springer-Verlag, New York, 1986
  * [21] H. Gluck and W. Ziller, Existence of periodic motions of conservative systems, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 65–98.
  * [22] M. Grayson, Shortening embedded curves, Ann. of Math. (2) 129, no. 1 (1989) 71-111.
  * [23] A. Hatcher, A proof of the Smale conjecture, $\text{Diff}(S^{3})\equiv O(4)$. Ann. of Math. (2), 117(3):553–607, 1983.
  * [24] H. Hong, and A. Saturnino, Capillary surfaces: stability, index and curvature estimates, arXiv:2105.12662, (2021).
  * [25] G. Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2, 127–133 (1998)
  * [26] D. Ketover, Flipping Heegaard Splittings and Minimal Surfaces, arXiv:2211.03745 (2022).
  * [27] D. Ketover and Y. Liokumovich, On the existence of closed $C^{1,1}$ curves of constant curvature, arXiv:1810.09308 (2018).
  * [28] D. Ko, Morse Index Bound and Morse Inequalities of Simple Closed Geodesics on $2$-spheres, arXiv:2303.00644 (2023)
  * [29] D. Ko, Two free boundary geodesics on Riemannian 2-disks with convex boundary with Morse Index bound, In preparation.
  * [30] M. Langford and J. Zhu, A distance comparison principle for curve shortening flow with free boundary, arXiv:2302.14258 (2023).
  * [31] M. Li, Chord shortening flow and a theorem of Lusternik and Schnirelmann. Pacific J. Math. 299 (2019), no. 2, 469-488.
  * [32] C. Li, A polyhedron comparison theorem for 3-manifolds with positive scalar curvature. Invent. Math., 219(1):1–37, 2020.
  * [33] C. Li, The dihedral rigidity conjecture for n-prisms. arXiv:1907.03855.
  * [34] C. Li, X. Zhou, and J. Zhu, Min-max theory for capillary surfaces, arXiv:2111.09924 (2021).
  * [35] L. Lusternik, and L. Schnirelmann, Méthodes topologiques dans les problèmes variationnels. Inst. for Math. and Mechanics, Moscow, 1930. (In Russian)
  * [36] F. C. Marques and A. Neves, Existence of infinitely many minimal hypersurfaces in positive Ricci curvature. Invent. Math. 209, 577–616 (2017).
  * [37] F. C. Marques and A. Neves, Morse index and multiplicity of min-max minimal hypersurfaces, Camb. J. Math. 4 (2016), no. 4, 463–511. MR3572636
  * [38] F. C. Marques and A. Neves, Morse index of multiplicity one min-max minimal hypersurfaces, Advances in Mathematics 378 (2021), 107527.
  * [39] L. Mazurowski, CMC doublings of minimal surfaces via min-max. arXiv:2010.01094 (2020).
  * [40] L. Mazurowski and X. Zhou, The Half-Volume Spectrum of a Manifold, arXiv:2302.07722 (2023).
  * [41] S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk 37 (1982), no. 5(227), 3–49, 248.
  * [42] J. T. Pitts, Regularity and singularity of one dimensional stationary integral varifolds on manifolds arising from variational methods in the large, Symposia Mathematica, Vol. XIV (Convegno di Teoria Geometrica dell’Integrazione e Variet‘a Minimali, INDAM, Roma, Maggio 1973), pp. 465–472. Academic Press, London, (1974).
  * [43] J. T. Pitts, Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. Princeton University Press, 1981.
  * [44] L. Sarnataro and D. Stryker, Existence of closed embedded curves of constant curvature via min-max, Preprint.
  * [45] M. Schneider, Closed magnetic geodesics on $S^{2}$, J. Differential Geom. 87 (2011), no. 2, 343–388.
  * [46] L. Simon, Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis. Australian National University, 1984.
  * [47] S. Smale, Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc., 10:621–626, 1959.
  * [48] S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR0185604 (32 $\\#$3067)
  * [49] A. Song, Existence of infinitely many minimal hypersurfaces in closed manifolds, Ann. of Math. (2) 197 (2023), no. 3, 859–895.
  * [50] B. Staffa. Bumpy metrics theorem for geodesic networks. arXiv:2107.12446, (2021).
  * [51] A. Sun, Z. Wang and X. Zhou, Multiplicity one for min-max theory in compact manifolds with boundary and its applications. arXiv:2011.04136, (2020).
  * [52] Z. Wang, Existence of infinitely many free boundary minimal hypersurfaces, J. Differential Geom. to appear, arXiv:2001.04674, (2020).
  * [53] B. White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), no. 1, 161–200. MR1101226 (92i:58028)
  * [54] B. White, A strong minimax property of nondegenerate minimal submanifolds, J. Reine Angew. Math. 457 (1994), 203–218.
  * [55] X. Zhou, On the free boundary min-max geodesics. Int. Math. Res. Not. IMRN, (5):1447–1466, 2016.
  * [56] X. Zhou, On the multiplicity one conjecture in min-max theory, Ann. of Math. (2) 192 (2020), no. 3, 767–820.
  * [57] X. Zhou and J. J. Zhu, Min-max theory for networks of constant geodesic curvature, Adv. Math. 361 (2020), 106941, 16.