# A form of Schwarz’s lemma and a bound for the Kobayashi metric on convex
domains

Anwoy Maitra  Department of Mathematics, Indian Institute of Science,
Bangalore 560012, India<EMAIL_ADDRESS>
###### Abstract.

We present a form of Schwarz’s lemma for holomorphic maps between convex
domains $D_{1}$ and $D_{2}$. This result provides a lower bound on the
distance between the images of relatively compact subsets of $D_{1}$ and the
boundary of $D_{2}$. This is a natural improvement of an old estimate by
Bernal-González that takes into account the geometry of $\partial{D_{1}}$. We
also provide a new estimate for the Kobayashi metric on bounded convex
domains.

###### Key words and phrases:

Convex domains, Kobayashi metric, Schwarz’s lemma

###### 2010 Mathematics Subject Classification:

Primary: 32F45, 32H02

This work is supported by a scholarship from the Indian Institute of Science

##  1\. Introduction

In this paper, we shall prove two theorems concerning the Kobayashi geometry
of convex domains in $\mathbb{C}^{n}$. In this section, we introduce these
theorems and discuss some of the motivations behind them.

Our first theorem is motivated by the following result of Bernal-González:

######  Result 1.1 (Bernal-González [1]).

Let $E$, $F$ be two complex Banach spaces. Assume that $D_{1}\subseteq E$ and
$D_{2}\subseteq F$ are convex domains and that $D_{1}$ is bounded. Fix two
points $a\in D_{1}$ and $b\in D_{2}$ and a real number $r>0$. Then there
exists a real number $s=s(a,b,r)>0$ such that, for every holomorphic map
$\phi:D_{1}\to D_{2}$ satisfying $\phi(a)=b$,

$\mathrm{dist}\big{(}\phi(\\{z\in
D_{1}\mid\mathrm{dist}(z,D^{\mathsf{c}}_{1})>r\\}),D^{\mathsf{c}}_{2}\big{)}\geqslant
s,$

where

$s\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{dist}(b,D^{\mathsf{c}}_{2})\exp\left(-\frac{2\mu(a)}{\min\big{(}r,\mathrm{dist}(a,D^{\mathsf{c}}_{1})\big{)}}\right),$
(1.1)

and where
$\mu(a)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup(\\{\|z-a\|\mid
z\in D_{1}\\})$.

While this result is stated in a setting that is very general, the dependence
of the lower bound for $\mathrm{dist}\big{(}\phi(\\{z\in
D_{1}\mid\mathrm{dist}(z,D^{\mathsf{c}}_{1})>r\\}),D^{\mathsf{c}}_{2}\big{)}$
on the parameter $r$ seems to be overly conservative, given that

$\exp\left(-\frac{2\mu(a)}{\min(r,\mathrm{dist}(a,D^{\mathsf{c}}_{1}))}\right)$

decays extremely rapidly as $r\to 0$. One realizes that the dependence on $r$
could be improved, especially if $E$ and $F$ are taken to be finite-
dimensional. For instance, if we take $D_{1}=D_{2}=\mathbb{D}$ (in this paper,
$\mathbb{D}$ will denote the open unit disk centred at $0\in\mathbb{C}$),
$E=F=\mathbb{C}$ and $a$, $b$ to be two arbitrary points of $\mathbb{D}$ in
Result 1.1, then it follows from the Schwarz–Pick lemma that for any
holomorphic map $\phi:\mathbb{D}\to\mathbb{D}$ such that $\phi(a)=b$, and for
any $s\in(0,1)$,

$\mathrm{dist}\big{(}\phi(s\mathbb{D}),\mathbb{D}^{\mathsf{c}}\big{)}\geqslant\frac{(1-s)\,\mathrm{dist}(a,\mathbb{D}^{\mathsf{c}})\,\mathrm{dist}(b,\mathbb{D}^{\mathsf{c}})}{4},$
(1.2)

where $(1-s)$ serves as the parameter $r$ of Result 1.1. The dependence on $r$
suggested by (1.2) is the chief motivation behind Theorem 1.2 below. Our
result is stated in the finite-dimensional setting because much richer results
are obtainable in this setting (and there is much about holomorphic maps in
this setting that remains to be discovered). We must clarify that, in what
follows, $\|\boldsymbol{\cdot}\|$ will denote the Euclidean norm and the
expression $\mathrm{dist}(x,S)$ ($S$ being a non-empty set) must be understood
in terms of the Euclidean norm. With these remarks, we can state our first
result.

######  Theorem 1.2.

Let $D_{1}$ and $D_{2}$ be open convex subsets of $\mathbb{C}^{n}$ and
$\mathbb{C}^{m}$ respectively. Assume $D_{1}$ is bounded. Fix $a\in D_{1}$ and
$b\in D_{2}$. Then there exist constants $\alpha\geq 1$ and $C>0$ — where
$\alpha$ depends only on $D_{1}$ and $C$ depends only on $D_{1}$ and $a$ —
such that for every holomorphic map $\phi$ from $D_{1}$ to $D_{2}$ with
$\phi(a)=b$, and for every $r>0$,

$\mathrm{dist}\big{(}\phi(\\{z\in
D_{1}\mid\mathrm{dist}(z,D_{1}^{\mathsf{c}})\geqslant
r\\}),D_{2}^{\mathsf{c}}\big{)}\geqslant
C\,\mathrm{dist}(b,D_{2}^{\mathsf{c}})\,r^{\alpha}.$ (1.3)

We must remark that, while the dependence on $r$ of the lower bound in
$\eqref{E:improved_BG_lb}$ is a power — note how much this improves upon the
expression (1.1) — the exponent can _in general_ be large. In Section 2 we
provide a simple example in which the dependence on $r$ of
$\mathrm{dist}\big{(}\phi(\\{z\in
D_{1}\mid\mathrm{dist}(z,D^{\mathsf{c}}_{1})>r\\}),D^{\mathsf{c}}_{2}\big{)}$,
$\phi$ as in Theorem 1.2, is a certain high power of $r$ that cannot be
lowered. Just as discussed in [1], we may view Theorem 1.2 as a form of
Schwarz’s lemma for convex domains.

Bernal-González’s result relies upon a well-known estimate for the
Carathéodory distance. This estimate actually holds true on any bounded
domain, whereas it is possible to provide sharper estimates on bounded convex
domains. This is at the heart of our improvement of Result 1.1. It is more
convenient to work with the Kobayashi distance (which coincides with the
Carathéodory distance on bounded convex domains, but this is not relevant to
our proof). The improved estimate for the Kobayashi distance that we shall use
is due to Mercer [9]—see Section 3 for details. Our use of Mercer’s estimate
is quite similar to its use recently in [11, 2].

Before we move on to our second result, we need to introduce two pieces of
notation: $D(a,r)$ will denote the open disk in $\mathbb{C}$ with centre $a$
and radius $r$, and $\kappa_{D}(p,\boldsymbol{\cdot})$ will denote the
Kobayashi pseudo-metric of the domain $D\subseteq\mathbb{C}^{n}$ at the point
$p\in D$.

It is of interest in many applications to be able to estimate
$\kappa_{D}(p,\boldsymbol{\cdot})$. If nothing is assumed about $D$ beyond the
fact that it is a bounded convex open set, then the best result that seems to
be available is the following one by Graham:

######  Result 1.3 (Graham [5, Theorem 3], also see [6]).

Let $D\subseteq\mathbb{C}^{n}$ be a bounded convex open set. Given $p\in D$
and $\xi\in T_{p}^{(1,0)}D$, we let $r$ denote the supremum of the radii of
the disks centred at $p$, tangent to $\xi$, and included in $D$. Then

$\frac{\|\xi\|}{2r}\leqslant\kappa_{D}(p,\xi)\leqslant\frac{\|\xi\|}{r}.$
(1.4)

We ought to clarify that the non-trivial bound in (1.4) is the lower bound.
The upper bound is merely a consequence of the metric-decreasing property of
holomorphic mappings (with respect to the Kobayashi pseudo-metric). The upper
bound in (1.4) is achieved as an equality in rare cases. For example, if we
take $D=\mathbb{D}$ and $p$ to be any off-centre point ($p\neq 0$) then the
upper bound for $\kappa_{\mathbb{D}}(p,1)$ given by Result 1.3 is $1/(1-|p|)$,
whereas the actual value of $\kappa_{\mathbb{D}}(p,1)$ is $1/(1-|p|^{2})$,
which is less than $1/(1-|p|)$. This motivates us to find a better upper
bound, which can be stated (as is the case in Result 1.3) in terms of the
positioning of $(p,\xi)$. As the following theorem shows: an upper bound on
$\kappa_{D}(p,\xi)$ is available that is strictly smaller than that provided
by Result 1.3 for $(p,\xi)$ that is, in a certain sense, “generic” (see the
concluding sentence of the following theorem). This theorem also shows that
this more efficient bound is governed by one of two regimes, _both_ of which
do arise (see the examples in Section 2).

######  Theorem 1.4.

Let $D$ be a bounded convex open subset of $\mathbb{C}^{n}$. Let $p\in D$ and
let $\xi\in T_{p}^{(1,0)}D\setminus\\{0\\}$. Write

$\displaystyle D(\xi)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\Big{\\{}z\in\mathbb{C}\mid
p+z\frac{\xi}{\|\xi\|}\in D\Big{\\}},$ $\displaystyle r^{\bullet}(p,\xi)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup(\\{r>0\mid\exists\zeta\in
D(\xi)\text{ such that }0\in D(\zeta,r)\subseteq D(\xi)\\}).$

Let

$S^{\bullet}(p,\xi)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\Big{\\{}p+z\frac{\xi}{\|\xi\|}\mid
z\in D(\xi),\;0\in\overline{D(z,r^{\bullet}(p,\xi))}\text{ and
}D(z,r^{\bullet}(p,\xi))\subseteq D(\xi)\Big{\\}}.$ (1.5)

Also, for any $w\in(p+\mathbb{C}\,\xi)$, let $r(w,\xi)$ denote the supremum of
the radii of the discs centred at $w$, tangent to $\xi$, and included in $D$.
Then

  1. (1)

$S^{\bullet}(p,\xi)$ is a non-empty compact convex subset of $\mathbb{C}^{n}$
(indeed, of $D\cap(p+\mathbb{C}\,\xi)$) and there exists a unique point
$q(\xi)\in S^{\bullet}(p,\xi)$ such that

$\|q(\xi)-p\|=\mathrm{dist}(p,S^{\bullet}(p,\xi)).$

Write
$\beta\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=r^{\bullet}(p,\xi)-r(p,\xi)$
and
$\gamma\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\|q(\xi)-p\|^{2}-\beta^{2}$.

  1. (2)

Suppose $(2r(p,\xi)+\beta)\gamma\leqslant\beta r(p,\xi)^{2}$. Then

$\kappa_{D}(p,\xi)\,\leqslant\,\frac{r^{\bullet}(p,\xi)}{r^{\bullet}(p,\xi)^{2}-\|q(\xi)-p\|^{2}}\|\xi\|.$
(1.6)

  2. (3)

Suppose $(2r(p,\xi)+\beta)\gamma>\beta r(p,\xi)^{2}$. Then

$\kappa_{D}(p,\xi)\,\leqslant\,\frac{1}{2r(p,\xi)}\boldsymbol{\cdot}\frac{\beta^{2}}{\|q(\xi)-p\|(\|q(\xi)-p\|-\sqrt{\gamma})}\|\xi\|.$
(1.7)

The upper bounds occurring above are strictly smaller than $\|\xi\|/r(p,\xi)$
unless $q(\xi)=p$.

The above result is a part of the effort to provide more informative bounds
for $\kappa_{\mathbb{D}}(p,\boldsymbol{\cdot})$. There are works describing
the contribution of lower order terms in $1/\mathrm{dist}(p,D^{\mathsf{c}})$
to asymptotic expressions and to bounds for $\kappa_{D}(p,\boldsymbol{\cdot})$
when $D$ is a bounded strongly pseudoconvex domain; see [4, 8, 3]. The
description of the lower order terms in $1/\mathrm{dist}(p,D^{\mathsf{c}})$ is
in terms of certain geometric invariants of the ($\mathcal{C}^{2}$-smooth)
manifold $\partial D$. While we merely study convex domains
$D\Subset\mathbb{C}^{n}$, we make absolutely no assumptions about $\partial
D$, whence the latter descriptions make no sense in general. Instead, we have
the estimates of Theorem 1.4.

The conclusions of Theorem 1.4 may appear technical. However, they are often
easy to work with when a specific domain is given. Furthermore, these
inequalities are sharp. Indeed, the last sentence of Theorem 1.4 suggests that
there are instances where the upper bound provided by Result 1.3 is not sharp
whereas that provided by Theorem 1.4 is. We provide a class of examples
illustrating all of these points in Section 2.

In Section 3 we present (there are no proofs in this section) the lemmas that
will be needed to prove Theorem 1.2. Section 4 contains the proof of,
essentially, the planar version of Theorem 1.4, from which a substantial part
of Theorem 1.4 is derived. Finally, Section 5 and Section 6 contain,
respectively, the proofs of Theorem 1.2 and Theorem 1.4.

##  2\. Examples

We first present the example alluded to in the paragraph following Theorem
1.2.

######  Example 2.1.

An example of a bounded convex domain $\Omega_{h}\Subset\mathbb{C}$ and a
holomorphic map $\phi:\Omega_{h}\to\mathbb{D}$ for which the lowest possible
$\alpha$ for which the bound (1.3), with $D_{1}=\Omega_{h}$ and
$D_{2}=\mathbb{D}$, holds true is large.

Consider, for an arbitrary (small) $h>0$ the bounded convex region

$\Omega_{h}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=D\big{(}i(1-h),1\big{)}\cap
D\big{(}-i(1-h),1\big{)}.$

Write $C_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\partial
D\big{(}i(1-h),1\big{)}$ and
$C_{2}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\partial
D\big{(}-i(1-h),1\big{)}$. Then $C_{1}$ and $C_{2}$ intersect at two points
$c$ and $-c$, where

$c\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sqrt{2h-h^{2}}.$

Observe that $\Omega_{h}$ has the shape of the cross-section of a lens, with
vertices $\pm c\in\mathbb{R}$. Let us construct a biholomorphism $\phi$ from
$\Omega_{h}$ to $\mathbb{D}$. Consider the following four functions:

$\displaystyle f_{1}(z)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{z-c}\quad\forall\,z\in\mathbb{C}\setminus\\{c\\},$
$\displaystyle f_{2}(z)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\left(z+\frac{1}{2c}\right)\quad\forall\,z\in\mathbb{C},$
$\displaystyle f_{3}(z)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=z^{\alpha}\quad\forall\,z\text{
such that }\mathsf{Re}\,z>0,$ $\displaystyle f_{4}(z)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{z-1}{z+1}\quad\forall\,z\in\mathbb{C}\setminus\\{-1\\},$

where $f_{3}$ is the holomorphic branch of the $\alpha$th power that maps
$\mathbb{R}_{+}$ onto $\mathbb{R}_{+}$,

$\alpha\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\pi}{2A}$

and

$A\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\arctan\left(\frac{\sqrt{2h-h^{2}}}{1-h}\right).$

Here, $2A$ is the magnitude of the smaller angle that the circles $C_{1}$ and
$C_{2}$ make with one another at both $c$ and $-c$, and also (by conformality)
the acute angle between the lines $L_{1}$ and $L_{2}$, which the circles
$C_{1}$ and $C_{2}$ get mapped to, respectively, under $f_{1}$, at their point
of intersection, $-\frac{1}{2c}$. The following composition makes sense on
$\Omega_{h}$

$\phi\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f_{4}\circ
f_{3}\circ f_{2}\circ f_{1}$

and it follows from standard facts about Möbius transformations that it is a
biholomorphism from $\Omega_{h}$ to $\mathbb{D}$. The explicit expression for
$\phi$ is given by

$\phi(z)=\frac{\left(\frac{z+c}{2c(c-z)}\right)^{\alpha}-1}{\left(\frac{z+c}{2c(c-z)}\right)^{\alpha}+1}.$

Now, for $t>0$ small, consider the point $c-t$ of $\Omega_{h}$. A simple
geometric argument shows that the distance $d(t)$ of this point from
$\Omega_{h}^{\mathsf{c}}$ is

$d(t)=t\left(\frac{2c-t}{1+\sqrt{t^{2}-2ct+1}}\right).$

Therefore $d(t)\approx t$ as $t\to 0$. Consider the set

$\Omega_{h,t}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{z\in\Omega_{h}\mid\mathrm{dist}(z,\Omega_{h}^{\mathsf{c}})\geqslant
d(t)\\}.$

Certainly $c-t$ is a point of this set; in fact, it is a boundary point of
this set. Therefore

$\phi(c-t)\in\phi\big{(}\Omega_{h,t}\big{)};$

it, too, is a boundary point. Also,

$\phi(c-t)=\frac{\left(\frac{2c-t}{2ct}\right)^{\alpha}-1}{\left(\frac{2c-t}{2ct}\right)^{\alpha}+1}.$

The distance of $\phi(c-t)$ from $\mathbb{D}^{\mathsf{c}}$ is

$1-\phi(c-t)=\frac{2(2ct)^{\alpha}}{(2c-t)^{\alpha}+(2ct)^{\alpha}}.$

Therefore, denoting by $d^{\prime}(t)$ the distance of $\phi(c-t)$ from
$\mathbb{D}^{\mathsf{c}}$, we see that

$d^{\prime}(t)\approx t^{\alpha}\;\text{as }t\to 0.$

Now
$\mathrm{dist}\big{(}\phi(\Omega_{h,t}),\mathbb{D}^{\mathsf{c}}\big{)}\leqslant
d^{\prime}(t)$ and so
$\mathrm{dist}\big{(}\phi(\Omega_{h,t}),\mathbb{D}^{\mathsf{c}}\big{)}=O(t^{\alpha})$
as $t\to 0$. Therefore, since
$\mathrm{dist}(\Omega_{h,t},\Omega_{h}^{\mathsf{c}})=d(t)\approx t$, if we let
$g$, defined for all sufficiently small positive numbers, be such that, for
all $t$ sufficiently small,

$\mathrm{dist}\big{(}\phi(\Omega_{h,t}),\mathbb{D}^{\mathsf{c}}\big{)}\geqslant
g(d(t)),$

then it follows that $g(t)=O(t^{\alpha})$. Clearly, then, for any lower bound
for $\mathrm{dist}\big{(}\phi(\Omega_{h,t}),\mathbb{D}^{\mathsf{c}}\big{)}$
that is (comparable to) a power of $d(t)$, no exponent smaller than $\alpha$
would suffice (for small $t$). Finally, $\alpha$ can be made arbitrarily large
by taking $h$ to be sufficiently small. $\blacktriangleleft$

######  Example 2.2.

A family of examples for which the bound provided by Theorem 1.4 is sharp
while the upper bound provided by Result 1.3 is _strictly_ greater.

Consider $D=\mathbb{B}^{n}$, the unit Euclidean ball with centre
$0\in\mathbb{C}^{n}$. Let us consider a point
$p\in\mathbb{B}^{n}\setminus\\{0\\}$. By the fact that unitary transformations
are holomorphic automorphisms of $\mathbb{B}^{n}$, it suffices to consider
$(p,\xi)$ of the form $\big{(}x,(\|\xi\|,0,\ldots,0)\big{)}$,
$x\in\mathbb{B}^{n}$. Let us write
$v\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(\|\xi\|,0,\ldots,0)$.
We leave it to the reader to verify that

$\displaystyle D(v)=D\big{(}-x_{1},\sqrt{1-\|x^{\prime}\|^{2}}\big{)},$
$\displaystyle\;\quad r^{\bullet}(x,v)=\sqrt{1-\|x^{\prime}\|^{2}},$
$\displaystyle S^{\bullet}(x,v)=\\{(0,x^{\prime})\\},$ $\displaystyle\;\quad
q(v)=(0,x^{\prime}),$

where we write $x=(x_{1},x^{\prime})$. It is easy to see that the expression
$(2r(x,v)+\beta)\gamma$ reduces to $0$, whence, by item (2) of Theorem 1.4, we
get the bound (note that $\|\xi\|=\|v\|$)

$\kappa_{\mathbb{B}^{n}}(x,v)\leqslant\frac{\sqrt{1-\|x^{\prime}\|^{2}}}{1-\|x\|^{2}}\|\xi\|.$
(2.1)

The exact expression for $\kappa_{\mathbb{B}^{n}}(p,\xi)$ — see [7, Section
3.5], for instance — is as follows:

$\kappa_{\mathbb{B}^{n}}(p,\xi)=\kappa_{\mathbb{B}^{n}}(x,v)=\left(\frac{\|v\|^{2}}{1-\|x\|^{2}}+\frac{|\langle
v,x\rangle|^{2}}{(1-\|x\|^{2})^{2}}\right)^{1/2}=\frac{\big{(}1-(\|x\|^{2}-|x_{1}|^{2})\big{)}^{1/2}}{1-\|x\|^{2}}\|\xi\|.$

We see that whenever $p\neq 0$ and $\xi\in\mathbb{C}{p}$, the unitary
transformation that maps $\xi$ to $(\|\xi\|,0,\ldots,0)$ (and $p$ to $x$)
gives $x=(c\|p\|,0,\dots,0)$, where $c$ is some complex number with $|c|=1$.
Since, for any such $(p,\xi)$, $x^{\prime}=0$, it follows from (2.1) that the
bound for $\kappa_{\mathbb{B}^{n}}(p,\xi)$ provided by Theorem 1.4 is exactly
equal to $\kappa_{\mathbb{B}^{n}}(p,\xi)$, whereas the upper bound provided by
Result 1.3 is $\|\xi\|/(1-\|p\|)$, which is strictly greater.
$\blacktriangleleft$

######  Example 2.3.

An example showing that the condition appearing in Item 3 of Theorem 1.4 holds
in simple situations.

Before presenting this example, we observe that the condition appearing in
Item 2 of Theorem 1.4 holds for the family of examples discussed in Example
2.2. Now, consider the domain

$\displaystyle D\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=$
$\displaystyle\Big{\\{}z\in\mathbb{C}\mid(|z|<1)\text{ or
}\Big{(}2^{-1}<\mathsf{Re}(z)<2\text{ and
}|\mathsf{Im}(z)|<\frac{2}{\sqrt{3}}\big{(}1-2^{-1}\mathsf{Re}(z)\big{)}\,\Big{)}\Big{\\}}.$

This is the unit disk together with all those $z\in\mathbb{C}$ such that
$\mathsf{Re}(z)>1/2$ and such that $z$ lies in the angle formed by the tangent
lines to the unit circle at the points $(1/2,\sqrt{3}/2)$ and
$(1/2,-\sqrt{3}/2)$. Consider the open subset $U$ of $D$ given by

$\left\\{z=x+iy\in\mathbb{C}\mid\sqrt{3}/2<x<1,\,|y|<5/7\sqrt{3}\,\right\\}\cap\mathbb{D}.$

We will show that, for every $z\in U$, the condition in Item 3 of Theorem 1.4
holds for the pair $(z,1)$ (and therefore, since we are in one dimension, for
any pair $(z,\xi)$, where $\xi\in\mathbb{C}\setminus\\{0\\}$). By the symmetry
of $D$ about the real axis, it suffices to show that the condition holds for
every $z\in U$ with $y\geqslant 0$. For such a $z$, $r(z,1)$ equals the
distance from $z$ to the tangent to the unit circle at $(1/2,\sqrt{3}/2)$,
which is $(2-x-\sqrt{3}y)/2$. Also, for such a $z$, $r^{\bullet}(z,1)=1$ and
$S^{\bullet}(z,1)=\\{0\\}$. So, necessarily, $q(1)=0$. The quantity of our
interest is

$(2r(z,1)+\beta)\gamma-\beta r(z,1)^{2},$

which, after substituting the expressions for $\beta$ and $\gamma$, is:

$|q(1)-z|^{2}\big{(}r^{\bullet}(z,1)+r(z,1)\big{)}-r^{\bullet}(z,1)^{2}\big{(}r^{\bullet}(z,1)-r(z,1)\big{)}.$

Substituting the actual values, we get:

$\displaystyle(x^{2}+y^{2})\left(1+\frac{2-x-\sqrt{3}y}{2}\right)$
$\displaystyle-\left(1-\frac{2-x-\sqrt{3}y}{2}\right)$
$\displaystyle>\frac{(3/4)(4-x-\sqrt{3}y)-x-\sqrt{3}y}{2}$
$\displaystyle[\text{since }x>\sqrt{3}/2]$
$\displaystyle=\frac{12-7x-7\sqrt{3}y}{8}>\frac{5-7\sqrt{3}y}{8}$
$\displaystyle[\text{since }x<1]$ $\displaystyle\qquad\qquad\qquad\qquad\
\;>0\quad$ $\displaystyle[\text{since }y<\tfrac{5}{7\sqrt{3}}].$

This shows that for every $z\in U$, the condition appearing in Item 3 of
Theorem 1.4 holds for the pair $(z,1)$. $\blacktriangleleft$

##  3\. Preliminary lemmas

In order to prove Theorem 1.2, one needs to efficiently estimate the Kobayashi
distance on $D_{1}$. One of the most basic estimates, which holds true on any
bounded domain $\Omega$, is that, given a compact _convex_ subset $K$ of
$\Omega$, the Kobayashi distance $\mathsf{k}_{\Omega}$ has the following upper
bound:

$\mathsf{k}_{\Omega}(z,w)\leqslant\frac{1}{\mathrm{dist}(K,\Omega^{\mathsf{c}})}\|z-w\|\quad\forall\,z,w\in
K.$ (3.1)

This is essentially the estimate that is used by Bernal-González (he uses the
Carathéodory distance, for which an analogue of (3.1) holds). We need a more
efficient upper bound. By the nature of these estimates, this is a challenge
only close to $\partial D_{1}$. Now (3.1) arises from a comparison between the
Kobayashi _metrics_ — resulting from the metric-decreasing property of
holomorphic maps — of $\Omega$ and of an appropriate Euclidean ball embedded
into $\Omega$ by the inclusion map. This comparison yields the following
inequality:

$\kappa_{\Omega}(p,\xi)\leqslant\frac{\|\xi\|}{\mathrm{dist}(p,\Omega^{\mathsf{c}})},\quad
p\in\Omega,\ \xi\in T^{(1,0)}_{p}(\Omega).$ (3.2)

The above suggests that a more efficient estimate for $\mathsf{k}_{D_{1}}$
could, in principle, be obtained by a comparison between the Kobayashi metrics
of $D_{1}$ and of (the embedded image of) some class of planar regions that
are better adapted to the shape of $\partial D_{1}$. This leads us to appeal
to an idea described and used by Mercer [9]. We consider the class of regions
in $\mathbb{C}$ defined as follows: for every $\alpha>1$, let
$\Lambda_{\alpha}$ denote the image of $\mathbb{D}$ under the holomorphic
mapping

$f\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=z\mapsto(z+1)^{\frac{1}{\alpha}}:\\{w\in\mathbb{C}\mid\mathsf{Re}\,w>-1\\}\to\mathbb{C}.$

For $\alpha=2$, $\Lambda_{\alpha}$ is the interior of one loop of the
lemniscate. The following two results are proved in [9, pp. 203–204]:

######  Lemma 3.1 (Mercer, [9, Lemma 2.1]).

Let $z_{0}\in\Lambda_{\alpha}$. Then there exists a $C>0$ such that, for all
$z\in\Lambda_{\alpha}$,

$\mathsf{k}_{\Lambda_{\alpha}}(z_{0},z)\leqslant
C+\frac{\alpha}{2}\log\left(\frac{1}{\mathrm{dist}(z,\Lambda_{\alpha}^{\mathsf{c}})}\right).$

The above lemma; a result on how the domains $\Lambda_{\alpha}$ relate to a
given convex, planar domain; and a comparison between the Kobayashi distances
— based on the fact that holomorphic mappings are contractive relative to the
Kobayashi distance — of $\Omega$ (as below) and of a suitable affine embedding
of $\Lambda_{\alpha}$ into $\Omega$ yield the result that we need:

######  Lemma 3.2 (Mercer, [9, Proposition 2.3]).

Let $\Omega\subseteq\mathbb{C}^{n}$ be a bounded convex domain, and let
$z_{0}\in\Omega$. Then there are constants $\alpha>1$ and $C(z_{0})>0$ such
that, for every $z\in\Omega$,

$\mathsf{k}_{\Omega}(z_{0},z)\leqslant
C(z_{0})+\frac{\alpha}{2}\log\left(\frac{1}{\mathrm{dist}(z,\Omega^{\mathsf{c}})}\right).$

The bound in the above lemma can be tighter if $\Omega$, in addition to the
properties stated in Lemma 3.2, has $\mathcal{C}^{2}$-smooth boundary. In that
case, one can carry out the procedure hinted at prior to Lemma 3.2 with
$\Lambda_{\alpha}$ replaced by the unit disk $\mathbb{D}$. This argument is
very classical (and of which Mercer’s argument leading to Lemma 3.2 is an
adaptation) and widely known. Its first step is the analogue of Lemma 3.1 for
$\mathbb{D}$, which is just a direct calculation: fixing a
$z_{0}\in\mathbb{D}$, there exists a $C>0$ such that

$\mathsf{k}_{\mathbb{D}}(z_{0},z)\leqslant
C+\frac{1}{2}\log\left(\frac{1}{\mathrm{dist}(z,\mathbb{D}^{\mathsf{c}})}\right)$

for all $z\in\mathbb{D}$. This leads to the classical result:

######  Lemma 3.3.

Let $\Omega\subseteq\mathbb{C}^{n}$ be a bounded convex domain whose boundary
is $\mathcal{C}^{2}$-smooth. Let $z_{0}\in\Omega$. Then there is a constant
$C(z_{0})>0$ such that, for every $z\in\Omega$,

$\mathsf{k}_{\Omega}(z_{0},z)\leqslant
C(z_{0})+\frac{1}{2}\log\left(\frac{1}{\mathrm{dist}(z,\Omega^{\mathsf{c}})}\right).$

Finally, we have the following lemma that helps us to effectively estimate the
Kobayashi distance _from below_ on arbitrary convex domains.

######  Lemma 3.4.

Let $\Omega\varsubsetneq\mathbb{C}^{n}$ be a convex domain. Then

$\mathsf{k}_{\Omega}(z,w)\geqslant\frac{1}{2}\log\left(\frac{\mathrm{dist}(w,\Omega^{\mathsf{c}})}{\mathrm{dist}(z,\Omega^{\mathsf{c}})}\right)\quad\forall\,z,w\in\Omega.$

###### Proof.

Fix $z,w\in\Omega$. Choose $q\in\partial\Omega$ such that
$\mathrm{dist}(z,\Omega^{\mathsf{c}})=\|z-q\|$. By the convexity of $\Omega$,
we may choose a $\mathbb{C}$-linear functional $F:\mathbb{C}^{n}\to\mathbb{C}$
such that

$\Omega\subseteq\\{x\in\mathbb{C}^{n}\mid\mathsf{Im}(F(x-q))>0\\},$

i.e., such that

$H\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{x\in\mathbb{C}^{n}\mid\mathsf{Im}(F(x-q))=0\\}$

is a supporting hyperplane for $\Omega$ at $q$. In fact, we can choose $F$
such that, for every $x\in\mathbb{C}^{n}$,
$|\mathsf{Im}(F(x-q))|=\mathrm{dist}(x,H)$. Consider the $\mathbb{C}$-affine
function $T$ on $\mathbb{C}^{n}$ given by

$T(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=F(x-q)\quad\forall\,x\in\mathbb{C}^{n}.$

Then, $T$ maps $\Omega$ holomorphically into the upper half plane
$\mathbb{H}$. By the Kobayashi-distance-decreasing property of $T$ and the
formula for the Kobayashi distance in $\mathbb{H}$,

$\mathsf{k}_{\Omega}(z,w)\geqslant\mathsf{k}_{\mathbb{H}}(T(z),T(w))\geqslant\frac{1}{2}\log\left(\frac{\mathsf{Im}(T(w))}{\mathsf{Im}(T(z))}\right)=\frac{1}{2}\log\left(\frac{\mathsf{Im}(F(w-q))}{\mathsf{Im}(F(z-q))}\right).$

Recall that
$\mathsf{Im}(F(z-q))=\mathrm{dist}(z,H)=\mathrm{dist}(z,\Omega^{\mathsf{c}})$.
Furthermore,
$\mathsf{Im}(F(w-q))=\mathrm{dist}(w,H)\geqslant\mathrm{dist}(w,\Omega^{\mathsf{c}})$.
Therefore, the sequence of inequalities above gives

$\mathsf{k}_{\Omega}(z,w)\geqslant\frac{1}{2}\log\left(\frac{\mathrm{dist}(w,\Omega^{\mathsf{c}})}{\mathrm{dist}(z,\Omega^{\mathsf{c}})}\right).$

Since the points $z,w\in\Omega$ were arbitrarily chosen, we have the
conclusion desired. ∎

##  4\. Lemmas concerning planar convex domains

In this section we will state and prove a number of lemmas about planar convex
domains, which will be used to prove our second result. We abbreviate
$\mathrm{dist}(x,\Omega^{\mathsf{c}})$ to $\delta_{\Omega}(x)$ in this
section.

######  Lemma 4.1.

Let $\Omega\subseteq\mathbb{C}$ be an open convex set and let
$p,\zeta\in\Omega$. Let $R(p),R(\zeta)>0$ be such that
$D(p,R(p))\subseteq\Omega$ and $D(\zeta,R(\zeta))\subseteq\Omega$. Then, for
every $t\in[0,1]$, $\Omega$ includes the disk in $\mathbb{C}$ with centre
$(1-t)p+t\zeta$ and radius $(1-t)R(p)+tR(\zeta)$.

We omit the proof because it is straightforward. The main idea behind the
proof is to show that the disk described in the above lemma is contained in
the convex hull of the union of the disks $D(p,R(p))$ and $D(\zeta,R(\zeta))$.

######  Lemma 4.2.

Let $\Omega\subseteq\mathbb{C}$ be an open convex set and let $p\in\Omega$.
Suppose $\zeta\in\Omega$ is such that

$p\in\overline{D\big{(}\zeta,\delta_{\Omega}(\zeta)\big{)}}.$ (4.1)

Then, for every $t\in[0,1)$,

$p\in
D\big{(}(1-t)p+t\zeta,(1-t)\delta_{\Omega}(p)+t\delta_{\Omega}(\zeta)\big{)}\subseteq\Omega.$

Now suppose $\delta_{\Omega}(\zeta)>\delta_{\Omega}(p)$. Let
$r_{\zeta}(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(1-t)\delta_{\Omega}(p)+t\delta_{\Omega}(\zeta)\;\forall\,t\in[0,1]$,
let $\alpha\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=|\zeta-p|$
and let
$\beta\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\delta_{\Omega}(\zeta)-\delta_{\Omega}(p)$.
If we consider the mapping

$\phi_{\zeta}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=t\mapsto\frac{r_{\zeta}(t)}{r_{\zeta}(t)^{2}-t^{2}|\zeta-p|^{2}}:[0,1)\to\mathbb{R},$

then

  1. (1)

$\phi_{\zeta}$ is differentiable;

  2. (2)

If
$(\alpha^{2}-\beta^{2})(2\delta_{\Omega}(p)+\beta)\leqslant\beta\delta_{\Omega}(p)^{2}$
then $\phi_{\zeta}$ is continuously extendable to $[0,1]$ and the minimum
value of $\phi_{\zeta}$ is

$\frac{\delta_{\Omega}(\zeta)}{\delta_{\Omega}(\zeta)^{2}-|\zeta-p|^{2}};$

  3. (3)

If
$(\alpha^{2}-\beta^{2})(2\delta_{\Omega}(p)+\beta)>\beta\delta_{\Omega}(p)^{2}$
then $\phi_{\zeta}$ attains its minimum value at

$t(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\delta_{\Omega}(p)\frac{-(\alpha^{2}-\beta^{2})+\alpha\sqrt{\alpha^{2}-\beta^{2}}}{(\alpha^{2}-\beta^{2})\beta}\in(0,1)$

and its minimum value is

$\phi_{\zeta}\big{(}t(\zeta)\big{)}=\frac{1}{2\delta_{\Omega}(p)}\boldsymbol{\cdot}\frac{\beta^{2}}{\alpha\big{(}\alpha-\sqrt{\alpha^{2}-\beta^{2}}\big{)}}.$

  4. (4)

Finally, the minimum value of $\phi_{\zeta}$ is less than
$\frac{1}{\delta_{\Omega}(p)}$.

###### Proof.

In order to prove that

$p\in
D\big{(}(1-t)p+t\zeta,(1-t)\delta_{\Omega}(p)+t\delta_{\Omega}(\zeta)\big{)}\subseteq\Omega,$

we first have to prove that
$|p-\big{(}(1-t)p+t\zeta\big{)}|=t|\zeta-p|<(1-t)\delta_{\Omega}(p)+t\delta_{\Omega}(\zeta)$.
But by the condition (4.1),

$t|\zeta-p|\leqslant
t\delta_{\Omega}(\zeta)<(1-t)\delta_{\Omega}(p)+t\delta_{\Omega}(\zeta)\quad\forall\,t<1.$
(4.2)

The inclusion statement follows from Lemma 4.1.

Turning to $\phi_{\zeta}$, it is clear from (4.2) that it is well-defined and
differentiable on $[0,1)$.

Suppose first that
$(\alpha^{2}-\beta^{2})(2\delta_{\Omega}(p)+\beta)\leqslant\beta\delta_{\Omega}(p)^{2}$.
Then note that necessarily $|\zeta-p|<\delta_{\Omega}(\zeta)$. To see this,
suppose $|\zeta-p|=\delta_{\Omega}(\zeta)$. Then

$\displaystyle(\alpha^{2}-\beta^{2})(2\delta_{\Omega}(p)+\beta)$
$\displaystyle=\big{(}\delta_{\Omega}(\zeta)^{2}-(\delta_{\Omega}(\zeta)-\delta_{\Omega}(p))^{2}\big{)}(\delta_{\Omega}(\zeta)+\delta_{\Omega}(p))$
$\displaystyle=2\delta_{\Omega}(\zeta)^{2}\delta_{\Omega}(p)+\delta_{\Omega}(\zeta)\delta_{\Omega}(p)^{2}-\delta_{\Omega}(p)^{3}$
$\displaystyle>\beta\delta_{\Omega}(p)^{2},$

which is a contradiction. So $|\zeta-p|<\delta_{\Omega}(\zeta)$ and this shows
that the expression for $\phi_{\zeta}(t)$ makes sense for $t\in[0,1]$.
Moreover, $\phi_{\zeta}$ is differentiable on $[0,1]$ in this case. A
calculation shows that

$\phi_{\zeta}^{\prime}(t)=\frac{-\beta\delta_{\Omega}(p)^{2}+(\alpha^{2}-\beta^{2})(2\delta_{\Omega}(p)t+\beta
t^{2})}{\big{(}r_{\zeta}(t)^{2}-t^{2}\alpha^{2}\big{)}^{2}}.$

If $\alpha\leqslant\beta$ then clearly $\phi_{\zeta}^{\prime}(t)<0$ for all
$t\in[0,1]$. And if $\alpha>\beta$, then, for all $t\in[0,1)$,

$(\alpha^{2}-\beta^{2})(2\delta_{\Omega}(p)t+\beta
t^{2})<(\alpha^{2}-\beta^{2})(2\delta_{\Omega}(p)+\beta)\leqslant\beta\delta_{\Omega}(p)^{2},$

whence

$\phi_{\zeta}^{\prime}(t)<0\quad\forall\,t\in[0,1).$ (4.3)

So, in the case under consideration, one invariably has that $\phi_{\zeta}$
attains its minimum value at $1$, and the minimum value is

$\frac{\delta_{\Omega}(\zeta)}{\delta_{\Omega}(\zeta)^{2}-|\zeta-p|^{2}}.$

Furthermore, by (4.3), the minimum value above is less than
$\frac{1}{\delta_{\Omega}(p)}$.

Suppose now that
$(\alpha^{2}-\beta^{2})(2\delta_{\Omega}(p)+\beta)>\beta\delta_{\Omega}(p)^{2}$.
Then one necessarily has $\alpha>\beta$. Furthermore, one expects a critical
point of $\phi_{\zeta}$ in $(0,1)$. In this case, by calculating the critical
points of $\phi_{\zeta}$, we obtain that $\phi_{\zeta}$ attains its minimum
value at

$t(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\delta_{\Omega}(p)\frac{-(\alpha^{2}-\beta^{2})+\alpha\sqrt{\alpha^{2}-\beta^{2}}}{(\alpha^{2}-\beta^{2})\beta},$

which is a point of $(0,1)$, and that the minimum value of $\phi_{\zeta}$ is

$\phi_{\zeta}\big{(}t(\zeta)\big{)}=\frac{1}{2\delta_{\Omega}(p)}\boldsymbol{\cdot}\frac{\beta^{2}}{\alpha^{2}-\alpha\sqrt{\alpha^{2}-\beta^{2}}}.$

The following calculation shows that the minimum value above is _less than_
$\frac{1}{\delta_{\Omega}(p)}$:

$\displaystyle\frac{\beta^{2}}{\alpha^{2}-\alpha\sqrt{\alpha^{2}-\beta^{2}}}$
$\displaystyle<2$ $\displaystyle\iff$ $\displaystyle
2\alpha\sqrt{\alpha^{2}-\beta^{2}}$ $\displaystyle<(2\alpha^{2}-\beta^{2})$
$\displaystyle\iff$ $\displaystyle 4\alpha^{2}(\alpha^{2}-\beta^{2})$
$\displaystyle<(2\alpha^{2}-\beta^{2})^{2}$ (in this case
$\alpha^{2}-\beta^{2}>0$) $\displaystyle\iff$ $\displaystyle 0$
$\displaystyle<\beta^{4}.$

Since the last inequality is true, by taking into account what we obtained in
the other case, we get (4). ∎

Before we state our last lemma we need to make two definitions. For $\Omega$
an open _bounded_ convex subset of $\mathbb{C}$ and for $p\in\Omega$, we let

$r^{\bullet}_{\Omega}(p)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup\big{(}\\{\,r>0\mid\exists\zeta\in\Omega\text{
such that }p\in D(\zeta,r)\subseteq\Omega\,\\}\big{)}$ (4.4)

and we let

$S^{\bullet}_{\Omega}(p)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{\,\zeta\in\Omega\mid
p\in\overline{D(\zeta,r^{\bullet}_{\Omega}(p))}\text{ and
}D(\zeta,r^{\bullet}_{\Omega}(p))\subseteq\Omega\\}.$ (4.5)

######  Proposition 4.3.

Let $\Omega\subseteq\mathbb{C}$ be an open bounded convex set and let
$p\in\Omega$. For any $\zeta\in\Omega$ such that
$\delta_{\Omega}(\zeta)>\delta_{\Omega}(p)$, let $\phi_{\zeta}$ denote the
same function as in Lemma 4.2. Then

  1. (1)

$S^{\bullet}_{\Omega}(p)$ is a non-empty compact convex subset of $\Omega$.

  2. (2)

There exists a unique point $\zeta\in S^{\bullet}_{\Omega}(p)$ such that

$|\zeta-p|=\mathrm{dist}(p,S^{\bullet}_{\Omega}(p)).$ (4.6)

In the next two statements, $\zeta$ is the point in $S^{\bullet}_{\Omega}(p)$
introduced in (2).

  1. (3)

Suppose
$\big{(}|\zeta-p|^{2}-(r^{\bullet}_{\Omega}(p)-\delta_{\Omega}(p))^{2}\big{)}\big{(}r^{\bullet}_{\Omega}(p)+\delta_{\Omega}(p)\big{)}\leqslant(r^{\bullet}_{\Omega}(p)-\delta_{\Omega}(p))\delta_{\Omega}(p)^{2}$.
Then

$\kappa_{\Omega}(p,1)\leqslant\frac{r^{\bullet}_{\Omega}(p)}{r^{\bullet}_{\Omega}(p)^{2}-|\zeta-p|^{2}}\leqslant\frac{1}{\delta_{\Omega}(p)},$
(4.7)

where the latter is an equality if and only if $\zeta=p$.

  2. (4)

Suppose
$\big{(}|\zeta-p|^{2}-(r^{\bullet}_{\Omega}(p)-\delta_{\Omega}(p))^{2}\big{)}\big{(}r^{\bullet}_{\Omega}(p)+\delta_{\Omega}(p)\big{)}>(r^{\bullet}_{\Omega}(p)-\delta_{\Omega}(p))\delta_{\Omega}(p)^{2}$.
Then, with $\alpha$ and $\beta$ denoting the same quantities as in Lemma 4.2,

$\kappa_{\Omega}(p,1)\leqslant\frac{1}{2\delta_{\Omega}(p)}\boldsymbol{\cdot}\frac{\beta^{2}}{\alpha\big{(}\alpha-\sqrt{\alpha^{2}-\beta^{2}}\big{)}}<\frac{1}{\delta_{\Omega}(p)}.$
(4.8)

###### Proof.

First we prove that $S^{\bullet}_{\Omega}(p)$ is non-empty. Choose an
increasing sequence $(r_{\nu})_{\nu\geqslant 1}$ from the set occurring in
(4.4) such that $r_{\nu}>r^{\bullet}_{\Omega}(p)-\tfrac{1}{\nu}$. For each
$\nu$, there is a $\zeta_{\nu}\in\Omega$ such that

$p\in D(\zeta_{\nu},r_{\nu})\subseteq\Omega.$

By the boundedness of $\Omega$, there exists a $w\in\overline{\Omega}$ such
that (without loss of generality) $\zeta_{\nu}\to w$. Since, for each $\nu$,
$|p-\zeta_{\nu}|<r_{\nu}$, therefore, by taking the limit, $|p-w|\leqslant
r^{\bullet}_{\Omega}(p)$, i.e., $p\in\overline{D(w,r^{\bullet}_{\Omega}(p))}$.
Now suppose $x\in D(w,r^{\bullet}_{\Omega}(p))$. Let
$\epsilon\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(r^{\bullet}_{\Omega}(p)-|x-w|)/2$.
Choose $\nu^{\prime}\in\mathbb{Z}_{+}$ such that
$r^{\bullet}_{\Omega}(p)-r_{\nu^{\prime}}<\epsilon$. Then choose
$\nu\geqslant\nu^{\prime}$ such that $|w-\zeta_{\nu}|<\epsilon$. Then,

$|x-\zeta_{\nu}|\leqslant|x-w|+|w-\zeta_{\nu}|<|x-w|+\epsilon=r^{\bullet}_{\Omega}(p)-\epsilon<r_{\nu^{\prime}}\leqslant
r_{\nu}.$

So, $x\in D(\zeta_{\nu},r_{\nu})$, whence $x\in\Omega$. This shows that
$D(w,r^{\bullet}_{\Omega}(p))\subseteq\Omega$. So $w\in
S^{\bullet}_{\Omega}(p)$, whence $S^{\bullet}_{\Omega}(p)$ is non-empty. Hence
it makes sense to talk of points of $S^{\bullet}_{\Omega}(p)$ at least
distance from $p$. Also, note that if $w\in S^{\bullet}_{\Omega}(p)$ then
$\delta_{\Omega}(w)=r^{\bullet}_{\Omega}(p)$ (since
$\delta_{\Omega}(w)\geqslant r^{\bullet}_{\Omega}(p)$, and if strict
inequality held then the maximality of $r^{\bullet}_{\Omega}(p)$ would be
contradicted).

Now we prove the compactness of $S^{\bullet}_{\Omega}(p)$. Since
$S^{\bullet}_{\Omega}(p)$ is a bounded subset of $\mathbb{C}$ (it is included
in $\Omega$), it suffices to prove that it is a closed subset of $\mathbb{C}$.
So suppose $(\zeta_{\nu})_{\nu\geqslant 1}$ is a sequence of points of
$S^{\bullet}_{\Omega}(p)$ converging to $w\in\mathbb{C}$. That
$p\in\overline{D(w,r^{\bullet}_{\Omega}(p))}$ is obvious. Now we show that
$D(w,r^{\bullet}_{\Omega}(p))\subseteq\Omega$. Suppose $x\in
D(w,r^{\bullet}_{\Omega}(p))$. Let
$\epsilon\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=r^{\bullet}_{\Omega}(p)-|x-w|$.
Choose $\nu\in\mathbb{Z}_{+}$ such that $|w-\zeta_{\nu}|<\epsilon$. Then

$|x-\zeta_{\nu}|\leqslant|x-w|+|w-\zeta_{\nu}|<r^{\bullet}_{\Omega}(p).$

So $x\in D(\zeta_{\nu},r^{\bullet}_{\Omega}(p))$, whence $x\in\Omega$. As this
is true for any $x\in D(w,r^{\bullet}_{\Omega}(p))$, the latter is a subset of
$\Omega$. So $w\in S^{\bullet}_{\Omega}(p)$ and this argument shows that
$S^{\bullet}_{\Omega}(p)$ is a compact subset of $\Omega$.

Finally we prove that $S^{\bullet}_{\Omega}(p)$ is convex. To this end,
suppose $\zeta_{1},\zeta_{2}\in S^{\bullet}_{\Omega}(p)$ and that $t\in[0,1]$.
We want to prove that $(1-t)\zeta_{1}+t\zeta_{2}\in S^{\bullet}_{\Omega}(p)$.
In order to do this we have to prove that
$D\big{(}(1-t)\zeta_{1}+t\zeta_{2},r^{\bullet}_{\Omega}(p)\big{)}\subseteq\Omega$
and that
$p\in\overline{D\big{(}(1-t)\zeta_{1}+t\zeta_{2},r^{\bullet}_{\Omega}(p)\big{)}}$.
The first inclusion follows from Lemma 4.1. As for the second containment,

$\displaystyle|p-((1-t)\zeta_{1}+t\zeta_{2})|$
$\displaystyle=|((1-t)p+tp)-((1-t)\zeta_{1}+t\zeta_{2})|$
$\displaystyle\leqslant(1-t)|p-\zeta_{1}|+t|p-\zeta_{2}|$
$\displaystyle\leqslant(1-t)r^{\bullet}_{\Omega}(p)+tr^{\bullet}_{\Omega}(p)$
$\displaystyle=r^{\bullet}_{\Omega}(p).$

This shows that $(1-t)\zeta_{1}+t\zeta_{2}\in S^{\bullet}_{\Omega}(p)$ and
this in turn proves that $S^{\bullet}_{\Omega}(p)$ is convex. This completes
the proof of (1).

We equip $\mathbb{C}$ with the standard Hilbert space structure from which the
Euclidean norm arises. Note that the expressions in (4.6) are derived from the
Euclidean norm. Since $S^{\bullet}_{\Omega}(p)$ is closed and convex, it
follows from a theorem in the elementary theory of Hilbert spaces (see [10,
Theorem 4.10], for instance) that there is a unique $\zeta\in
S^{\bullet}_{\Omega}(p)$ such that (4.6) holds.

Now suppose that the condition in (3) holds. We divide the discussion into two
further sub-cases:

Sub-case (a) $r^{\bullet}_{\Omega}(p)=\delta_{\Omega}(p)$.

In this case, $p\in S^{\bullet}_{\Omega}(p)$ and so $\zeta$ must be $p$.
Consequently, in this case,

$\kappa_{\Omega}(p,1)\leqslant\frac{1}{\delta_{\Omega}(p)}=\frac{r^{\bullet}_{\Omega}(p)}{r^{\bullet}_{\Omega}(p)^{2}-|\zeta-p|^{2}},$

where we have used the estimate (3.2) to write the first inequality.

Sub-case (b) $r^{\bullet}_{\Omega}(p)>\delta_{\Omega}(p)$.

In this case, we note that, since
$\delta_{\Omega}(\zeta)=r^{\bullet}_{\Omega}(p)$ and therefore
$\delta_{\Omega}(\zeta)>\delta_{\Omega}(p)$, we can appeal to Lemma 4.2. By
that lemma we have, for an arbitrary $t\in[0,1)$, $p\in
D\big{(}(1-t)p+t\zeta,r_{\zeta}(t)\big{)}$. Now we estimate
$\kappa_{\Omega}(p,1)$. For $t\in[0,1)$ arbitrary, consider the holomorphic
function

$f_{t}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=z\mapsto(1-t)p+t\zeta+r_{\zeta}(t)z:\mathbb{D}\to\Omega.$

That $f_{t}(\mathbb{D})\subseteq\Omega$ follows from Lemma 4.1 with
$R(p)=\delta_{\Omega}(p)$ and $R(\zeta)=\delta_{\Omega}(\zeta)$. For every
$z\in\mathbb{D}$, $f^{\prime}_{t}(z)=r_{\zeta}(t)$. Write
$z_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=t(p-\zeta)/r_{\zeta}(t)\in\mathbb{D}$.
Then, by the metric-decreasing property of holomorphic mappings,

$\kappa_{\Omega}(p,1)=\kappa_{\Omega}\Big{(}f_{t}(z_{0}),f^{\prime}_{t}(z_{0})\frac{1}{r_{\zeta}(t)}\Big{)}\leqslant\kappa_{\mathbb{D}}\Big{(}z_{0},\frac{1}{r_{\zeta}(t)}\Big{)}=\frac{r_{\zeta}(t)}{r_{\zeta}(t)^{2}-t^{2}|\zeta-p|^{2}}=\phi_{\zeta}(t).$

Minimizing the right-hand side of the inequality above with respect to $t$
tells us that the minimum of the function $\phi_{\zeta}$ is an upper bound for
$\kappa_{\Omega}(p,1)$. Now we determine the minimum of $\phi_{\zeta}$. The
condition satisfied by $\zeta$ is simply a restatement of the condition
occurring in (2) of Lemma 4.2. So by that lemma, the minimum value of
$\phi_{\zeta}$ is

$\frac{r^{\bullet}_{\Omega}(p)}{r^{\bullet}_{\Omega}(p)^{2}-|\zeta-p|^{2}}<\frac{1}{\delta_{\Omega}(p)}$

and hence

$\kappa_{\Omega}(p,1)\leqslant\min_{t\in[0,1]}\phi_{\zeta}(t)=\frac{r^{\bullet}_{\Omega}(p)}{r^{\bullet}_{\Omega}(p)^{2}-|\zeta-p|^{2}}<\frac{1}{\delta_{\Omega}(p)}.$

Hence, the required inequalities hold in either sub-case.

Obviously, if $\zeta=p$ then the second inequality is an equality (because
$r^{\bullet}_{\Omega}(p)=\delta_{\Omega}(p)$). Suppose, conversely, that the
second inequality is an equality, and suppose, to get a contradiction, that
$\zeta\neq p$. Then it must be that
$r^{\bullet}_{\Omega}(p)>\delta_{\Omega}(p)$. Because, if not, then
$r^{\bullet}_{\Omega}(p)=\delta_{\Omega}(p)$, whence we have $\zeta=p$, as
argued in sub-case (a). This is a contradiction. So
$\delta_{\Omega}(\zeta)=r^{\bullet}_{\Omega}(p)>\delta_{\Omega}(p)$ and
therefore we can consider $\phi_{\zeta}$ and appeal to Lemma 4.2 to get that
the minimum value of $\phi_{\zeta}$ is

$\frac{r^{\bullet}_{\Omega}(p)}{r^{\bullet}_{\Omega}(p)^{2}-|\zeta-p|^{2}}<\frac{1}{\delta_{\Omega}(p)}.$

But that is a contradiction to the hypothesis, and this completes the proof of
(3).

Now suppose that the condition in (4) holds. In this case
$r^{\bullet}_{\Omega}(p)>\delta_{\Omega}(p)$. The reasoning is similar to what
occurs above. Therefore we can again consider $\phi_{\zeta}$, appeal to Lemma
4.2, and, reasoning as in the previous case, get that the minimum of
$\phi_{\zeta}$ is an upper bound for $\kappa_{\Omega}(p,1)$. But in this case,
since the condition satisfied by $\zeta$ is a restatement of the condition
occurring in (3) of Lemma 4.2, the minimum in question is

$\frac{1}{2\delta_{\Omega}(p)}\boldsymbol{\cdot}\frac{\beta^{2}}{\alpha\big{(}\alpha-\sqrt{\alpha^{2}-\beta^{2}}\big{)}}<\frac{1}{\delta_{\Omega}(p)}.$

This completes the proof of the proposition. ∎

##  5\. Proof of Theorem 1.2

Before we proceed with our proof, we point out that its basic idea is inspired
by the proof of Bernal-González [1], but with one significant departure. This
departure from the proof in [1] is the use of a refined estimate for
$\mathsf{k}_{D_{1}}$ as discussed in Section 3.

###### Proof.

Let

$D_{1}(r)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{z\in
D_{1}\mid\mathrm{dist}(z,D_{1}^{\mathsf{c}})\geqslant r\\}$

for every $r>0$ sufficiently small. Note that if $D_{2}=\mathbb{C}^{m}$, then
the conclusion of the theorem is trivially true. Therefore, we suppose that
$D_{2}\varsubsetneq\mathbb{C}^{m}$. Then, by Lemma 3.4,

$\mathsf{k}_{D_{2}}(b,w)\geqslant\frac{1}{2}\log\left(\frac{\mathrm{dist}(b,D^{\mathsf{c}}_{2})}{\mathrm{dist}(w,D^{\mathsf{c}}_{2})}\right)\quad\forall\,w\in
D_{2}.$

Let $\phi$ be as in the statement of Theorem 1.2. Then, for every $z\in
D_{1}(r)$,

$\mathsf{k}_{D_{2}}(b,\phi(z))\geqslant\frac{1}{2}\log\left(\frac{\mathrm{dist}(b,D^{\mathsf{c}}_{2})}{\mathrm{dist}(\phi(z),D^{\mathsf{c}}_{2})}\right).$
(5.1)

Let us now suppose that $\partial D_{1}$ has _lower than_ $\mathcal{C}^{2}$
regularity. In that case we have

$\mathsf{k}_{D_{2}}(b,\phi(z))\leqslant\mathsf{k}_{D_{1}}(a,z)\leqslant
C(a)+\frac{\alpha}{2}\log\left(\frac{1}{\mathrm{dist}(z,D^{\mathsf{c}}_{1})}\right).$
(5.2)

The second inequality follows from Lemma 3.2. Therefore, by (5.1), the above
inequality, and the fact that $z\in D_{1}(r)$,

$\frac{1}{2}\log\left(\frac{\mathrm{dist}(b,D^{\mathsf{c}}_{2})}{\mathrm{dist}(\phi(z),D^{\mathsf{c}}_{2})}\right)\leqslant
C(a)+\frac{\alpha}{2}\log\left(\frac{1}{r}\right).$

After exponentiating and a couple of computations, we get

$\mathrm{dist}(\phi(z),D^{\mathsf{c}}_{2})\geqslant
C\mathrm{dist}(b,D^{\mathsf{c}}_{2})r^{\alpha},$ (5.3)

where $C=e^{-2C(a)}$. Since $z\in D_{1}(r)$ was arbitrary, the above
inequality completes the proof under the assumption that $\partial D_{1}$ has
lower than $\mathcal{C}^{2}$ regularity. In this case, $\alpha$, as obtained
by our argument, is greater than $1$.

If $\partial D_{1}$ is $\mathcal{C}^{2}$-smooth, then, by Lemma 3.3, we may
take $\alpha=1$ in (5.2). Every subsequent step of the argument goes through,
and we arrive at the conclusion of Theorem 1.2 with $\alpha=1$. This completes
the proof. ∎

##  6\. Proof of Theorem 1.4

###### Proof.

For convenience we first define

$g\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=z\mapsto
p+z\frac{\xi}{\|\xi\|}:D(\xi)\to D.$ (6.1)

We note that

$S^{\bullet}(p,\xi)=\Big{\\{}p+z\frac{\xi}{\|\xi\|}\mid z\in
S(p,\xi)\Big{\\}},$ (6.2)

where

$S(p,\xi)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{z\in
D(\xi)\mid 0\in\overline{D(z,r^{\bullet}(p,\xi))}\text{ and
}D(z,r^{\bullet}(p,\xi))\subseteq D(\xi)\\}.$

The definition of $r^{\bullet}(p,\xi)$ tells us that, in the language of
Proposition 4.3, $r^{\bullet}(p,\xi)=r^{\bullet}_{D(\xi)}(0)$. We see from the
above that $S(p,\xi)$ is nothing but $S^{\bullet}_{D(\xi)}(0)$. Therefore, by
Proposition 4.3, $S(p,\xi)$ is a non-empty compact convex subset of $D(\xi)$
and there is a unique point $z_{0}$ of $S(p,\xi)$ such that

$|z_{0}|=\mathrm{dist}(0,S(p,\xi)).$ (6.3)

Then (6.2) implies that $S^{\bullet}(p,\xi)$ is also a non-empty compact
convex subset of $D\cap(p+\mathbb{C}\,\xi)$. Let
$q(\xi)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=p+z_{0}\tfrac{\xi}{\|\xi\|}$.
Then of course $q(\xi)\in S^{\bullet}(p,\xi)$ and

$\|q(\xi)-p\|=|z_{0}|=\mathrm{dist}(0,S(p,\xi))=\mathrm{dist}(p,S^{\bullet}(p,\xi)).$

The last equality holds because $g$ preserves Euclidean distances. The
uniqueness of $q(\xi)$ is also clear from the corresponding uniqueness of
$z_{0}$. We note that

$\|q(\xi)-p\|=|z_{0}|=\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}\alpha,$

that

$r\Big{(}p+z\frac{\xi}{\|\xi\|},\xi\Big{)}=\delta_{D(\xi)}(z)\quad\forall\,z\in
D(\xi)$

in the notation of Proposition 4.3, and that

$\displaystyle\beta=r^{\bullet}(p,\xi)-r(p,\xi)$
$\displaystyle=r^{\bullet}_{D(\xi)}(0)-\delta_{D(\xi)}(0)$
$\displaystyle=\delta_{D(\xi)}(z_{0})-\delta_{D(\xi)}(0),$

where the $\alpha$ and $\beta$ above _also_ equal the quantities denoted by
the same symbols in Proposition 4.3 with
$\Omega\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=D(\xi)$,
$p\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=0$ and
$\zeta\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=z_{0}$. (We note
that the third equality above comes from the last sentence of the first
paragraph of the proof of Proposition 4.3.)

Now suppose that the condition in (2) of Theorem 1.4 holds. By the above
observations, we can invoke Part (3) of Proposition 4.3 to get

$\kappa_{D(\xi)}(0,1)\leqslant\frac{r^{\bullet}_{D(\xi)}(0)}{r^{\bullet}_{D(\xi)}(0)^{2}-|z_{0}|^{2}}=\frac{r^{\bullet}(p,\xi)}{r^{\bullet}(p,\xi)^{2}-\|q(\xi)-p\|^{2}}.$

By the metric-decreasing property of holomorphic mappings,

$\kappa_{D}\Big{(}p,\frac{\xi}{\|\xi\|}\Big{)}=\kappa_{D}(g(0),g^{\prime}(0)\,1)\leqslant\kappa_{D(\xi)}(0,1)\leqslant\frac{r^{\bullet}(p,\xi)}{r^{\bullet}(p,\xi)^{2}-\|q(\xi)-p\|^{2}}.$

Now, using the homogeneity of $\kappa_{D}(p,\boldsymbol{\cdot})$, we get
(1.6). We also note that, by Proposition 4.3,

$\frac{r^{\bullet}_{D(\xi)}(0)}{r^{\bullet}_{D(\xi)}(0)^{2}-|z_{0}|^{2}}<\frac{1}{\delta_{D(\xi)}(0)},$

which translates to

$\frac{r^{\bullet}(p,\xi)}{r^{\bullet}(p,\xi)^{2}-\|q(\xi)-p\|^{2}}<\frac{1}{r(p,\xi)}$

if $z_{0}\neq 0$, i.e., the upper bound obtained is strictly smaller than
$\|\xi\|/r(p,\xi)$ if $q(\xi)\neq p$.

Now suppose that the condition in (3) of Theorem 1.4 holds. This time we can
invoke Part (4) of Proposition 4.3 and the inequality
$\kappa_{D}(p,\xi/\|\xi\|)\leqslant\kappa_{D(\xi)}(0,1)$ to get — by our
observation above about the quantities $\alpha$ and $\beta$ — the bound (1.7).
In this case the bound obtained is in fact strictly smaller that
$\|\xi\|/r(p,\xi)$. This completes the proof of the theorem. ∎

## Acknowledgements

I am indebted to my thesis adviser, Gautam Bharali, for suggesting some of the
ideas in this paper. I would also like to thank him for providing valuable
help during the writing of this paper. I am also indebted to the anonymous
referee of an earlier version of this paper for suggesting a way to simplify
one of the key steps in the proof of Theorem 1.2.

## References

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# Component Separation method for CMB using Convolutional Neural Networks

A. Quintana1,2,3 B. Ruiz-Granados4 P. Ruiz-Lapuente1,2 1Instituto de Física
Fundamental (IFF-CSIC), Madrid, Spain,
2Institut de Ciències del Cosmos (UB-IEEC), Barcelona, Spain
3Universidad Internacional de Valencia, Valencia, Spain
4Universidad de Córdoba, Córdoba, Spain

###### Abstract

The aim of this project is to recover the CMB anisotropies maps in temperature
and polarized intensity by means of a deep convolutional neural network (CNN)
which, after appropiate training, can remove the foregrounds from Planck and
QUIJOTE data. The results are then compared with those obtained by COMMANDER,
based on Bayesian parametric component separation. The CNN successfully
recovered the CMB signal for both All Sky and Partial Sky maps showing
frequency dependant results, being optimum for central frequencies where there
is less contamination by foregrounds emissions such as galactic synchrotron
and thermal dust emissions. Recovered maps in temperature are consistent with
those obtained by Planck Collaboration, while polarized intensity has been
recovered as a new observable. The polarized intensity maps recovered from
QUIJOTE experiment are novel and of potential interest to the scientific
community for the detection of primordial gravitational waves. The way forward
will be to recover the maps at higher NSIDE and make them available to the
scientific community.

##  1 Introduction

Previous works [1] have successfully demonstrated the use of Deep Learning
techniques, such as Convolutional Neural Networks (CNN), for the extraction of
CMB temperature maps, in particular with U-Net architecture [2]. In this work
a U-Net CNN is trained from simulated observational maps, with the aim of
recovering the CMB signal in temperature and polarized intensity from Planck
111https://www.cosmos.esa.int/web/planck and QUIJOTE
222https://www.iac.es/es/proyectos/experimento-quijote-cmb observational maps.
In order to validate the methodology, the recovered CMB maps are compared with
those obtained by Planck Collaboration with COMMANDER as a different Component
Separation Method.

##  2 Methodology and Results

A CNN with U-Net architecture is developed and trained in order to be able to
recover the CMB signal. To train the CNN, 803 mock maps are simulated at NSIDE
64 at different frequency bands, both in temperature and polarized intensity
for Planck and QUIJOTE maps, meaning that the CNN has to be trained four
times. A different dataset of 209 mock maps are also generated in order to
validate the CNN, and finally 11 mock maps are generated for testing the CNN.
For each frequency, a different realization of the clean CMB signal generated
with PySM is provided to the CNN. Once the CNN has been proved to be properly
trained with the validation and test dataset, it is directly applied to
resolve the Planck and QUIJOTE maps.

Figure 1: Left: Recovered CMB maps in temperature at frequencies 30, 44, 70,
100, 143, 217 and 353 GHz from Planck. Right: Recovered CMB maps in polarized
intensity at frequencies 11, 13, 17 and 19 GHz from QUIJOTE.

Figure 2: Comparison between CMB signal in temperature recovered from Planck
(left) and in polarized intensity from QUIJOTE (right). CMB recovered by the
CNN (up), by COMMANDER (middle) and residuals (down) are shown.

In Figure 1, it is observed that below 44 GHz and above 217 GHZ the CNN
struggles to recover the CMB signal in temperature since synchrotron and
thermal dust foregrounds emissions, respectively, are dominant. In Figure 2,
on the left side it is seen that recovered map in temperature is consistent
with the CMB obtained by Planck. On the right side, while deviations from
results in polarized intensity with respect to Planck are apparent by eye, it
can be proved that the map recovered by the CNN is significantly more Gaussian
than Planck, which could be a hint of a better CMB signal; deep statistical
analysis is yet to be performed though.

##  3 Conclusions

CMB anisotropies maps have been successfully recovered by a CNN for both All
Sky and Partial Sky maps. Recovered maps are frequency dependant, with optimum
results for central frequencies. It is the first time that polarized intensity
defined as a scalar $IP=\sqrt{Q^{2}+U^{2}}$ is recovered as an observable.
While all simulations have been run at NSIDE 64, the next step will be to
repeat the procedure at NSIDE 512.

## References

## References

  * [1] G.J. Wang, H.L. Shi, Y.P. Yan, J.Q. Xia, Y.Y. Zhao, S.Y. Li, and J.F. Li, Astrophysical Journal Supplement Series 260(1), 13 (2022)
  * [2] O.Ronneberger, P. Fischer, and T.Brox, arxiv: 1505.04597
  * [3] Planck Collaboration. Astronomy and Astrophysics 641, A1 (2020)
  * [4] Rubiño-Martín et al. Monthly Notices of the Royal Astronomical Society 519(3), 3383 (2023)


$\mathfrak{sl}(n+1,\mathbb{C})$ | $\\{1,p\\}$ | $2\phi_{p}$ |  | $(\alpha_{1},\alpha_{p^{\prime}})$
$\mathfrak{su}(q,n-q+1)$ | $\\{1,2\\}$ |  | $-\frac{2}{3}\phi_{1}$ | $(\alpha_{2},\alpha_{1})$
$\mathfrak{so}(3,4),\mathfrak{so}(7,\mathbb{C})$ | $\\{2,3\\}$ |  | $\sqrt[3]{1}$ | $(\alpha_{3},\alpha_{2})$
$\mathfrak{so}(2,5)$, $\mathfrak{so}(3,4)$, $\mathfrak{so}(7,\mathbb{C})$ |  |  |  |
$\mathfrak{so}(2,6)$, $\mathfrak{so}(3,5)$, | $\\{1,2\\}$ |  | $1$ | $(\alpha_{2},\alpha_{1})$
$\mathfrak{so}(4,4)$, $\mathfrak{so}(7,\mathbb{C})$ |  |  |  |
$\mathfrak{so}(3,5)$ | $\\{2,3\\}$ |  | $\sqrt[3]{1}$ | $(\alpha_{3},\alpha_{2})$
$\mathfrak{so}(q,n-q)$, $\mathfrak{so}(n,\mathbb{C})$ | $\\{1,2\\}$ |  | $1$ | $(\alpha_{2},\alpha_{1})$
$\mathfrak{sp}(4,\\{\mathbb{R,C}\\})$ | $\\{1,2\\}$ | $\sqrt[3]{1}$ |  | $(\alpha_{1},\alpha_{2})$
$\mathfrak{sp}(4,\mathbb{C})$ | $\\{1,2\\}$ | $2\phi_{2}$ |  | $(\alpha_{1},\alpha_{2^{\prime}})$
$\mathfrak{sp}(4,\mathbb{C})$ | $\\{1,2\\}$ | $-\frac{2}{5}\phi_{2}$ |  | $(\alpha_{1^{\prime}},\alpha_{2^{\prime}})$
$\mathfrak{sp}(6,\\{\mathbb{R,C}\\})$ | $\\{1,3\\}$ | $1$ |  | $(\alpha_{1},\alpha_{3})$
$\mathfrak{sp}(6,\\{\mathbb{R,C}\\})$ | $\\{2,3\\}$ | $1$ |  | $(\alpha_{2},\alpha_{3})$
$\mathfrak{sp}(6,\mathbb{C})$ | $\\{1,3\\}$ | $2\phi_{3}$ |  | $(\alpha_{1},\alpha_{3^{\prime}})$
$\mathfrak{sp}(6,\mathbb{C})$ | $\\{2,3\\}$ | $-\frac{2}{5}\phi_{3}$ |  | $(\alpha_{2^{\prime}},\alpha_{3^{\prime}})$
$\mathfrak{sp}(2n,\\{\mathbb{R,C}\\})$ | $\\{1,n\\}$ | $1$ |  | $(\alpha_{1},\alpha_{n})$
$\mathfrak{sp}(2n,\\{\mathbb{R,C}\\})$ | $\\{n-1,n\\}$ | $1$ |  | $(\alpha_{n-1},\alpha_{n})$
$\mathfrak{sp}(2n,\mathbb{C})$ | $\\{1,n\\}$ | $2\phi_{n}$ |  | $(\alpha_{1},\alpha_{n^{\prime}})$
$\mathfrak{sp}(2n,\mathbb{C})$ | $\\{n-1,n\\}$ | $-\frac{2}{5}\phi_{n}$ |  | $(\alpha_{n-1^{\prime}},\alpha_{n^{\prime}})$
$\mathfrak{g}_{2}(\\{2,\mathbb{C}\\})$ | $\\{1,2\\}$ | $\sqrt[4]{1}$ |  | $(\alpha_{1},\alpha_{2})$
Table 11: Theorem 5.12, part with $|\Sigma|=3$. $\mathfrak{g}$ | $\Sigma$ | $j_{i_{1}}$ | $j_{i_{2}}$ | $j_{i_{3}}$ | $\mu$ | PR
---|---|---|---|---|---|---
$\mathfrak{sl}(4,\\{\mathbb{R,C}\\})$ | $\\{1,2,3\\}$ |  |  | $j_{1}^{2}$ | $(\alpha_{1},\alpha_{2})$ |
$\mathfrak{sl}(n+1,\\{\mathbb{R,C}\\})$ | $\\{1,2,3\\}$ |  |  | $j_{1}^{2}$ | $(\alpha_{1},\alpha_{2})$ |
$\mathfrak{sl}(n+1,\\{\mathbb{R,C}\\})$ |  $\\{1,2,p\\}$, $3<p<n$ |  |  | $j_{1}^{2}$ | $(\alpha_{1},\alpha_{2})$ | $j_{1}=\sqrt{1}$
$\mathfrak{sl}(n+1,\\{\mathbb{R,C}\\})$ | $\\{1,2,n\\}$ |  |  | $j_{1}^{2}$ | $(\alpha_{1},\alpha_{2})$ |
$\mathfrak{sl}(n+1,\\{\mathbb{R,C}\\})$ | $\\{1,2,n\\}$ |  | $j_{1}j_{n}$ |  | $(\alpha_{1},\alpha_{n})$ |
$\mathfrak{so}(3,4)$, $\mathfrak{so}(7,\mathbb{C})$ | $\\{1,2,3\\}$ | $j_{3}^{3}$ |  |  | $(\alpha_{3},\alpha_{2})$ |
Table 12: Theorem 5.12, part with $|\Sigma|=4$. $\mathfrak{g}$ | $\Sigma$ | $j_{i_{1}}$ | $j_{i_{2}}$ | $j_{i_{3}}$ | $j_{i_{4}}$ | $\mu$ | PR
---|---|---|---|---|---|---|---
$\mathfrak{sl}(n+1,\\{\mathbb{R,C}\\})$ | $\\{1,2,3,n\\}$ |  |  |  | $j_{1}j_{2}^{2}j_{p}^{-1}$ | $(\alpha_{2},\alpha_{1})$ | $j_{1}=1$

######  Example 5.11.

Let us continue in the discussion of generalized path geometries from Example
5.4. The case when the harmonic curvature $\kappa_{(\alpha_{2},\alpha_{1})}$
vanishes and the harmonic torsion $\kappa_{(\alpha_{1},\alpha_{2})}$ does not
vanish can be found in Table 10. There are several possible situations
depending on the eigenvalues of $s\in Z(G_{0})$.

If $j_{1}=1$, then we are precisely in the situation which is not covered by
any of the previous theorems and we can apply only the results of Propositions
4.3 and 5.12.

If $j_{1}=-1$ and $j_{2}=1$, then we can apply Theorem 5.5 and we are in the
situation of a generalized path geometry on the projectivized cotangent space
of an affine locally symmetric space.

If $j_{1}=-1$ and $j_{2}=-1$, then $\mathfrak{g}_{-}^{s}(1)=\mathfrak{g}_{-2}$
and we need the assumptions of Theorem 5.7 to show that we are in the
situation of a generalized path geometry on a (locally) homogeneous
$(n-1)$-dimensional fiber bundle over an affine locally symmetric space.

Finally, if $j_{1}=-1$ and $j_{2}\neq\sqrt{1}$, then we can apply Theorem 5.3.

The properties of these geometries are as follows.

######  Proposition 5.12.

Assume $(\mathfrak{g},\mathfrak{p},\mu)$ is prolongation rigid outside of the
$1$-eigenspace of $s$ for some $s\in Z(G_{0})$. Assume the harmonic curvature
$\kappa_{H}$ of a $($locally$)$ $s$-symmetric parabolic geometry
$(\mathcal{G}\to M,\omega)$ of type $(G,P)$ has a non-zero component in $\mu$
at all $x\in M$ and $S$ is the unique system of $($local$)$ $s$-symmetries on
$M$. Then the distribution $TM^{s}(1)$ is integrable and for each $x\in M$,
the leaf $\mathcal{F}_{x}$ of the foliation $\mathcal{F}$ of $TM^{s}(1)$
through $x$ is a totally geodesic submanifold for arbitrary Weyl connection.

Let $n\colon U_{x}\to N$ be a sufficiently small local leaf space of
$TM^{s}(1)$.

  * •

There is a unique local diffeomorphism $\underline{S}(n(y))$ of the local leaf
space $N$ at each $n(y)\in N$ such that $\underline{S}(n(y))\circ
n=n\circ\underline{S}(x)$ holds for all $y\in U_{x}$, and

  * •

for each eigenvalue $a$, $T_{y}n(T_{y}M^{s}(a))$ is the $a$-eigenspace of
$T_{n(y)}\underline{S}(n(y))$ for all $y\in U_{x}$.

###### Proof.

The proof is analogous to the proof of Claim (6) of Theorem 5.7, but, instead
of an $S$-invariant Weyl structure $\sigma$, we need to consider some almost
$S$-invariant Weyl structure invariant at $x$ from Claim (3) of Proposition
4.3 for each $x\in M$. ∎

###  5.7 Parabolic geometries that do not admit non-flat examples

There are triples $(\mathfrak{g},\mathfrak{p},\mu)$ that are prolongation
rigid outside of the $1$-eigenspace of $s$ for some $s\in Z(G_{0})$, but they
admit only flat (locally) $s$-symmetric parabolic geometries due to the
structure of the harmonic curvature and [12, Lemma 2.2]. These are contained
in Table 13.

Table 13: Flat geometries. $\mathfrak{g}$ | $\Sigma$ | $j_{i_{1}}$ | $j_{i_{2}}$ | $\mu$
---|---|---|---|---
$\mathfrak{sl}(3,\mathbb{C})$ | $\\{1\\}$ | $\sqrt[3]{1}$ | $1$ | $(\alpha_{1},\alpha_{2})$
$\mathfrak{sl}(3,\\{\mathbb{R},\mathbb{C}\\})$ | $\\{1,2\\}$ | $j_{1}^{4}\neq 1,j_{1}^{5}\neq 1$ | $j_{1}^{3}$ | $(\alpha_{1},\alpha_{2})$
$\mathfrak{so}(5,\mathbb{C})$ | $\\{1\\}$ | $\sqrt[3]{1}$ | $1$ | $(\alpha_{1},\alpha_{2})$
$\mathfrak{so}(2,3),\mathfrak{so}(5,\mathbb{C})$ | $\\{1,2\\}$ | $j_{1}^{4}\neq 1,j_{1}^{5}\neq 1,j_{1}^{7}\neq 1$ | $j_{1}^{3}$ | $(\alpha_{1},\alpha_{2})$

###  5.8 Parabolic geometries with more non-zero components
of the harmonic curvature

Let us also look at the parabolic geometries that allow a harmonic curvature
$\kappa_{H}$ with several non-zero components $\mu_{i}$ such that for each
$\mu_{i}$ the triple $(\mathfrak{g},\mathfrak{p},\mu_{i})$ is not prolongation
rigid outside of the $1$-eigenspace of $s$. In Table 14, we present the
complete classification of all triples $(\mathfrak{g},\mathfrak{p},\mu_{i})$
that are not prolongation rigid outside of the $1$-eigenspace of $s$ for the
same $s\in Z(G_{0})$, but for which $\mathfrak{a}_{i}$ in Proposition 3.5 is
contained in the $1$-eigenspaces of $s$ when the harmonic curvature has non-
zero component in each $\mu_{i}$. Geometric properties of the geometries from
Table 14 can be deduced from the previous sections depending on the position
and shape of $\mathfrak{g}_{-}^{s}(1)$ inside of $\mathfrak{g}_{-}$.

Table 14: More non-zero components of the harmonic curvature. $\mathfrak{g}$ | $\Sigma$ | eigenvalues | $\mu$
---|---|---|---
$\mathfrak{sl}(4,\mathbb{C})$ | $\\{1,2\\}$ | $j_{1}=\sqrt[4]{1}^{2},j_{2}=\sqrt[4]{1}$ |  $(\alpha_{2},\alpha_{3})$, $(\alpha_{2},\alpha_{1})$
$\mathfrak{sl}(n+1,\mathbb{C})$ | $\\{1,n-1\\}$ | $j_{1}=\sqrt[3]{1},j_{n-1}=\sqrt[3]{1}^{2}$ |  $(\alpha_{1},\alpha_{2})$, $(\alpha_{n-1},\alpha_{n})$
$\mathfrak{sl}(n+1,\mathbb{C})$ | $\\{2,n-1\\}$ | $j_{2}=\sqrt[3]{1},j_{n-1}\sqrt[3]{1}^{2}$ |  $(\alpha_{2},\alpha_{1})$, $(\alpha_{n-1},\alpha_{n})$
$\mathfrak{sl}(4,\mathbb{C})$ | $\\{1,2,3\\}$ | $j_{2}=\sqrt[4]{1},j_{3}=j_{1}(\sqrt[4]{1})^{2}$ |  $(\alpha_{2},\alpha_{1})$, $(\alpha_{2},\alpha_{3})$
$\mathfrak{sl}(n+1,\\{\mathbb{R,C}\\})$ | $\\{1,2,n-1,n\\}$ | $j_{n-1}=j_{2}^{-1},j_{n}=j_{1}j_{2}^{3}$ |  $(\alpha_{2},\alpha_{1})$, $(\alpha_{n-1},\alpha_{n})$

###  5.9 Remaining parabolic geometries with $\boldsymbol{\mu}$ in the
1-eigenspace of $\boldsymbol{s}$

For the sake of completeness, let us remark that there are triples
$(\mathfrak{g},\mathfrak{p},\mu)$ that are not prolongation rigid outside of
the $1$-eigenspace of $s$ for any $s$ such that $\mu$ is in the $1$-eigenspace
of $s$. These are contained in Table 15.

Table 15: Remaining parabolic geometries with $\mu$ in the $1$-eigenspace of $s$. $\mathfrak{g}$ | $\Sigma$ | $j_{i_{1}}$ | $j_{i_{2}}$ | $\mu$
---|---|---|---|---
$\mathfrak{sl}(n+1,\\{\mathbb{R,C}\\})$ |  $\\{p,p+1\\}$, $n-1>p>1$ |  | $1$ | $(\alpha_{p+1},\alpha_{p})$
$\mathfrak{so}(q,n-q),\mathfrak{so}(n,\mathbb{C})$ | $\\{2,3\\}$ |  | $1$ | $(\alpha_{3},\alpha_{2})$

##  Appendix A A construction of locally homogeneous locally
$\boldsymbol{s}$-symmetric
parabolic geometries

It is proved in [12, Section 2] how to algebraically construct and classify
all homogeneous $s$-symmetric parabolic geometries. Part of the classification
is done in [6, 9] using the classification of semisimple symmetric spaces.
There is the result from [7, Section 1.3] and [12, Lemma 2.2] stating that for
the construction and the classification of locally homogeneous locally
$s$-symmetric parabolic geometries, it is sufficient to find the following
data:

  * •

an extension $(\alpha,i)$ of the Klein geometry $(K,H)$ to $(G,P)$ such that
the action of $s$ preserves $\alpha(\mathfrak{k})\subset\mathfrak{g}$, and $s$
acts trivially on the tensor
$[\cdot,\cdot]-\alpha([\alpha^{-1}(\cdot),\alpha^{-1}(\cdot)])$ in
$\wedge^{2}\mathfrak{g}/\mathfrak{p}^{*}\otimes\mathfrak{g}$, and

  * •

the subset $\mathcal{A}$ of $P$ consisting of elements $g\in P$, which act as
local automorphisms on the parabolic geometry $(K\times_{i}P\to
K/H,\omega_{\alpha})$ of type $(G,P)$ given by the extension $(\alpha,i)$.

If $U$ and $V$ are open subsets of $K/H$ such that there are $k\in K$,
$g\in\mathcal{A}$ and a maximal open subset $W$ of $U$ such that $kg(W)\subset
V$, then we can glue $K\times_{i}P|_{U}\to U$ with $K\times_{i}P|_{V}\to V$ by
identifying $w\in W\subset U$ with $kg(w)\in V$, and glue the Cartan
connection $\omega_{\alpha}|_{U}$ with the pullback connection
$(kg)^{*}\omega_{\alpha}|_{V}=\omega_{\alpha}|_{(kg)^{*}(V)}$. Of course, we
can without loss of generality assume that $U$, $V$ and $W$ are simply
connected, because we can always choose coverings of our manifolds by open
sets satisfying this condition. Therefore, we can also assume that the
automorphism $k$ is given by the flow of a local infinitesimal automorphism of
$(K\times_{i}P\to K/H,\omega_{\alpha})$. Then we obtain the following result
as a consequence of the construction in [13, Section 3] and [7, Section 1.3].

######  Theorem A.1.

Let $(\mathcal{G}\to M,\omega)$ be a locally homogeneous locally $s$-symmetric
parabolic geometry, let $\mathfrak{k}$ be the Lie algebra of the local
infinitesimal automorphisms and denote by $\alpha$ the inclusion of
$\mathfrak{k}$ into $\mathfrak{g}$ given by $\omega(u)$ at some
$u\in\mathcal{G}$. Then:

  1. $1)$

${\rm Ad}(s)(\mathfrak{k})\subset\mathfrak{k}$ is an automorphism of the Lie
algebra $\mathfrak{k}$,

  2. $2)$

there exist $($see [13, Section 3] for the explicit construction$)$

     * •

a Klein geometry $(K,H)$ such that $\mathfrak{k}$ is the Lie algebra of $K$,

     * •

an extension $(\alpha,i)$ of $(K,H)$ to $(G,P)$,

     * •

an open covering $U_{a}$ of $M$, and

     * •

isomorphisms $\phi_{a}\colon U_{i}\to K/H$ of parabolic geometries
$(\mathcal{G}|_{U_{a}}\to U_{i},\omega|_{U_{b}})$ and
$(K\times_{i}P|_{\phi_{a}(U_{a})}\to\phi_{a}(U_{a}),\omega_{\alpha}|_{\phi_{a}(U_{a})})$
of type $(G,P)$ such that $\phi_{a}\circ\phi_{b}^{-1}$ is the restriction of
the left action of some element of $K$ for each $a$, $b$.

### Acknowledgments

JG supported by the Grant agency of the Czech Republic under the grant
GBP201/12/G028. The authors would like to thank the anonymous referees for
their valuable comments which helped to improve the manuscript.

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