# On the partial uniform ellipticity and complete conformal metrics with
prescribed curvature functions on manifolds with boundary

Rirong Yuan  School of Mathematics, South China University of Technology,
Guangzhou 510641, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider the problem of finding complete conformal metrics with prescribed
curvature functions of Einstein tensor and more general modified Schouten
tensors on manifolds with boundary. The asymptotic behavior of complete
conformal metrics near boundary is also formulated. Our method is based on
partial uniform ellipticity. Finally, we further discuss Hessian and
Weingarten equations by confirming a key assumption.

The author was supported by NSFC Grant 11801587.

##  1\. Introduction

Let $(M,g)$ be a compact Riemannian manifold of dimension $n$, $n\geq 3$, with
smooth boundary $\partial M$, $\bar{M}=M\cup\partial M$, and with Levi-Civita
connection $\nabla$. ($g$ is smooth in $\bar{M}$). Let $\mathrm{Ric}_{g}$ and
$\mathrm{R}_{g}$ denote the Ricci and scalar curvature of $g$. Denote the
Einstein tensor on $(M,g)$ by $G_{g}$,

$G_{g}:=\mathrm{Ric}_{g}-\frac{1}{2}\mathrm{R}_{g}g.$

In [1], Aviles-McOwen extended a result of Loewner-Nirenberg [36] and further
proved a nice theorem: Such $(M,g)$ admits a complete conformal Riemannian
metric with constant negative scalar curvature.

We are concerned with a fully nonlinear version of their results: Deforming
$g$ conformally to find a complete metric ${g}_{\infty}$ on $M$ with

(1.1) $\displaystyle F(G_{{g}_{\infty}})\equiv
f(\lambda_{{g}_{\infty}}(G_{{g}_{\infty}}))=\psi\mbox{ in }M,$

where $\lambda_{{g}_{\infty}}(G_{{g}_{\infty}})$ denote eigenvalues of
$G_{{g}_{\infty}}$ with respect to ${g}_{\infty}$. As in [4], $f$ is a smooth
symmetric function of $n$ real variables defined in an open symmetric and
convex cone $\Gamma\subset\mathbb{R}^{n}$ with vertex at origin,
$\partial\Gamma\neq\emptyset$, $\Gamma_{n}\subseteq\Gamma\subset\Gamma_{1},$
where $\Gamma_{k}$ is the $k$-th Gårding’s cone defined by
$\Gamma_{k}=\left\\{\lambda\in\mathbb{R}^{n}:\sigma_{j}(\lambda)>0,\mbox{
}\forall 1\leq j\leq k\right\\},$ $\sigma_{j}$ is the $j$-th elementary
symmetric function. In addition, $f$ shall satisfy the fundamental conditions:

(1.2) $f_{i}(\lambda):=\frac{\partial f}{\partial\lambda_{i}}(\lambda)>0\mbox{
in }\Gamma,\ 1\leq i\leq n,$ (1.3) $f\mbox{ is concave in }\Gamma,$ (1.4)
$\displaystyle f(t\lambda)=tf(\lambda),\mbox{ }f(\lambda)>0,\mbox{ for any
}\lambda\in\Gamma,\mbox{ }t>0.$

We assume throughout the article $f|_{\partial\Gamma}\equiv 0$,
$f|_{\Gamma}>0$ and $f(\vec{\bf 1})=1$, whenever $f$ is homogeneous of degree
one, i.e. $f$ satisfies (1.4), where and hereafter

$\vec{\bf 1}:=(1,\cdots,1)\in\mathbb{R}^{n}.$

We prove the main results via continuity method. The starting metric
$\underline{g}$ satisfies

(1.5) $\displaystyle\lambda_{\underline{g}}(G_{\underline{g}})\in\Gamma\mbox{
in }\bar{M},\,$ $\displaystyle\underline{g}={e^{2\underline{u}}g}\mbox{ for
some $\underline{u}\in C^{2}(\bar{M})$}.$

(We call such a metric an admissible conformal metric for simplicity).

The existence theorems can be stated as follows.

######  Theorem 1.1.

Assume (1.2), (1.3), (1.4) hold. Suppose in addition that
$\Gamma\neq\Gamma_{n}$ and there exists an $C^{2}$-admissible conformal metric
satisfying (1.5). Then for each smooth positive function $\psi$ on $\bar{M}$,
there is on $M$ a smooth complete metric ${g}_{\infty}=e^{2u_{\infty}}g$ that
is conformal to $g$, satisfying (1.1) and
$\lambda_{{g}_{\infty}}(G_{{g}_{\infty}})\in\Gamma$ in $M$.

In the special case when $\psi$ is a positive constant when restricted to
boundary, we further study the uniqueness and asymptotic behavior of complete
conformal metrics.

######  Theorem 1.2.

Let $\psi>0$, $\psi|_{\partial M}\equiv 1$ and we assume other assumptions in
Theorem 1.1 hold. There exists a unique smooth complete conformal metric
$g_{\infty}=e^{2u_{\infty}}g$ on $M$ to satisfy
$f(\lambda_{g_{\infty}}(G_{g_{\infty}}))=\psi$,
$\lambda_{{g}_{\infty}}(G_{{g}_{\infty}})\in\Gamma$ in $M$, and

(1.6) $\displaystyle\lim_{x\rightarrow\partial
M}(u_{\infty}(x)+\log\mathrm{d}(x))=\frac{1}{2}\log\frac{(n-1)(n-2)}{2}.$

Where and hereafter $\mathrm{d}(x)$ denotes the distance function from $x$ to
$\partial M$, i.e. $\mathrm{d}(x)=\mathrm{dist}_{g}(x,\partial M)$.

Denote by $[g]=\left\\{e^{2v}g:v\in C^{\infty}(\bar{M})\right\\}$ the
conformal class of $g$. We define

(1.7) $\displaystyle\mathcal{A}(\bar{M},[g],\Gamma):=\left\\{u\in
C^{2}(\bar{M}):\lambda_{\tilde{g}}(\Delta_{\tilde{g}}u\tilde{g}-\tilde{\nabla}^{2}u)\in\Gamma\mbox{
at which }\tilde{\nabla}u(x_{0})=0\mbox{ for }\tilde{g}\in[g]\right\\}.$

(It is well defined since $\mathcal{A}(\bar{M},[g],\Gamma)$ is independent of
choice of $\tilde{g}$ in $[g]$. So $\mathcal{A}(\bar{M},[g],\Gamma)$ is
conformal invariant). In Proposition 6.12 we prove admissible conformal
metrics exists if $\mathcal{A}(\bar{M},[g],\Gamma)\neq\emptyset$.
Consequently, the existence of admissible conformal metrics connect closely
with the zeros of $C^{1}$-gradient fields over underlying manifolds.

Consequently, on certain manifolds including smooth bounded domains and warped
products, admissible conformal metrics exist (see Lemma 6.13 for more
discussion).

######  Theorem 1.3.

Let $(M,g)=(X\times\Omega,g)$ be a Riemannian manifold, where $\Omega$ is a
smooth and bounded domain in $\mathbb{R}^{k}$ and $X$ is a closed Riemannian
manifold of dimension $n-k$ ($1\leq k\leq n$ and $g$ is not necessary the
product metric). Given a smooth positive function $\psi$ on $\bar{M}$, there
exists a smooth complete conformal Riemannian metric ${g}_{\infty}$ on $M$ to
satisfy (1.1) and $\lambda_{{g}_{\infty}}(G_{{g}_{\infty}})\in\Gamma$,
provided that $\Gamma\neq\Gamma_{n}$ and $f$ satisfies (1.2), (1.3) and (1.4).

In order to prove Theorem 1.1 it suffices to solve the Dirichlet problem for

(1.8) $\displaystyle f(\lambda(\Delta
ug-\nabla^{2}u+\frac{1}{n-2}G_{g}+\frac{n-3}{2}|\nabla u|^{2}g+du\otimes
du))=\frac{\psi e^{2u}}{n-2},$ (1.9) $\displaystyle u|_{\partial M}=+\infty,$

according to the formula for conformal transform $\tilde{g}=e^{2u}g$ (see e.g.
[2]),

(1.10) $\displaystyle\frac{1}{n-2}G_{\tilde{g}}=\frac{1}{n-2}G_{g}+\Delta
ug-\nabla^{2}u+\frac{n-3}{2}|\nabla u|^{2}g+du\otimes du,\mbox{ if }n\geq 3,$

where and hereafter $\nabla u$ and $\nabla^{2}u$ are gradient and Hessian of
$u$ with respect to $g$, $\Delta$ is Laplacian operator with $\Delta
u=\mathrm{tr}_{g}(\nabla^{2}u)$, $|\nabla u|^{2}=g(\nabla u,\nabla u)$, and
$\lambda(A)$ $(=\lambda_{g}(A))$ denote the eigenvalues of
$``A^{\prime\prime}$ with respect to background metric $g$.

The proof of Theorem 1.1 is similar to that used in [36] (see also [14, 22]).
In the proof we apply approximation to Dirichlet problem (1.8) and (1.9).
Aviles-McOwen’s aforementioned result can be viewed as a supersolution of
approximating Dirichlet problems, thereby, on each compact subset, providing a
uniform upper bound for approximating solutions. The key ingredient is to
derive interior estimates for (1.8). With the interior estimates at hand, we
obtain $C^{2}$-estimate on compact subsets, thus allowing us to apply Evans-
Krylov theorem [10, 30] and classical Schauder theory.

However, deriving the desired interior estimates is rather hard as described
later.

We start our strategy with partial uniform ellipticity. Throughout this
article, we call the function $f$ $k$-uniform ellipticity, if there exists a
uniformly positive constant $\theta$, such that for any $\lambda\in\Gamma$
with the order $\lambda_{1}\leq\cdots\leq\lambda_{n}$, there holds

(1.11) $\displaystyle
f_{i}(\lambda)\geq\theta\sum_{j=1}^{n}f_{j}(\lambda)\mbox{ in }\Gamma,\mbox{
}\forall 1\leq i\leq k.$

In particular, $n$-uniform ellipticity is also called fully uniform
ellipticity. Accordingly, we have an analogous notion of partial uniform
ellipticity for a second order equation, if its linearized operator satisfies
a similar condition.

Next, we define $\kappa_{\Gamma}$ for $\Gamma$ to measure partial uniform
ellipticity for $f$ (defined in $\Gamma$).

######  Definition 1.4.

For the cone $\Gamma$ stated above, we define

$\displaystyle\kappa_{\Gamma}:=\max\left\\{l:(-\alpha_{1},\cdots,-\alpha_{l},\alpha_{l+1},\cdots,\alpha_{n})\in\Gamma,\mbox{
where }\alpha_{j}>0,\mbox{ }\forall j=1,\cdots,n\right\\}.$

Clearly, $\kappa_{\Gamma}$ is well defined; in addition,

  * •

$\kappa_{\Gamma}$ is an integer with $0\leq\kappa_{\Gamma}\leq n-1$.

  * •

If $\Gamma^{(1)}\subseteq\Gamma^{(2)}$ then
$\kappa_{\Gamma^{(1)}}\leq\kappa_{\Gamma^{(2)}}$.

  * •

$\kappa_{\Gamma}\geq 1$ if and only if $\Gamma\neq\Gamma_{n}$.

  * •

$\kappa_{\Gamma}=n-k\mbox{ if }\Gamma=\Gamma_{k}$, $2\leq k\leq n$. In
particular, $\kappa_{\Gamma_{n}}=0$, $\kappa_{\Gamma_{2}}=n-2$.

  * •

$\kappa_{\Gamma}=n-1\mbox{ if and only if $\Gamma$ is of type 2 in the sense
of \cite[cite]{[\@@bibref{}{CNS3}{}{}]}}.$

######  Theorem 1.5.

In addition to (1.2) and (1.3), $f$ also satisfies

(1.12) $\displaystyle\mbox{For each $\sigma<\sup_{\Gamma}f$ and
}\lambda\in\Gamma,\mbox{ }\lim_{t\rightarrow+\infty}f(t\lambda)>\sigma.$

Then there is a universally positive constant $\vartheta$ depending only on
$\Gamma$, such that for each $\lambda\in\Gamma$ with order
$\lambda_{1}\leq\cdots\leq\lambda_{n}$,

$\displaystyle
f_{{i}}(\lambda)\geq\vartheta\sum_{j=1}^{n}f_{j}(\lambda),\mbox{ for all
}1\leq i\leq\kappa_{\Gamma}+1.$

Theorem 1.5 states that $f$ is exactly $(\kappa_{\Gamma}+1)$-uniform
ellipticity, which is sharp as a consequence of Proposition 2.3 below. Namely,

######  Theorem 1.6.

The $(\kappa_{\Gamma}+1)$-uniform ellipticity asserted in Theorem 1.5 cannot
be further improved.

Using the partial uniform ellipticity, we will derive interior estimates for
more general equations of $(n-1)$-type than (1.8),

(1.13) $\displaystyle f(\lambda(\Delta ug-\varrho\nabla^{2}u+W(x,u,\nabla
u)))=\psi(x,u,\nabla u)$

under an assumption that $\varrho$ is a constant and satisfies

(1.14) $\displaystyle\varrho<\frac{1}{1-\kappa_{\Gamma}\vartheta}\mbox{ and
}\varrho\neq 0,\mbox{ where $\vartheta$ is as in Theorem \ref{yuan-k+1}.}$

(Throughout this article, $\vartheta$ always stands for the constant claimed
in Theorem 1.5, $\kappa_{\Gamma}$ denotes the constant defined in Definition
1.4).

It is worth while to note that when $\Gamma\neq\Gamma_{n}$, besides critical
case $\varrho=1$ that applies to Einstein tensor, assumption (1.14) allows
supercritical case $1<\varrho<\frac{1}{1-\kappa_{\Gamma}\vartheta}$. In
addition, as stated in Remark 1.9 below, there is a topological obstruction to
derive the interior estimates in the case $\Gamma=\Gamma_{n}$, thereby showing
that such an assumption (1.14) is rather appropriate and delicate.

The interior estimates, stated in Section 3, further applies to conformally
deforming modified Schouten tensors
$S_{g}^{\tau}:=\frac{1}{n-2}(\mathrm{Ric}_{g}-\frac{\tau}{2(n-1)}\mathrm{R_{g}}g)$
which obey the formula

$\displaystyle S_{\tilde{g}}^{\tau}=S_{g}^{\tau}+\frac{\tau-1}{n-2}\Delta
ug-\nabla^{2}u+\frac{\tau-2}{2}|\nabla u|^{2}g+du\otimes du,\,$
$\displaystyle\tilde{g}=e^{2u}g.$

Einstein tensor corresponds exactly to $\tau=n-1$ $(G_{g}=(n-2)S_{g}^{n-1})$,
the equation accordingly invokes the critical case $\varrho=1$. When $\tau=1$,
$S_{g}^{1}$ is the classical Schouten tensor which arises naturally from fully
nonlinear version of Yamabe problem [40] and has drawn much attention (see
e.g. [3, 6, 5, 7, 8, 9, 11, 13, 18, 19, 20, 28, 31, 24, 23, 25, 26, 37, 41,
42]). For $\tau>n-1$ it was also studied in [32]. If $\tau=0$ it is the Ricci
tensor, and we refer to [14, 22, 24] for its conformal deformation and to [22]
for its relation to Poincaré-Einstein metrics.

We prove a more general result than Theorems 1.1 and 1.2. For the modified
Schouten tensors $S_{g}^{\tau}$ with

(1.15) $\displaystyle\tau>1+(n-2)(1-\kappa_{\Gamma}\vartheta)$

which is rather appropriate for the parameter $\tau$ as we will see in Remark
7.1 below, we can find complete conformal metrics by solving the curvature
equation

(1.16) $\displaystyle
f(\lambda_{\tilde{g}}(S_{\tilde{g}}^{\tau}))=\frac{\psi}{n-2}.$

######  Theorem 1.7.

Assume there is an $C^{2}$-conformal metric with

(1.17) $\displaystyle\lambda(S_{e^{2\underline{u}}g}^{\tau})\in\Gamma\mbox{ in
}\bar{M}\mbox{ for some }\underline{u}\in C^{2}(\bar{M}).$

Suppose (1.2), (1.3), (1.4) hold and that $\psi\in C^{\infty}(\bar{M})$,
$\psi>0$ in $\bar{M}$. If, in addition, the parameter $\tau$ satisfies (1.15)
then for the corresponding $S_{g}^{\tau}$, there is on $M$ a smooth complete
conformal metric ${g}_{\infty}$ satisfying (1.16) and
$\lambda_{{g}_{\infty}}(S^{\tau}_{{g}_{\infty}})\in\Gamma$ in $M$.

In addition, if $\psi|_{\partial M}\equiv 1$ then there is a unique smooth
complete conformal metric $g_{\infty}=e^{2u_{\infty}}g$ satisfying (1.16),
$\lambda_{{g}_{\infty}}(S^{\tau}_{{g}_{\infty}})\in\Gamma$ in $M$, and

(1.18) $\displaystyle\lim_{x\rightarrow\partial
M}(u_{\infty}(x)+\log\mathrm{d}(x))=\frac{1}{2}\log\frac{n\tau+2-2n}{2}.$

In particular, for products as in Theorem 1.3 we have the existence result:

######  Theorem 1.8.

Let $(M,g)=(X\times\Omega,g)$ be as in Theorem 1.3. Suppose (1.2), (1.3),
(1.4) and (1.15) hold. For any smooth positive function $\psi$ on $\bar{M}$,
there is a smooth complete conformal metric ${g}_{\infty}$ on $M$ to satisfy
(1.16), $\lambda_{{g}_{\infty}}(S^{\tau}_{{g}_{\infty}})\in\Gamma$ in $M$.

The methods used here work for
$f(\lambda_{\tilde{g}}(-S_{\tilde{g}}^{\tau}))=\psi$ with $\tau<1$,
accordingly. We shall remark here that for $\varrho<0$ or $0<\varrho<1$ the
equation (1.13) is automatically uniformly elliptic; while the value
$\varrho=1$ is critical in the procedure where the fully uniform ellipticity
of corresponding equation seems breaks down (it seems only $(n-1)$-uniform
ellipticity), which causes a huge obstacle for deriving interior estimates. We
observe that a characterization of $f$, initially proposed in previous article
[44], leads naturally to the partial uniform ellipticity, thereby providing an
effective way to understand the structure of the linearized operator of
corresponding equations. As a result, surprisingly, we can employ it to show
that equation (1.13) is in effect fully uniform ellipticity provided (1.14)
holds.

A somewhat surprising fact to us is that the measure of partial uniform
ellipticity depends only on the cone $\Gamma$ instead specifically on the
function $f$. Namely, the constants $\vartheta$, $\kappa_{\Gamma}+1$ as
claimed in Theorem 1.5 are both universal, and each $f$ as assumed there is
$(\kappa_{\Gamma}+1)$-uniform ellipticity with same constant $\vartheta$. This
is new even for homogenous functions, thus providing a surprising and useful
generalization from $f=(\sigma_{k})^{\frac{1}{k}}$ proved in [35] to general
contexts.

######  Remark 1.9.

It is a remarkable fact that the condition $\Gamma\neq\Gamma_{n}$ imposed in
Theorems 1.1, 1.2 and 1.3 is crucial and cannot be further dropped.

For a $3$-manifold $(M,g)$, fix $x\in M$, let $\Sigma\subset T_{x}M$ be a
tangent $2$-plane, $\vec{\bf n}\in T_{x}M$ be a unit normal vector to
$\Sigma$, and $K_{g}(\Sigma)$ be the section curvature of $(M,g)$ with respect
to the $2$-plane $\Sigma$, Gursky-Streets-Warren formulated in [21] a formula

(1.19) $\displaystyle G_{g}(\vec{\bf n},\vec{\bf n})=-K_{g}(\Sigma)$

connecting Einstein tensor with sectional curvature on 3-manifolds. Also, from
(1.19) our results can be applied to deform sectional curvature on
3-manifolds. Therefore, if Theorem 1.1 held for $\Gamma_{n}$ (so did Theorems
1.2 and 1.3), then for any conformal class $[g]$ on
$\mathbb{S}^{2}\times(0,1)$ (not necessary the standard one), there would
exist in $[g]$ a complete conformal metric with negative sectional curvature
according to Theorem 1.3, which contradicts to the Cartan-Hadamard theorem.

This topological obstruction, due to the Cartan-Hadamard theorem, was noted
previously by Gursky-Streets-Warren and motivated them to study a question
asking how close one can come to negative curvature. By studying a Monge-
Ampère type equation as a special case of (1.13) with $\Gamma=\Gamma_{n}$ and
$\varrho<1$, they [21] proved any 3-manifold with boundary admits a complete
conformal metric satisfying a pinching condition, which is similar to
Hamilton-Ivey pinching estimate of Ricci flow and roughly means the curvature
is “almost negative” in a scale-invariant sense.

The paper is organized as follows. In Section 2 we investigate the partial
uniform ellipticity. In Section 3 we state main interior estimates for (1.13).
In Section 4 we verify that equation (1.13) is fully uniform ellipticity
provided (1.14) holds. The key issue is the partial uniform ellipticity. In
Section 5 we derive interior estimates for a class of Hessian type equations.
In Section 6 we prove main results and briefly discuss the asymptotic
properties of complete conformal metrics as well as the connection of
$\mathcal{A}(\bar{M},[g],\Gamma)$ and the existence for admissible conformal
metrics. In Section 7 the existences of conformal metrics determined by more
general modified Schouten tensors are further obtained. Finally, we will show
another application of partial uniform ellipticity. More precisely, we prove
Proposition 8.1 confirming a key assumption used in various literatures
including [34, 39] and then discuss gradient estimate for Hessian and
Weingarten equations in Section 8.

##  2\. The partial uniform ellipticity

This section is devoted to the investigation of partial uniform ellipticity.
Throughout this paper, we denote

$\displaystyle Df(\lambda):=$
$\displaystyle(f_{1}(\lambda),\cdots,f_{n}(\lambda)),$
$\displaystyle\Gamma^{\sigma}:=$
$\displaystyle\\{\lambda\in\Gamma:f(\lambda)>\sigma\\},\mbox{
}\partial\Gamma^{\sigma}:=\\{\lambda\in\Gamma:f(\lambda)=\sigma\\},$
$\displaystyle\Gamma^{\infty}_{\mathbb{R}^{k}}:=$
$\displaystyle\left\\{\lambda^{\prime}\in\mathbb{R}^{k}:(\lambda^{\prime},c,\cdots,c)\in\Gamma\mbox{
for some }c>0\right\\}.$

Clearly,
$\Gamma^{\infty}_{\mathbb{R}^{\kappa_{\Gamma}}}=\mathbb{R}^{\kappa_{\Gamma}}$
if $\kappa_{\Gamma}\geq 1$.

The following lemma is the starting point of our partial uniform ellipticity,
among which the equivalence of the first two was asserted in Lemma 3.4 of
[44]. It is a characterization of $f$ satisfying (1.2), (1.3) and (1.12).

######  Lemma 2.1 ([44]).

For the function $f$ satisfying (1.2) and (1.3) in $\Gamma$, the following
statements are equivalent each other.

  1. (1)

$f$ further satisfies (1.12).

  2. (2)

For each $\lambda$, $\mu\in\Gamma,$ $\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}>0.$

  3. (3)

$f(\lambda+\mu)>f(\lambda)$ for any $\lambda$, $\mu\in\Gamma$.

  4. (4)

$f(\lambda+c\vec{\bf 1})>f(c\vec{\bf 1})$ for all $\lambda\in\Gamma$, $c>0$.

###### Proof.

$\mathrm{(1)}\Rightarrow\mathrm{(2)}$: Condition (1.12) yields for any
$\lambda$, $\mu\in\Gamma$, there is $T\geq 1$ such that for each $t>T$, there
holds $t\mu\in\Gamma^{f(\lambda)}$. Together with the convexity of level sets,
one gets $Df(\lambda)\cdot(t\mu-\lambda)>0$. Thus, $Df(\lambda)\cdot\lambda>0$
(if one takes $\mu=\lambda$) and $Df(\lambda)\cdot\mu>0$.

$\mathrm{(2)}\Rightarrow\mathrm{(3)}$: The proof uses

(2.1) $\displaystyle f(\mu)\geq
f(\lambda)+\sum_{i=1}^{n}f_{i}(\mu)(\lambda_{i}-\mu_{i}),\mbox{
}\forall\lambda,\mbox{ }\mu\in\Gamma.$

$\mathrm{(4)}\Rightarrow\mathrm{(1)}$: Given $\sigma<\sup_{\Gamma}f$. Let
$c_{\sigma}$ be the positive constant with

(2.2) $\displaystyle f(c_{\sigma}\vec{\bf 1})=\sigma.$

For any $\lambda\in\Gamma$, $t\lambda-c_{\sigma}\vec{\bf 1}\in\Gamma$ for
$t>\frac{\sqrt{n}c_{\sigma}}{\mbox{dist}(\lambda,\partial\Gamma)}$. Thus
$f(t\lambda)>\sigma$ for such $t$. ∎

When $\Gamma=\Gamma_{n}$ $(\mbox{i.e. }\kappa_{\Gamma}=0)$ Theorem 1.5
immediately follows. For general $\Gamma$, it is a consequence of the
following proposition.

######  Proposition 2.2.

Assume $\Gamma\neq\Gamma_{n}$ and $f$ satisfies (1.2), (1.3) and (1.12). Let
$\kappa_{\Gamma}$ be as defined in Definition 1.4, and
$\alpha_{1},\cdots,\alpha_{n}$ be $n$ strictly positive constants such that

$(-\alpha_{1},\cdots,-\alpha_{\kappa_{\Gamma}},\alpha_{\kappa_{\Gamma}+1},\cdots,\alpha_{n})\in\Gamma.$

In addition, assume $\alpha_{1}\geq\cdots\geq\alpha_{\kappa_{\Gamma}}$. Then
for each $\lambda\in\Gamma$ with order $\lambda_{1}\leq\cdots\leq\lambda_{n}$,

(2.3) $\displaystyle
f_{\kappa_{\Gamma}+1}(\lambda)\geq\frac{\alpha_{1}}{(\sum_{i=\kappa_{\Gamma}+1}^{n}\alpha_{i}-\sum_{i=2}^{\kappa_{\Gamma}}\alpha_{i})}f_{1}(\lambda).$

Furthermore, $f_{1}(\lambda)\geq\frac{1}{n}\sum_{i=1}^{n}f_{i}(\lambda).$

###### Proof.

By (1.3), one has $f_{1}(\lambda)\geq f_{2}(\lambda)\geq\cdots\geq
f_{n}(\lambda)\mbox{ and
}f_{1}(\lambda)\geq\frac{1}{n}\sum_{i=1}^{n}f_{i}(\lambda)$ for
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}$. By Lemma 2.1,

(2.4)
$\displaystyle-\sum_{i=1}^{\kappa_{\Gamma}}\alpha_{i}f_{i}(\lambda)+\sum_{i=\kappa_{\Gamma}+1}^{n}\alpha_{i}f_{i}(\lambda)>0,$

which simply yields
$f_{\kappa_{\Gamma}+1}(\lambda)>\frac{\alpha_{1}}{\sum_{i=\kappa_{\Gamma}+1}^{n}\alpha_{i}}f_{1}(\lambda)$.
In addition, by (2.4) and iteration, one derives (2.3).

∎

######  Proposition 2.3.

In addition to (1.2) and (1.3), we assume that $f$ is $(k+1)$-uniform
ellipticity for some $1\leq k\leq n-1$. Then
$\Gamma^{\infty}_{\mathbb{R}^{k}}=\mathbb{R}^{k}.$

###### Proof.

For $\epsilon,R>0$, set
$\lambda_{\epsilon,R}=(\epsilon,\cdots,\epsilon,R,\cdots,R)$,
$(\epsilon,\cdots,\epsilon)\in\mathbb{R}^{k}$,
$(R,\cdots,R)\in\mathbb{R}^{n-k}$. Fix
$\sup_{\partial\Gamma}f<\sigma<\sup_{\Gamma}f$, let $a=1+c_{\sigma}$,
$0<\epsilon<a$, here $c_{\sigma}$ obeys (2.2). Thus,

$\displaystyle f(\lambda_{\epsilon,R})\geq$ $\displaystyle f(a\vec{\bf
1})+\epsilon\sum_{i=1}^{k}f_{i}(\lambda_{\epsilon,R})+R\sum_{i=k+1}^{n}f_{i}(\lambda_{\epsilon,R})-a\sum_{i=1}^{n}f_{i}(\lambda_{\epsilon,R})$
$\displaystyle>$ $\displaystyle f(a\vec{\bf
1})+(R\theta-a)\sum_{i=1}^{n}f_{i}(\lambda_{\epsilon,R})\mbox{ (using
$(k+1)$-uniform ellipticity)}$ $\displaystyle=$ $\displaystyle f(a\vec{\bf
1})\mbox{ (by setting }R=\frac{a}{\theta})$ $\displaystyle>$
$\displaystyle\sigma.$

Here $\theta$ is the constant as in (1.11). So
$\lambda_{\epsilon,R}=(\epsilon,\cdots,\epsilon,R,\cdots,R)\in\Gamma^{f(a\vec{\bf
1})}$. Notice $R=\frac{a}{\theta}$ is chosen uniformly. Let
$\epsilon\rightarrow 0^{+}$, we know
$(0,\cdots,0,R,\cdots,R)\in\overline{\Gamma^{f(a\vec{\bf
1})}}\subset\Gamma^{\sigma}\subset\Gamma$. By the openness of $\Gamma$,
$\Gamma^{\infty}_{\mathbb{R}^{k}}=\mathbb{R}^{k}$. ∎

###### Proof of Theorem 1.6.

If $f$ is $(\kappa_{\Gamma}+2)$-uniform ellipticity then
$\Gamma^{\infty}_{\mathbb{R}^{\kappa_{\Gamma}+1}}=\mathbb{R}^{\kappa_{\Gamma}+1}$
according to Proposition 2.3, which contradicts to the definition of
$\kappa_{\Gamma}$.

∎

######  Corollary 2.4.

In addition to (1.2) and (1.3), we assume $f$ is fully uniform ellipticity.
Then the corresponding cone $\Gamma$ is of type 2.

######  Corollary 2.5.

If $\Gamma$ supposes $f$ satisfying (1.2), (1.3) and (1.12) then the following
statements are equivalent each other.

  * •

$\Gamma$ is of type 2.

  * •

$f$ is fully uniform ellipticity.

######  Remark 2.6.

The results on partial uniform ellipticity and interior estimates obtained in
this paper allow $n\geq 2$; while the theorems concerning conformal metrics
hold for $n\geq 3$.

##  3\. Statement of interior estimates

In this section we state our main estimates for solutions of (1.13); while the
proof will be given by combining the results obtained in Sections 4 and 5
below.

We shall focus on solutions in the class of admissible functions satisfying

(3.1) $\displaystyle\lambda(\Delta wg-\varrho\nabla^{2}w+W(x,w,\nabla
w))\in\Gamma\mbox{ in }\bar{M},\mbox{ }w\in C^{2}(\bar{M}).$

As a result, (1.2) and (1.3) correspond respectively to the ellipticity and
concavity of equation (1.13) on admissible solutions. For this purpose we
assume

(3.2) $\displaystyle\sup_{\partial\Gamma}f<\psi(x,w,\nabla
w)<\sup_{\Gamma}f:=\lim_{t\rightarrow+\infty}f(t\vec{\bf 1}),$

for all $x\in\bar{M}$, for each $w\in C^{2}(\bar{M})$, where

$\sup_{\partial\Gamma}f:=\sup_{\lambda_{0}\in\partial\Gamma}\limsup_{\lambda\rightarrow\lambda_{0}}f(\lambda).$

We obtain the following interior estimates for second derivatives.

######  Theorem 3.1.

Let $B_{r}\subset M$ be a geodesic ball of radius $r$. Assume (1.2), (1.3),
(1.12), (1.14) and (3.2) hold. Then any admissible solution $u\in
C^{4}(\bar{B}_{r})$ to (1.13) in $B_{r}$ satisfies

$\displaystyle\sup_{B_{\frac{r}{2}}}|\nabla^{2}u|\leq\frac{C}{r^{2}},$

where $C$ depends on $(1-\varrho(1-\kappa_{\Gamma}\vartheta))^{-1}$,
$|u|_{C^{1}(B_{r})}$ and other known data.

We will prove the interior gradient estimate under the asymptotic assumptions:

(3.3) $\displaystyle|D_{p}A|\leq\gamma(x,z)|p|,\mbox{
}g(D_{p}A,p)\leq\gamma(x,z)|p|^{2}g,$ $\displaystyle
D_{z}A+\frac{1}{|p|^{2}}g(\nabla^{\prime}A,p)\leq\beta(x,z,|p|)|p|^{2}g,$
$\displaystyle|D_{p}\psi|\leq\gamma(x,z)|p|,\mbox{
}-g(D_{p}\psi,p)\leq\gamma(x,z)|p|^{2},$ $\displaystyle-
D_{z}\psi-\frac{1}{|p|^{2}}g(\nabla^{\prime}\psi,p)\leq\beta(x,z,|p|)|p|^{2},$

where $\beta=\beta(x,z,r)$ and $\gamma=\gamma(x,z)$ are positive continuous
functions with

$\displaystyle\lim_{r\rightarrow+\infty}\beta(x,z,r)=0,\,$
$\displaystyle(x,z,r)\in\bar{M}\times\mathbb{R}\times\mathbb{R}^{+}$

and

$\displaystyle
A(x,z,p):=\frac{\mathrm{tr}_{g}W(x,z,p)}{\varrho(n-\varrho)}g-\frac{W(x,z,p)}{\varrho}.$

Throughout this paper, $\psi(x,z,p)$ and $A(x,z,p)$ (as well as $W(x,z,p)$)
denote respectively smooth function and smooth symmetric $(0,2)$-type tensor
of variables $(x,z,p)$, $(x,p)\in T^{*}\bar{M}$, $z\in\mathbb{R}$. For
simplicity $\nabla^{\prime}A$ and $\nabla^{\prime}\psi$ denote the partial
covariant derivative of $A$ and $\psi$, respectively, when viewed as depending
on $x\in M$, and the meanings of $D_{z}A$, $D_{z}\psi$, $D_{p}A$ and
$D_{p}\psi$ are obvious.

######  Theorem 3.2.

Let $B_{r}\subset M$ and $u\in C^{3}(\bar{B}_{r})$ be an admissible solution
to (1.13) in $B_{r}$. In addition to (1.2), (1.3), (1.12), (3.2), (1.14) and
(3.3), we assume furthermore that either $\psi[u]=\psi(x,u)$ or $f$ is a
homogeneous function of degree one. Then

$\sup_{B_{\frac{r}{2}}}|\nabla u|\leq\frac{C}{r},$

where $C$ is a positive constant depending on
$(1-\varrho(1-\kappa_{\Gamma}\vartheta))^{-1}$, $|u|_{C^{0}(B_{r})}$ and other
known data.

##  4\. Assumption (1.14) implies fully uniform ellipticity

The purpose of this section is to verify (1.14) implies the fully uniform
ellipticity of (1.13). First of all, we can check that if $\varrho=1$ then for
the cone $\widetilde{\Gamma}$ given by (4.1) below,

$\displaystyle\kappa_{\widetilde{\Gamma}}=n-1\mbox{ if
}\Gamma\neq\Gamma_{n};\mbox{ while }\kappa_{\widetilde{\Gamma_{n}}}=n-2.$

From Theorem 1.5, equation (1.8) is in fact fully uniform ellipticity if
$\Gamma\neq\Gamma_{n}$.

We have a more general result. Notice (1.14) yields $\varrho<n$, $\varrho\neq
0$. (Clearly $\vartheta\leq\frac{1}{n}$, $\kappa_{\Gamma}\leq n-1$). For $u$
we denote

$\displaystyle U[u]=\Delta ug-\varrho\nabla^{2}u+W(x,u,\nabla u),\,$
$\displaystyle\mathfrak{g}[u]=\nabla^{2}u+A(x,u,\nabla u),$

and as above $A(x,u,\nabla u)=\frac{\mathrm{tr}_{g}W(x,u,\nabla
u)}{\varrho(n-\varrho)}g-\frac{W(x,u,\nabla u)}{\varrho}.$ We also denote
eigenvalues by

$\displaystyle\lambda(U[u])=(\mu_{1},\cdots,\mu_{n}),\,$
$\displaystyle\lambda(\mathfrak{g}[u])=(\lambda_{1},\cdots,\lambda_{n}).$

Then one can check $\mu_{i}=\sum_{j=1}^{n}\lambda_{j}-\varrho\lambda_{i}$. Let
$P^{\prime}:\Gamma\longrightarrow P^{\prime}(\Gamma)=:\widetilde{\Gamma}$ be a
map

(4.1)
$\displaystyle(\mu_{1},\cdots,\mu_{n})\longrightarrow(\lambda_{1},\cdots,\lambda_{n})=(\mu_{1},\cdots,\mu_{n})Q^{-1},$

where $Q=(q_{ij})$ and $q_{ij}=1-\varrho\delta_{ij}$ ($Q$ is symmetric). Here
$Q^{-1}$ is well defined, since
$\mathrm{det}Q=(-1)^{n-1}\varrho^{n-1}(n-\varrho)\neq 0$. So
$\widetilde{\Gamma}$ is also an open symmetric convex cone of
$\mathbb{R}^{n}$, and we then define

$\displaystyle f(\mu)=\tilde{f}(\lambda).$

The equation (1.13) takes the form

(4.2) $\displaystyle\tilde{f}(\lambda(\mathfrak{g}[u]))=\psi(x,u,\nabla u).$

One can verify that if $f$ satisfies (1.2), (1.3) and (1.12) in ${\Gamma}$,
then so does $\tilde{f}$ in $\widetilde{\Gamma}$.

######  Proposition 4.1.

Equation (4.2) is fully uniform ellipticity provided (1.14) holds.

###### Proof.

Case 1: $0<\varrho<\frac{1}{1-\kappa_{\Gamma}\vartheta}$. By Theorem 1.5,
$\sum_{j=1}^{n}\frac{\partial f}{\partial\mu_{j}}-\frac{\partial
f}{\partial\mu_{i}}\geq\kappa_{\Gamma}\vartheta\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}$, i.e.,

$\displaystyle\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}\geq(1-\varrho(1-\kappa_{\Gamma}\vartheta))\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}+\varrho\frac{\partial f}{\partial\mu_{i}}$

for all $1\leq i\leq n$. A straightforward computation shows

$\displaystyle\frac{\partial\tilde{f}}{\partial\lambda_{i}}=\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}\frac{\partial\mu_{j}}{\partial\lambda_{i}}=\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}-\varrho\frac{\partial f}{\partial\mu_{i}}\geq$
$\displaystyle\frac{1-\varrho(1-\kappa_{\Gamma}\vartheta)}{n-\varrho}\sum_{j=1}^{n}\frac{\partial\tilde{f}}{\partial\lambda_{j}}.$

Case 2: $\varrho<0$. One has

$\frac{\partial\tilde{f}}{\partial\lambda_{i}}=\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}-\varrho\frac{\partial
f}{\partial\mu_{i}}\geq\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}=\frac{1}{n-\varrho}\sum_{j=1}^{n}\frac{\partial\tilde{f}}{\partial\lambda_{j}}.$

Hence, (4.2) (equivalently (1.13)) is fully uniform ellipticity if $\varrho$
obeys (1.14).

∎

##  5\. Interior and boundary estimates for certain Hessian type equations

In this section we focus on the Hessian type equations

(5.1) $\displaystyle f(\lambda(\nabla^{2}u+A(x,u,\nabla u)))=\psi(x,u,\nabla
u)=:\psi[u]$

of fully uniform ellipticity

(5.2) $\displaystyle
f_{i}(\lambda)\geq\theta\sum_{j=1}^{n}f_{j}(\lambda)\mbox{ in }\Gamma\mbox{
for some positive constant }\theta,\mbox{ for all }i$

on compact Riemannian manifolds possibly with boundary.

We derive interior and boundary estimates for equation (5.1).

######  Theorem 5.1.

Let $B_{r}\subset M$ and $u\in C^{3}(\bar{B}_{r})$ be an admissible solution
to (5.1) in $B_{r}$. In addition to (1.2), (1.3) and (3.2), we assume (5.2)
and (3.3) hold. Suppose furthermore that there is a positive constant
$\kappa_{0}$ depending not on $\nabla u$ such that

(5.3) $\displaystyle F^{ij}g_{ij}\geq\kappa_{0}.$

Then there is a positive constant $C$ depending on $|u|_{C^{0}(B_{r})}$,
$\theta^{-1}$ and other known data such that

$\sup_{B_{\frac{r}{2}}}|\nabla u|\leq\frac{C}{r}.$

######  Remark 5.2.

According to Lemma 2.1 and (2.1), if (1.2), (1.3), (1.12) hold then

(5.4) $\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)>\frac{f(R\vec{\bf
1})-f(\lambda)}{R}\mbox{ for }R>0,$

moreover, if $f$ is homogeneous of degree one then
$\sum_{i=1}^{n}f_{i}(\lambda)\geq f(\vec{\bf 1})>0.$ Thus, assumption (5.3)
clearly holds if $f$ is a homogeneous function of degree one, or $f$ satisfies
(1.12) and $\psi[u]=\psi(x,u)$.

######  Theorem 5.3.

Let $B_{r}\subset M$. Assume (1.2), (1.3), (1.12), (3.2) and (5.2) hold. Then
for any admissible solution $u\in C^{4}(\bar{B}_{r})$ to equation (5.1) in
$B_{r}$, we have

(5.5) $\displaystyle\sup_{B_{\frac{r}{2}}}|\nabla^{2}u|\leq\frac{C}{r^{2}},$

where $C$ depends on $|u|_{C^{1}(B_{r})}$, $\theta^{-1}$ and other known data.

######  Theorem 5.4.

Suppose (1.2), (1.3), (1.12), (3.2) and (5.2) hold. Let $u\in C^{3}(M)\cap
C^{2}(\bar{M})$ be an admissible solution to Dirichlet problem of equation
(5.1) with $u=\varphi$ on $\partial M$ for $\varphi\in C^{3}(\bar{M})$. Then

$\sup_{\partial M}|\nabla^{2}u|\leq C$

holds for a uniformly positive constant depending on $\theta^{-1}$,
$|u|_{C^{1}(\bar{M})}$, $|\varphi|_{C^{3}(\bar{M})}$ and other known data.

###  5.1. Preliminaries

On a Riemannian manifold $(M,g)$, one defines the curvature tensor by

$R(X,Y)Z=-\nabla_{X}\nabla_{Y}Z+\nabla_{Y}\nabla_{X}Z+\nabla_{[X,Y]}Z.$

Let $e_{1},...,e_{n}$ be a local frame on $M$. We denote
$g_{ij}=g(e_{i},e_{j}),({g^{ij}})=({g_{ij}})^{-1}$. Under Levi-Civita
connection $\nabla$ of $(M,g)$, $\nabla_{e_{i}}e_{j}=\Gamma_{ij}^{k}e_{k}$,
and $\Gamma_{ij}^{k}$ denote the Christoffel symbols and
$\Gamma_{ij}^{k}=\Gamma_{ji}^{k}$. The curvature coefficients are given by
$R_{ijkl}=g(e_{i},R(e_{k},e_{l})e_{j})$.

For simplicity we shall write

$\nabla_{i}=\nabla_{e_{i}},\nabla_{ij}=\nabla_{i}(\nabla_{j})-\Gamma_{ij}^{k}\nabla_{k},\nabla_{ijk}=\nabla_{i}(\nabla_{jk})-\Gamma_{ij}^{l}\nabla_{lk}-\Gamma^{l}_{ik}\nabla_{jl},\mbox{
etc}.$

Denote $\mathfrak{g}[u]=\nabla^{2}u+A(x,u,\nabla u)$ for $u$. Furthermore, for
solution $u$, $\mathfrak{g}=\mathfrak{g}[u]$,
$\mathfrak{g}_{\xi\eta}=\mathfrak{g}(\xi,\eta)$,
$A_{\xi\eta}(x,z,p)=A(x,z,p)(\xi,\eta)$ for $\xi,\eta\in T_{x}^{*}\bar{M}$,
$F^{ij}=\frac{\partial F}{\partial a_{ij}}(\mathfrak{g}[u])$. We now have
$\mathfrak{g}_{ij}=\nabla_{ij}u+A_{ij}(x,u,\nabla u)$ and

(5.6)
$\displaystyle\nabla_{k}\mathfrak{g}_{ij}=\nabla_{kij}u+\nabla^{\prime}_{k}A_{ij}(x,u,\nabla
u)+A_{ij,z}(x,u,\nabla u)\nabla_{k}u+A_{ij,p_{l}}(x,u,\nabla u)\nabla_{kl}u,$

and by differentiating equation (5.1),

(5.7) $\displaystyle
F^{ij}\nabla_{k}\mathfrak{g}_{ij}=\nabla^{\prime}_{k}\psi+\psi_{z}\nabla_{k}u+\psi_{p_{l}}\nabla_{kl}u.$

The linearized operator of equation (5.1) is given by

(5.8)
$\displaystyle\mathcal{L}v=F^{ij}\nabla_{ij}v+(F^{ij}A_{ij,p_{l}}-\psi_{p_{l}})\nabla_{l}v\mbox{
for }v\in C^{2}(\bar{M}).$

Let $\lambda=\lambda(\mathfrak{g})$. The matrix $\\{F^{ij}\\}$ has eigenvalues
$f_{1},\cdots,f_{n}$ which satisfy

$\displaystyle F^{ij}g_{ij}=\sum_{i=1}^{n}f_{i},\mbox{ }\,$ $\displaystyle
F^{ij}\mathfrak{g}_{ij}=\sum_{i=1}^{n}f_{i}\lambda_{i}.$

###  5.2. Interior gradient estimate

Let $w=|\nabla u|^{2}$. We consider the quantity

$Q:=\eta we^{\phi}$

where $\phi$ is determined later. Following [19] let $\eta$ be a smooth
function with compact support in $B_{r}\subset M$ and

(5.9) $\displaystyle 0\leq\eta\leq 1,\mbox{ }\eta|_{B_{\frac{r}{2}}}\equiv
1,\mbox{ }|\nabla\eta|\leq\frac{C\sqrt{\eta}}{r},\mbox{
}|\nabla^{2}\eta|\leq\frac{C}{r^{2}}.$

The quantity $Q$ must attain its maximum at an interior point $x_{0}\in M$. We
may assume $|\nabla u|(x_{0})\geq 1$. By maximum principle at $x_{0}$

(5.10)
$\displaystyle\frac{\nabla_{i}\eta}{\eta}+\frac{\nabla_{i}w}{w}+\nabla_{i}\phi=0,\,$
$\displaystyle\mathcal{L}(\log\eta+\log w+\phi)\leq 0.$

Around $x_{0}$ we choose a local orthonormal frame $e_{1},\cdots,e_{n}$; and
for simplicity, we further $e_{1},\cdots,e_{n}$ has been chosen such that at
$x_{0}$, $\Gamma_{ij}^{k}=0$, $\mathfrak{g}_{ij}$ is diagonal (so is
$F^{ij}$).

#### Step 1. Computation and estimation for $\mathcal{L}(\log w)$

Using (5.7), $\nabla_{ij}u=\nabla_{ji}u$ and
$\nabla_{ijk}u=\nabla_{kij}u-R_{ljik}\nabla_{l}u$,

(5.11) $\displaystyle
F^{ij}\nabla_{ijk}u+(F^{ij}A_{ij,p_{l}}-\psi_{p_{l}})\nabla_{lk}u$
$\displaystyle=$
$\displaystyle\nabla^{\prime}_{k}\psi+\psi_{z}\nabla_{k}u-F^{ij}R_{ljik}\nabla_{l}u-F^{ij}(\nabla^{\prime}_{k}A_{ij}+A_{ij,z}\nabla_{k}u).$

Together with straightforward computation

(5.12) $\displaystyle\nabla_{i}w=2\nabla_{k}u\nabla_{ik}u,\,$
$\displaystyle\nabla_{ij}w=2\nabla_{ik}u\nabla_{jk}u+2\nabla_{k}u\nabla_{ijk}u,$

one has

(5.13) $\displaystyle\mathcal{L}w=$ $\displaystyle
2F^{ij}\nabla_{ik}u\nabla_{jk}u+2\left\\{F^{ij}\nabla_{ijk}u+(F^{ij}A_{ij,p_{l}}-\psi_{p_{l}})\nabla_{lk}u\right\\}\nabla_{k}u$
$\displaystyle=$ $\displaystyle
2F^{ij}\nabla_{ik}u\nabla_{jk}u-2F^{ij}R_{ljik}\nabla_{k}u\nabla_{l}u$
$\displaystyle+2\nabla^{\prime}_{k}\psi\nabla_{k}u+2\psi_{z}|\nabla
u|^{2}-2F^{ij}(\nabla^{\prime}_{k}A_{ij}\nabla_{k}u+A_{ij,z}|\nabla u|^{2}).$

By (5.12) and Cauchy-Schwarz inequality, one derives $|\nabla_{i}w|\leq
4|\nabla u|^{2}|\nabla_{ik}u|^{2}$,

$\displaystyle F^{ij}\nabla_{i}w\nabla_{j}w\leq 4|\nabla
u|^{2}F^{ij}\nabla_{ik}u\nabla_{jk}u,$ $\displaystyle
F^{ij}\frac{\nabla_{i}w\nabla_{j}w}{w^{2}}\leq(1+\epsilon)\left(\frac{1}{\epsilon}F^{ij}\frac{\nabla_{i}\eta\nabla_{j}\eta}{\eta^{2}}+F^{ij}\nabla_{i}\phi\nabla_{j}\phi\right),$

which yields

$\displaystyle
F^{ij}\frac{\nabla_{i}w\nabla_{j}w}{w^{2}}\leq(1-\epsilon^{2})\left(\frac{1}{\epsilon}F^{ij}\frac{\nabla_{i}\eta\nabla_{j}\eta}{\eta^{2}}+F^{ij}\nabla_{i}\phi\nabla_{j}\phi\right)+\frac{4\epsilon}{w}F^{ij}\nabla_{ik}u\nabla_{jk}u.$

Set $0<\epsilon\leq\frac{1}{4}$. We now obtain

(5.14) $\displaystyle\mathcal{L}(\log w)\geq$
$\displaystyle\frac{1}{w}F^{ij}\nabla_{ik}u\nabla_{jk}u+2(\psi_{z}+\frac{1}{w}\nabla^{\prime}_{k}\psi\nabla_{k}u)$
$\displaystyle-2F^{ij}(A_{ij,z}+\frac{1}{w}\nabla^{\prime}_{k}A_{ij}\nabla_{k}u)-\frac{1-\epsilon^{2}}{\epsilon}F^{ij}\frac{\nabla_{i}\eta\nabla_{j}\eta}{\eta^{2}}$
$\displaystyle-(1-\epsilon^{2})F^{ij}\nabla_{i}\phi\nabla_{j}\phi-C_{0}\sum
F^{ii}.$

#### Step 2. Construction and computation of $\phi$

As in [14] (see also [17]) let $v=u-\inf_{B_{r}}u+2$ ($v\geq 2$ in $B_{r}$)
and $\phi=v^{-N}$ where $N\geq 1$ is an integer that is chosen later.

(5.15) $\displaystyle\nabla_{i}\phi=-Nv^{-N-1}\nabla_{i}u,\,$
$\displaystyle\nabla_{ij}\phi=N(N+1)v^{-N-2}\nabla_{i}u\nabla_{j}u-Nv^{-N-1}\nabla_{ij}u$

which implies

(5.16) $\displaystyle\mathcal{L}\phi=$ $\displaystyle
N(N+1)v^{-N-2}F^{ij}\nabla_{i}u\nabla_{j}u-Nv^{-N-1}F^{ij}\nabla_{ij}u$
$\displaystyle-Nv^{-N-1}(F^{ij}A_{ij,p_{l}}-\psi_{p_{l}})\nabla_{l}u.$

#### Step 3. Coumputation for $\mathcal{L}(\log\eta)$

(5.17)
$\displaystyle\mathcal{L}(\log\eta)=\frac{F^{ij}\nabla_{ij}\eta}{\eta}-\frac{F^{ij}\nabla_{i}\eta\nabla_{j}\eta}{\eta^{2}}+(F^{i\bar{j}}A_{ij,p_{l}}-\psi_{p_{l}})\frac{\nabla_{l}\eta}{\eta}.$

#### Step 4. Completion of the proof

By (5.2),

(5.18) $\displaystyle
F^{ij}\nabla_{ik}u\nabla_{jk}u\geq\theta|\nabla_{ik}u|^{2}\sum F^{ii},\,$
$\displaystyle F^{ij}\nabla_{i}u\nabla_{j}u\geq\theta w\sum F^{ii}.$

By Cauchy-Schwarz inequality again, one derives

$\displaystyle\frac{1}{8}N^{2}v^{-N-2}\theta
w+\frac{\theta}{2w}|\nabla_{ik}u|^{2}\geq\frac{1}{2}N\theta|\nabla_{ik}u|v^{-\frac{N}{2}-1}.$

We choose $N\gg 1$ so that

$\displaystyle\frac{1}{4}N\theta v^{-\frac{N}{2}-1}\geq Nv^{-N-1},\,$
$\displaystyle N(N+1)v^{-N-2}-N^{2}v^{-2N-2}\geq\frac{N^{2}}{2}v^{-N-2}.$

Suppose furthermore that $\frac{1}{8}Nv^{-N-2}\theta w\geq C_{0}$ where
$C_{0}$ is as in (5.14) (otherwise we are done). Together with (5.10), (5.14),
(5.16) and (5.17), we derive

(5.19) $\displaystyle 0\geq$
$\displaystyle\frac{1}{w}F^{ij}\nabla_{ik}u\nabla_{jk}u+2(\psi_{z}+\frac{1}{w}\nabla^{\prime}_{k}\psi\nabla_{k}u)-2F^{ij}(A_{ij,z}+\frac{1}{w}\nabla^{\prime}_{k}A_{ij}\nabla_{k}u)$
$\displaystyle+\\{N(N+1)v^{-N-2}-(1-\epsilon^{2})N^{2}v^{-2N-2}\\}F^{ij}\nabla_{i}u\nabla_{j}u-Nv^{-N-1}F^{ij}\nabla_{ij}u$
$\displaystyle+(F^{ij}A_{ij,p_{l}}-\psi_{p_{l}})\frac{\nabla_{l}\eta}{\eta}+\frac{F^{ij}\nabla_{ij}\eta}{\eta}-\frac{1+\epsilon-\epsilon^{2}}{\epsilon}F^{ij}\frac{\nabla_{i}\eta\nabla_{j}\eta}{\eta^{2}}$
$\displaystyle-Nv^{-N-1}(F^{ij}A_{ij,p_{l}}-\psi_{p_{l}})\nabla_{l}u-C_{0}\sum
F^{ii}$ $\displaystyle\geq$ $\displaystyle\frac{1}{4}\theta N^{2}wv^{-N-2}\sum
F^{ii}+\frac{1}{4}N\theta|\nabla_{ik}u|v^{-\frac{N}{2}-1}\sum
F^{ii}+2(\psi_{z}+\frac{1}{w}\nabla^{\prime}_{k}\psi\nabla_{k}u)$
$\displaystyle-2F^{ij}(A_{ij,z}+\frac{1}{w}\nabla^{\prime}_{k}A_{ij}\nabla_{k}u)-Nv^{-N-1}(F^{ij}A_{ij,p_{l}}-\psi_{p_{l}})\nabla_{l}u$
$\displaystyle+(F^{ij}A_{ij,p_{l}}-\psi_{p_{l}})\frac{\nabla_{l}\eta}{\eta}+\frac{F^{ij}\nabla_{ij}\eta}{\eta}-\frac{1+\epsilon-\epsilon^{2}}{\epsilon}F^{ij}\frac{\nabla_{i}\eta\nabla_{j}\eta}{\eta^{2}}.$

Using (5.9) and the asymptotic assumption (3.3)

(5.20) $\displaystyle 0\geq$ $\displaystyle\frac{1}{4}\theta
N^{2}wv^{-N-2}\sum
F^{ii}+\frac{1}{4}N\theta|\nabla_{ik}u|v^{-\frac{N}{2}-1}\sum F^{ii}$
$\displaystyle-\left(\beta(x,u,|\nabla
u|)w+CNv^{-N-1}w+\frac{C\sqrt{w}}{r\sqrt{\eta}}\right)\left(1+\sum
F^{ii}\right)-\frac{C}{r^{2}\eta}\sum F^{ii}.$

This gives $\eta w\leq\frac{C}{r^{2}}$ and completes the proof.

###  5.3. Interior estimates for second derivatives

Let’s consider the function

$P(x)=\max_{\xi\in T_{x}\bar{M},|\xi|=1}\eta\mathfrak{g}(\xi,\xi)e^{\varphi},$

where $\eta$ is the cutoff function as given by (5.9) and $\varphi$ is a
function to be chosen later. One knows $P$ achieves maximum at an interior
point $x_{0}\in B_{r}$ and for $\xi\in T_{x_{0}}\bar{M}$. Around $x_{0}$ we
choose a smooth orthonormal local frame $e_{1},\cdots,e_{n}$ such that
$e_{1}(x_{0})=\xi$, $\Gamma_{ij}^{k}(x_{0})=0$ and
$\\{\mathfrak{g}_{ij}(x_{0})\\}$ is diagonal (so is $\\{F^{ij}(x_{0})\\}$). We
may assume $\mathfrak{g}_{11}(x_{0})\geq 1$. At $x_{0}$ one has

(5.21)
$\displaystyle\nabla_{i}\varphi+\frac{\nabla_{i}\mathfrak{g}_{11}}{\mathfrak{g}_{11}}+\frac{\nabla_{i}\eta}{\eta},\,$
$\displaystyle\mathcal{L}(\varphi+\log\eta+\log\mathfrak{g}_{11})\leq 0.$

#### Step 1. Estimation for $\mathcal{L}(\mathfrak{g}_{11})$

Differentiating the equation, one has

$\displaystyle F^{ii}\nabla_{11}\mathfrak{g}_{ii}=$
$\displaystyle-F^{ij,kl}\nabla_{1}\mathfrak{g}_{ij}\nabla_{1}\mathfrak{g}_{kl}+\nabla^{\prime}_{11}\psi+2\nabla^{\prime}_{1}\psi_{z}\nabla_{1}u+2\nabla^{\prime}_{1}\psi_{p_{l}}\nabla_{1l}u+\psi_{z}\nabla_{11}u$
$\displaystyle+2\psi_{zp_{l}}\nabla_{1l}u\nabla_{1}u+\psi_{p_{l}p_{m}}\nabla_{1l}u\nabla_{1m}u+\psi_{p_{l}}\nabla_{11l}u+\psi_{zz}|\nabla_{1}u|^{2},$

which then follows

(5.22) $\displaystyle
F^{ii}\nabla_{11}\mathfrak{g}_{ii}\geq\psi_{p_{l}}\nabla_{l}\mathfrak{g}_{11}-C_{0}\mathfrak{g}_{11}^{2}.$

The straightforward computation shows

(5.23) $\displaystyle
F^{ii}(\nabla_{ii}\mathfrak{g}_{11}-\nabla_{11}\mathfrak{g}_{ii})\geq$
$\displaystyle
F^{ii}\nabla_{l}\mathfrak{g}_{ii}A_{11,p_{l}}-F^{ii}A_{ii,p_{l}}\nabla_{l}\mathfrak{g}_{11}-C_{1}\mathfrak{g}_{11}^{2}\sum_{i=1}^{n}F^{ii}$
$\displaystyle\geq$
$\displaystyle-F^{ii}A_{ii,p_{l}}\nabla_{l}\mathfrak{g}_{11}-C_{2}\mathfrak{g}_{11}^{2}\sum_{i=1}^{n}F^{ii}-C_{2}\mathfrak{g}_{11}.$

Here we use
$F^{ii}\nabla_{l}\mathfrak{g}_{ii}=\nabla^{\prime}_{l}\psi+\psi_{z}\nabla_{l}u+\psi_{p_{k}}\nabla_{lk}u$.
Combing (5.22) and (5.23),

$\displaystyle\mathcal{L}(\mathfrak{g}_{11})=$ $\displaystyle
F^{ii}\nabla_{ii}\mathfrak{g}_{11}+(F^{ii}A_{ii,p_{l}}-\psi_{p_{l}})\nabla_{l}\mathfrak{g}_{11}$
$\displaystyle\geq$ $\displaystyle-
C_{3}\mathfrak{g}_{11}^{2}(1+\sum_{i=1}^{n}F^{ii}).$

Using (5.21) and Cauchy-Schwarz inequality,

$\displaystyle
F^{ii}\frac{|\nabla_{i}\mathfrak{g}_{11}|^{2}}{\mathfrak{g}_{11}^{2}}\leq\frac{3}{2}F^{ii}|\nabla_{i}\varphi|^{2}+3F^{ii}\frac{|\nabla_{i}\eta|^{2}}{\eta^{2}}.$

Hencefore,

(5.24) $\displaystyle\mathcal{L}(\log\mathfrak{g}_{11})\geq-
C_{3}\mathfrak{g}_{11}(1+\sum_{i=1}^{n}F^{ii})-\frac{3}{2}F^{ii}|\nabla_{i}\varphi|^{2}-3F^{ii}\frac{|\nabla_{i}\eta|^{2}}{\eta^{2}}.$

#### Step 2. Construction and estimate for $\mathcal{L}\varphi$

Following [14] we set

$\varphi=\varphi(w)=(1-\frac{w}{2N})^{-\frac{1}{2}}\mbox{ where $w=|\nabla
u|^{2}$ and $N=\sup_{\\{\eta>0\\}}|\nabla u|^{2}$}.$

One can check

$\displaystyle\varphi^{\prime}=\frac{\varphi^{3}}{4N},\,$
$\displaystyle\varphi^{\prime\prime}=\frac{3\varphi^{5}}{16N^{2}},\mbox{ }$
$\displaystyle 1\leq\varphi\leq\sqrt{2}.$

By (5.13), (5.2) and Cauchy-Schwarz inequality, we obtain

$\displaystyle\mathcal{L}w\geq\frac{1}{2}F^{ii}\mathfrak{g}_{ii}^{2}-C_{4}\sum_{i=1}^{n}F^{ii}\geq(\frac{\theta}{2}\mathfrak{g}_{11}^{2}-C_{4})\sum_{i=1}^{n}F^{ii}.$

Finally,

(5.25) $\displaystyle\mathcal{L}\varphi=$
$\displaystyle\varphi^{\prime\prime}F^{ii}|\nabla_{i}w|^{2}+\varphi^{\prime}\mathcal{L}w=\frac{3\varphi^{5}}{16N^{2}}F^{ii}|\nabla_{i}w|^{2}+\frac{\varphi^{3}}{4N}\mathcal{L}w$
$\displaystyle\geq$
$\displaystyle\frac{3\varphi^{5}}{16N^{2}}F^{ii}|\nabla_{i}w|^{2}+\frac{\varphi^{3}}{4N}(\frac{\theta}{2}\mathfrak{g}_{11}^{2}-C_{4})\sum_{i=1}^{n}F^{ii}.$

#### Step 3. Completion of the proof

From (5.9),

$\displaystyle
F^{ii}\frac{|\nabla_{i}\eta|^{2}}{\eta^{2}}\leq\frac{C}{r^{2}\eta}\sum_{i=1}^{n}F^{ii},\,$
$\displaystyle\mathcal{L}(\log\eta)\geq-\frac{C}{r^{2}\eta}\sum_{i=1}^{n}F^{ii}-\frac{C}{r\sqrt{\eta}}(1+\sum_{i=1}^{n}F^{ii}).$

We next verify

$\displaystyle\frac{3}{2}F^{ii}|\nabla_{i}\varphi|^{2}=\frac{3\varphi^{6}}{32N^{2}}F^{ii}|\nabla_{i}w|^{2}\leq\frac{3\sqrt{2}\varphi^{5}}{32N^{2}}F^{ii}|\nabla_{i}w|^{2}\leq\frac{3\varphi^{5}}{16N^{2}}F^{ii}|\nabla_{i}w|^{2}.$

Finally, we achieve our goal, at $x_{0}$,

(5.26) $\displaystyle 0\geq$
$\displaystyle\frac{3}{4N}\left(\frac{\theta}{2}\mathfrak{g}_{11}^{2}-\frac{4NC_{3}}{3}\mathfrak{g}_{11}-C_{4}\right)\sum_{i=1}^{n}F^{ii}-C_{3}\mathfrak{g}_{11}$
$\displaystyle-\frac{C}{r^{2}\eta}\sum_{i=1}^{n}F^{ii}-\frac{C}{r\sqrt{\eta}}(1+\sum_{i=1}^{n}F^{ii})$

Combining with (5.4) we have

$\displaystyle\eta\mathfrak{g}_{11}\leq\frac{C}{r^{2}}\mbox{ at }x_{0}$

and therefore finish the proof.

###  5.4. Boundary estimates

For a boundary point $x_{0}$ we choose a smooth orthonormal local frame
$e_{1},\cdots,e_{n}$ around $x_{0}$ such that $e_{n}$ is unit outer normal
vector field when restricted to $\partial M$. As in Theorem 1.2,
$\mathrm{d}(x)$ is the distance function from $x$ to $\partial M$. Denote by
$\rho(x)$ the distance function from $x$ to $x_{0}$,

(5.27) $\displaystyle\rho(x):=\mathrm{dist}_{g}(x,x_{0}).$

Then $\mathrm{d}$ is smooth near the boundary (whenever $\partial M$ is
smooth), i.e., $\mathrm{d}$ is smooth and $|\nabla\mathrm{d}|\geq\frac{1}{2}$
in $M_{\delta}$ for small $\delta>0$, where

$\displaystyle M_{\delta}:=\\{x\in M:\rho(x)<\delta\\}.$

#### Case 1. Pure tangential derivatives

From the boundary value condition, on $\partial M$

(5.28) $\displaystyle\nabla_{\alpha}u=\nabla_{\alpha}\varphi,\,$
$\displaystyle\nabla_{\alpha\beta}u=\nabla_{\alpha\beta}\varphi+\nabla_{n}(u-\varphi)\mathrm{II}(e_{\alpha},e_{\beta})$

for $1\leq\alpha,\beta<n$, where $\mathrm{II}$ is the fundamental form of
$\partial M$. Which also gives the bound of second estimates for pure
tangential derivatives

(5.29) $\displaystyle|\mathfrak{g}_{\alpha\beta}(x_{0})|\leq C.$

#### Case 2. Mixed derivatives

For $1\leq\alpha<n$ the local barrier function $\Psi$ on $M_{\delta}$
($\delta$ is small) is given by

(5.30)
$\displaystyle\Psi=\pm\nabla_{\alpha}(u-\varphi)+A_{1}(\frac{N\mathrm{d}^{2}}{2}-t\mathrm{d})-A_{2}\rho^{2}+A_{3}\sum_{k=1}^{n-1}|\nabla_{k}(u-\varphi)|^{2}.$

Here $A_{1}$, $A_{2}$, $A_{3}$, $N$, $t$ are all positive constants to be
determined, and

$N\delta-2t\leq 0.$

Next, we collect and derive some useful inequalities used here. Clearly,

$\displaystyle\mathcal{L}(\mathrm{d})\leq C_{d}(1+\sum F^{ii}),\,$
$\displaystyle\mathcal{L}(\rho^{2})\leq C_{\rho}(1+\sum F^{ii}).$

To estimate the local barrier functions, we need the following formula (see
e.g. [15])

$\displaystyle\nabla_{ij}(\nabla_{k}u)=\nabla_{ijk}+\Gamma_{ik}^{l}\nabla_{jl}u+\Gamma_{jk}^{l}\nabla_{il}u+\nabla_{\nabla_{ij}e_{k}}u.$

Combining with (5.11) one derives

$\displaystyle|\mathcal{L}(\nabla_{k}(u-\varphi))|\leq
C_{1}(1+\sum_{i=1}^{n}f_{i}+\sum_{i=1}^{n}f_{i}|\lambda_{i})|.$

As in (5.13) we can prove

$\displaystyle\mathcal{L}(|\nabla_{k}(u-\varphi)|^{2})\geq
F^{ij}\mathfrak{g}_{ik}\mathfrak{g}_{jk}-C_{1}(1+\sum_{i=1}^{n}f_{i}+\sum_{i=1}^{n}f_{i}|\lambda_{i})|.$

By Proposition 2.19 of [15], there exists an index $1\leq r\leq n$

$\displaystyle\sum_{k=1}^{n-1}F^{ij}\mathfrak{g}_{ik}\mathfrak{g}_{jk}\geq\frac{1}{2}\sum_{i\neq
r}f_{i}\lambda_{i}^{2}.$

Since $f$ satisfies (5.2), i.e., $f$ is of fully uniform ellipticity

$f_{i}\geq\theta\sum_{j=1}^{n}f_{j}\mbox{ for all }i,$

as in [16], we can prove there are $c_{0}$, $C_{0}$ depending on $\theta$ and
other known data so that

(5.31) $\displaystyle\sum_{i\neq r}f_{i}\lambda_{i}^{2}\geq
c_{0}|\lambda|^{2}\sum_{i=1}^{n}f_{i}-C_{0}\sum_{i=1}^{n}f_{i}.$

More precisely, if $\lambda_{r}\leq 0$ then
$|\lambda_{r}|=-\lambda_{r}<\sum_{i\neq r}\lambda_{i}$ and $\sum_{i\neq
r}\lambda_{i}^{2}\geq\frac{|\lambda|^{2}}{n}.$ For
$\lambda\in\partial\Gamma^{\sigma}$, if $\lambda_{r}>0$ then
$(\theta\sum_{i=1}^{n}f_{i})^{2}\lambda_{r}^{2}\leq(f_{r}\lambda_{r})^{2}\leq(-\sum_{i\neq
r}f_{i}\lambda_{i}+c_{\sigma}\sum_{i=1}^{n}f_{i})^{2}$ which implies

$\sum_{i\neq
r}\lambda_{i}^{2}\geq\frac{\theta^{2}|\lambda^{2}|}{2(n-1)+\theta^{2}}-\frac{2c_{\sigma}^{2}}{2(n-1)+\theta^{2}},$

here as denoted above $f(c_{\sigma}\vec{\bf 1})=\sigma.$ In fact we get
(5.31).

On the other hand

$\displaystyle\mathcal{L}(\frac{N\mathrm{d}^{2}}{2}-t\mathrm{d})=$
$\displaystyle
NF^{ij}\nabla_{i}\mathrm{d}\nabla_{j}\mathrm{d}+(N\mathrm{d}-t)\mathcal{L}\mathrm{d}$
$\displaystyle\geq$
$\displaystyle\frac{N\theta}{4}\sum_{i=1}^{n}f_{i}-C_{d}|N\mathrm{d}-t|(1+\sum
F^{ii}),\mbox{ }N>0.$

By (5.4) there is $\kappa>0$ so that $\sum_{i=1}^{n}f_{i}=\sum
F^{ii}\geq\kappa>0.$ Thus

$\displaystyle\mathcal{L}(\frac{N\mathrm{d}^{2}}{2}-t\mathrm{d})\geq$
$\displaystyle\frac{N\kappa\theta}{4(1+\kappa)}(1+\sum_{i=1}^{n}f_{i})-C_{d}|N\mathrm{d}-t|(1+\sum
F^{ii})$ $\displaystyle\geq$
$\displaystyle\frac{N\kappa\theta}{8(1+\kappa)}(1+\sum_{i=1}^{n}f_{i})$

if $\delta$ and $t$ are chosen small enough so that

$C_{d}\max\\{|N\delta-t|,t\\}\leq\frac{N\kappa\theta}{8(1+\kappa)}.$

Putting inequalities above together

$\displaystyle\mathcal{L}(\Psi)\geq$
$\displaystyle\frac{A_{1}N\theta\kappa}{8(1+\kappa)}(1+\sum_{i=1}^{n}f_{i})+\frac{A_{3}c_{0}}{2}|\lambda|^{2}\sum_{i=1}^{n}f_{i}-C_{1}(1+A_{3})\sum_{i=1}^{n}f_{i}|\lambda_{i}|$
$\displaystyle-\frac{A_{3}C_{0}}{2}\sum_{i=1}^{n}f_{i}-(A_{2}C_{\rho}+C_{1}(1+A_{3}))(1+\sum_{i=1}^{n}f_{i})$
$\displaystyle\geq$
$\displaystyle\frac{A_{1}N\kappa\theta}{8(1+\kappa)}(1+\sum_{i=1}^{n}f_{i})-(\frac{A_{3}C_{0}}{2}+\frac{C_{1}^{2}(1+A_{3})^{2}}{2A_{3}c_{0}})\sum_{i=1}^{n}f_{i}$
$\displaystyle-(A_{2}C_{\rho}+C_{1}(1+A_{3}))(1+\sum_{i=1}^{n}f_{i})>0\mbox{
in }M_{\delta}$

if $A_{1}\gg A_{2}$, $A_{1}\gg A_{3}>1$. On the other hand $\Psi|_{\partial
M}=-A_{2}\rho^{2}\leq 0$; while on $\partial M_{\delta}\setminus\partial M$,
$\rho=\delta$, so $\Psi\leq 0$ if $N\delta-2t\leq 0$, $A_{2}\gg 1$ and
$A_{2}\gg 1$.

Note that $\Psi(x_{0})=0$. By Hopf lemma one derives the second estimates for
mixed derivatives

(5.32) $\displaystyle|\mathfrak{g}_{\alpha n}(x_{0})|\leq C\mbox{ for
}1\leq\alpha\leq n-1$

#### Case 3. Double normal derivatives

Fix $x_{0}\in\partial M$. As above $c_{\psi[u](x_{0})}$ denotes the positive
constant such that

$f(c_{\psi[u](x_{0})}\vec{\bf 1})=\psi[u](x_{0}).$

Since $\mathrm{tr}_{g}(\mathfrak{g})>0$, we have $\mathfrak{g}_{nn}\geq-C$
since (5.29) and (5.32). In what follows we assume
$\mathfrak{g}_{nn}(x_{0})\geq c_{\psi[u](x_{0})}$ (otherwise we are done).
Since (5.2), $\\{F^{ij}\\}\geq\theta\sum_{k=1}^{n}f_{k}\delta_{ij}$. (Here
$g_{ij}(x_{0})=\delta_{ij}$). By the concavity of equation, (5.29) and (5.32)

$\displaystyle 0=F(\mathfrak{g})-F(c_{\psi[u](x_{0})}g)\geq$ $\displaystyle
F^{ij}(\mathfrak{g})(\mathfrak{g}_{ij}-c_{\psi[u](x_{0})}\delta_{ij})$
$\displaystyle\geq$
$\displaystyle-C^{\prime}\sum_{i=1}^{n}f_{i}(\lambda)+\theta(\mathfrak{g}_{nn}-c_{\psi[u](x_{0})})\sum_{i=1}^{n}f_{i}(\lambda),$

which further gives $\mathfrak{g}_{nn}(x_{0})\leq C.$

######  Remark 5.5.

In proof of interior and boundary estimates we do not use subsolutions.

######  Remark 5.6.

The cone $\widetilde{\Gamma}$, corresponding to $\Gamma$ constructed as in
(4.1), is an open, symmetric and convex cone in $\mathbb{R}^{n}$ with vertex
at origin. It should be notable that $\widetilde{\Gamma}$ contains the
positive cone $\Gamma_{n}$ if $\varrho\leq 1$ and $\varrho\neq 0$ (thus, it is
of type 2 according to Corollary 2.4), while it may not contain $\Gamma_{n}$
when $1<\varrho<\frac{1}{1-\kappa_{\Gamma}\vartheta}.$ Fortunately, from the
proof of interior and boundary estimates above, we can check that this does
not affect the proof. So the estimates obtained there still hold for (1.13)
even $1<\varrho<\frac{1}{1-\kappa_{\Gamma}\vartheta}$.

##  6\. Existence and asymptotic behavior of complete conformal metrics

In addition to (1.2) and (1.3), we assume throughout this section that $\nu$
is the outer unit normal vector field along $\partial M$, $\psi\in
C^{\infty}(\bar{M}),\psi>0\mbox{ in }\bar{M}$, and $f$ is homogeneous of
degree one. Using (2.1), $\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}=f(\lambda)$,
$f(\vec{\bf 1})=1$, we derive $f_{i}(\vec{\bf 1})=\frac{1}{n}$ for all $i$,
and

(6.1) $\displaystyle nf(\lambda)\leq\sum_{i=1}^{n}\lambda_{i},\,$
$\displaystyle\forall\lambda\in\Gamma.$

We observe that (6.1) is a key ingredient in the study of asymptotic behavior.
Also, we denote

$U[u]:=\Delta ug-\nabla^{2}u+\frac{n-3}{2}|\nabla u|^{2}g+du\otimes du.$

From (1.10),
$\mathfrak{g}[u]=\frac{1}{n-2}G_{\tilde{g}}=\frac{1}{n-2}G_{g}+U[u]\mbox{ for
}\tilde{g}=e^{2u}g.$ Without loss of generality, we assume the background
metric $g$ is admissible with $\lambda(G_{g})\in\Gamma$ in $\bar{M}$.

###  6.1. Proof of Theorem 1.1

First, we are going to study Dirichlet problem for equation (1.8):

(6.2) $\displaystyle
f(\lambda(\mathfrak{g}[{u}])=\frac{\psi}{n-2}e^{2{u}}\mbox{ in }M,\,$
$\displaystyle u=\varphi\mbox{ on }\partial M.$

######  Proposition 6.1.

Given $\varphi\in C^{\infty}(\partial M)$. The Dirichlet problem (6.2) admits
a unique smooth admissible solution provided that there is an admissible
function $\underline{u}\in C^{2}(\bar{M})$.

The conformal metrics with prescribed boundary metric also follows.

######  Theorem 6.2.

Let $\psi\in C^{\infty}(\bar{M})$, $\psi>0$ in $\bar{M}$. Suppose (1.2),
(1.3), (1.4), (1.5) and $\Gamma\neq\Gamma_{n}$ hold. For a Riemannian metric
$h$ on $\partial M$ which is conformal to $g|_{\partial M}$, there is a smooth
conformal metric $\tilde{g}$ with prescribed boundary condition
$\tilde{g}|_{\partial M}=h$ to satisfy (1.1). Moreover,
$\lambda_{\tilde{g}}(G_{\tilde{g}})\in\Gamma\mbox{ in }\bar{M}$.

We let $\eta\equiv 1$ in proof of interior estimates, there is a positive
constant $C$ depending on $|u|_{C^{0}(\bar{M})}$ such that

(6.3) $\displaystyle\sup_{M}|\nabla u|^{2}+\sup_{M}|\nabla^{2}u|\leq
C(1+\sup_{\partial M}|\nabla u|^{2}+\sup_{\partial M}|\nabla^{2}u|).$

The boundary estimates for second derivatives are already obtained in previous
section. It remains to prove

######  Lemma 6.3.

For any $C^{2}$-admissible solution $u$ to Dirichlet problem (6.2), we have

$\displaystyle\sup_{M}|u|\leq C_{0}^{\prime},$

where $C_{0}^{\prime}$ is a uniformly positive constant.

###### Proof.

Without loss of generality, we assume $g$ is admissible, i.e.
$\lambda_{g}(G_{g})\in\Gamma$. If $u(x_{0})=\min_{\bar{M}}u$ for some
$x_{0}\in M$ then $\frac{\psi(x_{0})}{n-2}e^{2u(x_{0})}\geq
f(\lambda(\frac{1}{n-2}G_{g}))$. We get $u(x_{0})\geq-C^{\prime}$, and
$\inf_{M}u\geq-\min\\{C^{\prime},\min_{\partial M}\varphi\\}$. Similarly,
$\sup_{M}u\leq C^{\prime\prime}.$ ∎

######  Lemma 6.4.

Let $u$ be a $C^{2}$-admissible solution to Dirichlet problem (6.2), there is
a uniformly positive constant $C$ such that $\nabla_{\nu}u|_{\partial M}\leq
C.$

######  Lemma 6.5.

For any $C^{2}$-admissible solution $u$ to Dirichlet problem (6.2), we have

$\displaystyle-C_{1}^{\prime}\leq\nabla_{\nu}u|_{\partial M}.$

where $C_{1}^{\prime}$ is a uniformly positive constant.

The proof of Lemmas 6.4 and 6.5 is standard by combining maximum principle and
the existence of local lower and upper barriers near boundary (see e.g. [12]).
The construction here is inspired by [14].

Let $\mathrm{d}(x)$ be as defined in Theorem 1.2 the distance function to
boundary. Then we know $\mathrm{d}(x)$ is smooth near the boundary (whenever
$\partial M$ is smooth), i.e., for small $\delta>0$, $\mathrm{d}(x)$ is smooth
in

$\Omega_{\delta}:=\\{x\in M:0<\mathrm{d}(x)<\delta\\}.$

Let $\underline{h}$ be of the form $\underline{h}=\underline{h}(\mathrm{d})$.
By direct computation one obtains

######  Lemma 6.6.

$U[\underline{h}]=(\underline{h}^{\prime\prime}+\frac{(n-3)\underline{h}^{\prime
2}}{2})|\nabla\mathrm{d}|^{2}g+(\underline{h}^{\prime
2}-\underline{h}^{\prime\prime})d\mathrm{d}\otimes
d\mathrm{d}+\underline{h}^{\prime}(\Delta\mathrm{d}g-\nabla^{2}\mathrm{d}).$

We hope to construct a (local) barrier $\underline{h}$ defined in
$\bar{\Omega}_{\delta}$ satisfying, for $0<\delta\ll 1$,

  1. (1)

$\underline{h}^{\prime 2}\geq\underline{h}^{\prime\prime},$
$\underline{h}^{\prime\prime}+\frac{n-3}{2}\underline{h}^{\prime
2}>c_{0}\underline{h}^{\prime 2}$ in $\Omega_{\delta}$ for some $c_{0}>0$.

  2. (2)

In $\Omega_{\delta}$, $\underline{h}^{\prime 2}e^{-2\underline{h}}\gg 1$ and
$\underline{h}^{\prime 2}\gg 1$.

  3. (3)

$\underline{h}(\delta)\leq\inf_{M}(u-\varphi)$.

It is interesting to find $\underline{h}$ such that
$\underline{h}^{\prime\prime}$ and $\underline{h}^{\prime 2}$ are comparable.
Following [14] we set

$\displaystyle
h_{k}(\mathrm{d})=\log\frac{k\delta^{2}}{k\mathrm{d}+\delta^{2}}.$

The $h_{k}$ satisfies the above requirements, as one checks that for any
$k\geq 1$,

(6.4) $\displaystyle h^{\prime}_{k}(\mathrm{d})=$
$\displaystyle-\frac{k}{k\mathrm{d}+\delta^{2}},\mbox{
}h^{\prime\prime}_{k}(\mathrm{d})=\frac{k^{2}}{(k\mathrm{d}+\delta^{2})^{2}},$
$\displaystyle h_{k}^{\prime 2}(\mathrm{d})=$ $\displaystyle
h_{k}^{\prime\prime}(\mathrm{d}),\mbox{ }h_{k}^{\prime
2}e^{-2h_{k}}=\delta^{-4},\mbox{ }h_{k}(\delta)\leq\log\delta.$

######  Lemma 6.7.

Let $\varphi\in C^{2}(\bar{M})$ and $k>0$ fixed. Then for $0<\delta\ll 1$

(6.5) $\displaystyle
U[h_{k}]\geq\frac{(n-1)k^{2}}{8(k\mathrm{d}+\delta^{2})^{2}}g,\,$
$\displaystyle
U[h_{k}+\varphi]\geq\frac{(n-1)k^{2}}{16(k\mathrm{d}+\delta^{2})^{2}}g,\mbox{
in }\Omega_{\delta},$

###### Proof.

Since
$-h_{k}^{\prime}=\frac{k}{k\mathrm{d}+\delta^{2}}\geq\frac{k}{k\delta+\delta^{2}}$
in $\Omega_{\delta}$, so

$-h_{k}^{\prime}\gg 1\mbox{ whenever }0<\delta\ll 1;$

moreover $|\nabla\mathrm{d}|\geq\frac{1}{\sqrt{2}}$ on such $\Omega_{\delta}$.
Notice $\Delta\mathrm{d}g-\nabla^{2}\mathrm{d}$ is bounded in $\bar{M}$, we
can get the first inequaliry of (6.5). Next, using

$\displaystyle
U[h_{k}+\varphi]=U[h_{k}]+U[\varphi]+h_{k}^{\prime}(d\mathrm{d}\otimes
d\varphi+d\varphi\otimes
d\mathrm{d}+(n-3)g(\nabla\mathrm{d},\nabla\varphi)g),$

we have the second one of (6.5) by the same argument.

∎

For $0<\delta\ll 1$ satisfying
$\frac{G_{g}}{n-2}\leq\frac{(n-1)k^{2}}{32(k\mathrm{d}+\delta^{2})^{2}}g$, we
have

$\displaystyle f(\lambda(\mathfrak{g}[h_{k}+\varphi]))\geq
F(\frac{(n-1)k^{2}}{32(k\mathrm{d}+\delta^{2})^{2}}g)=\frac{(n-1)k^{2}}{32(k\mathrm{d}+\delta^{2})^{2}}.$

Observe as in (6.4) that
$\frac{k^{2}}{(k\mathrm{d}+\delta^{2})^{2}}e^{-2h_{k}}=\delta^{-4}\rightarrow+\infty\mbox{
as }\delta\rightarrow 0^{+}.$ So

(6.6) $\displaystyle
f(\lambda(\mathfrak{g}[h_{k}+\varphi]))\geq\frac{\psi}{n-2}e^{2(h_{k}+\varphi)}\mbox{
in }\Omega_{\delta},\mbox{ for }0<\delta\ll 1.$

###### Proof of Lemma 6.4.

Let $k=1$ $({h}_{1}(0)=0)$ and

$\underline{w}=h_{1}+\varphi=\log\frac{\delta^{2}}{\mathrm{d}+\delta^{2}}+\varphi.$

Near the boundary for small $\delta$

(6.7) $\displaystyle\underline{w}=\log\frac{\delta}{1+\delta}+\varphi\leq
u\mbox{ on }\\{\mathrm{d}=\delta\\}.$

Such $\delta$ exists according to Lemma 6.3. Together with (6.6), (6.7), and
$\underline{w}=\varphi$ on $\partial M$, we apply maximum principle to the
equation on $\Omega_{\delta}$ to obtain

(6.8) $\displaystyle u\geq\underline{w}\mbox{ in }\Omega_{\delta}.$

Couple with $u=\underline{w}=\varphi$ on $\partial M$, we get
$\nabla_{\nu}(u-\underline{w})\leq 0$ on $\partial M$. ∎

###### Proof of Lemma 6.5.

It suffices to construct (local) upper barrier near boundary. Here we give a
local upper barrier analogous to that used in the proof of Lemma 6.4. Near
boundary, we let

$\bar{w}=\bar{h}+\varphi$

for $\bar{h}=\bar{h}(\mathrm{d})$ to be picked later. Straightforward
computation gives

$\displaystyle\frac{1}{n-1}\mathrm{tr}_{g}(U[\bar{h}])=\bar{h}^{\prime}\Delta\mathrm{d}+(\bar{h}^{\prime\prime}+\frac{n-2}{2}\bar{h}^{\prime
2})|\nabla\mathrm{d}|^{2},$ $\displaystyle
U[\bar{h}+\varphi]=U[\bar{h}]+U[\varphi]+\bar{h}^{\prime}[d\mathrm{d}\otimes
d\varphi+d\varphi\otimes
d\mathrm{d}+(n-3)g(\nabla\mathrm{d},\nabla\varphi)g].$

The (local) barrier $\bar{h}$ would be obtained if one could find $\bar{h}$
satisfying $\mathrm{tr}_{g}(\mathfrak{g}[\bar{h}+\varphi])\leq 0$ in
$\Omega_{\delta}$, $\bar{h}|_{\partial M}\geq 0$,
$(\bar{h}+\varphi-u)|_{\\{\mathrm{d}=\delta\\}}\geq 0$. It suffices to
construct $\bar{h}$ satisfying

  1. (1)

$\bar{h}^{\prime\prime}+\frac{n-2}{2}\bar{h}^{\prime 2}<-c_{1}\bar{h}^{\prime
2}$ in $\Omega_{\delta}$ for some $c_{1}>0$.

  2. (2)

$\bar{h}^{\prime 2}\rightarrow+\infty$ in $\Omega_{\delta}$ as
$\delta\rightarrow 0^{+}$.

  3. (3)

$\bar{h}(0)=0$, $\bar{h}(\delta)\geq\sup_{M}(u-\varphi)$ for $0<\delta\ll 1$.

Similarly, $-\bar{h}^{\prime\prime}$ and $\bar{h}^{\prime 2}$ shall be
comparable. Such conditions are all fulfilled if

$\bar{h}(d)=\frac{1}{n-2}\log(1+\frac{\mathrm{d}}{\delta^{2}}),\mbox{ or
}\bar{h}(d)=\frac{1}{n-2}\log(1+\frac{k}{\delta}\mathrm{d})\mbox{ for }k\geq
e^{(n-2)\sup_{M}(u-\varphi)}-1.$

The barriers of this type can also be found in textbooks (see e.g. [12]). ∎

###### Proof of Proposition 6.1.

Let $\underline{\psi}=f(\lambda(G_{g}))$. For $t\in[0,1]$ we consider

(6.9) $\displaystyle
f(\lambda(\frac{1}{n-2}G_{e^{2u^{t}}g})=\frac{1}{{n-2}}\left({t\psi}+(1-t)\underline{\psi}\right)e^{2u^{t}}\mbox{
in }M,\,$ $\displaystyle{u}^{t}=t\varphi\mbox{ on }\partial M.$

Set ${\bf S}=\\{t\in[0,1]:\mbox{For $t$ the Dirichlet problem
\eqref{Dirichlet-t} is classically solvable}\\}.$

Clearly $0\in{\bf S}$ ($u^{0}=0$). The openness is classical. By Proposition
4.1 the linearized operator
$\mathcal{L}_{t}:C^{k+2,\alpha}(\bar{M})\rightarrow C^{k,\alpha}(\bar{M})$ is
invertible ($t\in{\bf S}$), thus implying the openness of ${\bf S}$. Combining
with Lemmas 6.3, 6.4, 6.5, Theorem 5.4, and (5.28), (6.3), we have

$|u|_{C^{2}(\bar{M})}\leq C,\mbox{ independent of $u$ and its derivatives}.$

Using Evans-Krylov theorem and Schauder theory, ${\bf S}$ is closed. So ${\bf
S}=[0,1]$. ∎

The following theorem plays an important role in the proof of Theorem 1.1.

######  Theorem 6.8 ([1]).

For any constant $-c$ $(c>0)$, there is a smooth function $\tilde{u}$ in $M$
satisfying

(6.10)
$\displaystyle\mathrm{tr}_{g}(\mathfrak{g}[\tilde{u}])=\mathrm{tr}_{g}(\frac{G_{g}}{n-2}+U[\tilde{u}])=\frac{c}{2}e^{2\tilde{u}}\mbox{
in }M,\,$ $\displaystyle\tilde{u}=+\infty\mbox{ on }\partial M.$

Geometrically, $\tilde{g}=e^{2\tilde{u}}g$ is a complete conformal metric of
negative constant scalar curvature $-c$.

###### Proof of Theorem 1.1.

Let’s consider approximating Dirichlet problems

(6.11) $\displaystyle
f(\lambda(\mathfrak{g}[{u_{k}}])=\frac{\psi}{n-2}e^{2{u_{k}}}\mbox{ in }M,\,$
$\displaystyle u_{k}=\log k\mbox{ on }\partial M.$

The Dirichlet problems are solvability in class of smooth admissible functions
according to Proposition 6.1.

Let $c=\frac{2n}{n-2}\inf_{M}\psi$ and $\tilde{u}$ be the corresponding
function in Theorem 6.8. By (6.1),
$\frac{n\psi}{n-2}e^{2u_{k}}\leq\mathrm{tr}_{g}(\mathfrak{g}[u_{k}])$. The
maximum principle yields

(6.12) $\displaystyle u_{k}\leq u_{k+1}\leq\tilde{u}\mbox{ in }M\mbox{ for
}k\geq 1.$

As a result, for any compact subset $K\subset\subset M$,

$|u_{k}|_{C^{0}(K)}\leq C_{1}(K)\mbox{ for all }k,\mbox{ for $C_{1}(K)$ being
independent of $k$}.$

Given any compact subset $K\subset\subset M$, we choose a compact subset
$K_{1}\subset\subset M$ such that $K\subset\subset K_{1}$. By interior
estimates established above and Evans-Krylov theorem,

$|u_{k}|_{C^{2,\alpha}(K)}\leq C_{2}(K,K_{1})\mbox{ for all }k,\mbox{ where
$C_{2}(K,K_{1})$ depends not on $k$}.$

Combining with Schauder theory, let $k\rightarrow+\infty$, there is a smoothly
admissible function $u_{\infty}$ in $M$ to solve

(6.13) $\displaystyle
f(\lambda(\mathfrak{g}[{u_{\infty}}])=\frac{\psi}{n-2}e^{2{u_{\infty}}}\mbox{
in }M,\,$ $\displaystyle u_{\infty}=+\infty\mbox{ on }\partial M.$

Indeed, $u_{\infty}(x)=\lim_{k\rightarrow+\infty}u_{k}(x)$ for all $x\in M$.
Again, maximum principle yields
$u_{k}\geq\log\frac{k\delta^{2}}{k\mathrm{d}+\delta^{2}}$ near boundary. Thus
$u_{\infty}+\log{\mathrm{d}}\geq-C_{0}$ for some $C_{0}$ near the boundary.
The metric $g_{\infty}=e^{2u_{\infty}}g$ is complete.

∎

###  6.2. Asymptotic properties of complete conformal metrics

It would be interesting to investigate the asymptotic behavior when
$x\rightarrow\partial M$.

For conformal scalar curvature equation (6.10) (with prescribed scalar
curvature $-c$) on smooth bounded domains in Euclidean spaces, Loewner-
Nirenberg [36] proved that
$\mathrm{d}(x)^{2}e^{2\tilde{u}(x)}\rightarrow\frac{n(n-1)}{c}$ as
$x\rightarrow\partial M$; while on curved manifolds, in the proof of their
Lemma 5.2 in [22], Gursky-Streets-Warren constructed local supersolution of
(6.10). From their construction one can see $\tilde{u}$ obeys

(6.14) $\displaystyle\lim_{x\rightarrow\partial
M}\mathrm{d}(x)^{2}e^{2\tilde{u}(x)}\leq\frac{n(n-1)}{c}.$

Please refer to [36, 22] for more details.

For the complete conformal metrics obtained in Theorem 1.1, we further prove
the following asymptotic properties.

######  Proposition 6.9.

Let $g_{\infty}=e^{2u_{\infty}}g$ be the complete metric in the statement of
Theorem 1.1. Then

(6.15) $\displaystyle\lim_{x\rightarrow\partial
M}(u_{\infty}(x)+\log\mathrm{d}(x))\leq\frac{1}{2}\log\frac{(n-1)(n-2)}{2\inf_{\partial
M}\psi}.$

###### Proof.

Let $0<\epsilon\ll 1$. Fix $\epsilon$. There is a positive constant $\delta$
(small) such that $\Omega_{\delta}$ is smooth, and in $\Omega_{\delta}$,
$\psi\geq\inf_{\partial M}\psi-\epsilon>0.$ From Aviles-McOwen’s result, there
is on $\Omega_{\delta}$ a smooth complete conformal metric
$g_{\infty}^{\delta}=e^{2u_{\infty}^{\delta}}g$ with constant scalar curvature
$-1$, that is
$\mathrm{tr}_{g}(\mathfrak{g}[u_{\infty}^{\delta}])=\frac{1}{2}e^{2u_{\infty}^{\delta}},u_{\infty}^{\delta}|_{\partial\Omega_{\delta}}=+\infty.$
By (6.14),

$\lim_{x\rightarrow\partial
M}(u_{\infty}^{\delta}+\log\mathrm{d})\leq\frac{1}{2}\log[n(n-1)].$

For the solution $u_{k}$ of (6.11),
$\mathrm{tr}_{g}(\mathfrak{g}[u_{k}])\geq\frac{n\psi}{n-2}e^{2u_{k}}$
according to (6.1). The maximum principle implies that
$u_{\infty}^{\delta}\geq
u_{k}+\frac{1}{2}\log(\frac{n}{n-2}\inf_{\Omega_{\delta}}\psi)$ in
$\Omega_{\delta}$. Putting them together, $\lim_{x\rightarrow\partial
M}(u_{\infty}(x)+\log\mathrm{d}(x))\leq\frac{1}{2}\log\frac{(n-1)(n-2)}{2(\inf_{\partial
M}\psi-\epsilon)}.$ Thus (6.15) follows. ∎

The proof above indeed gives the following proposition:

######  Proposition 6.10.

Suppose the smooth conformal metric $\widetilde{g}=e^{2w}g$ has strictly
negative scalar curvature $R_{\widetilde{g}}$ in $M$. Assume
$R_{\widetilde{g}}$ can be continuously extended to boundary, still denoted
$R_{\widetilde{g}}$. Then $\lim_{x\rightarrow\partial
M}\mathrm{d}(x)^{2}e^{2w(x)}\leq\frac{n(n-1)}{\inf_{\partial
M}(-R_{\widetilde{g}})}.$

By constructing local subsolutions, we prove

######  Proposition 6.11.

The complete metric $g_{\infty}=e^{2u_{\infty}}g$ obtained in Theorem 1.1
satisfies

(6.16) $\displaystyle\lim_{x\rightarrow\partial
M}(u_{\infty}(x)+\log\mathrm{d}(x))\geq\frac{1}{2}\log\frac{(n-1)(n-2)}{2\sup_{\partial
M}\psi}.$

In addition if $\inf_{\partial M}\psi=\sup_{\partial M}\psi$, i.e. $\psi$ is
constant when restricted to boundary, then

(6.17) $\displaystyle\lim_{x\rightarrow\partial
M}(u_{\infty}(x)+\log\mathrm{d}(x))=\frac{1}{2}\log\frac{(n-1)(n-2)}{2\sup_{\partial
M}\psi}$

and the (admissible) complete conformal metric is unique.

###### Proof.

Given a constant $0<\epsilon<1$. Fix $\epsilon$. We consider

(6.18) $\left\\{\begin{aligned}
\,&f(\lambda(\mathfrak{g}[u_{k,\epsilon,\delta}]))=\frac{\psi}{n-2}e^{2u_{k,\epsilon,\delta}}\mbox{
in }M,\\\
\,&u_{k,\epsilon,\delta}=\log{k}+\frac{1}{2}\log\frac{(1-\epsilon)^{2}(n-1)(n-2)}{2(\sup_{\partial
M}\psi+\epsilon)}\mbox{ on }\partial M\end{aligned}\right.$

where $\delta$ is a positive constant to be determined later. The solvability
is obtained by Proposition 6.1. The maximum principle implies

(6.19)
$\displaystyle\inf_{M}u_{k,\epsilon,\delta}\geq\min\left\\{\frac{1}{2}\inf_{M}\log\frac{f(\lambda(G_{g}))}{\psi},\log
k+\frac{1}{2}\log\frac{(1-\epsilon)^{2}(n-1)(n-2)}{2(\sup_{\partial
M}\psi+\epsilon)}\right\\}.$

Obviously, if $k\gg 1$ then

$\displaystyle\min\left\\{\frac{1}{2}\inf_{M}\log\frac{f(\lambda(G_{g}))}{\psi},\log
k+\frac{1}{2}\log\frac{(1-\epsilon)^{2}(n-1)(n-2)}{2(\sup_{\partial
M}\psi+\epsilon)}\right\\}=\frac{1}{2}\inf_{M}\log\frac{f(\lambda(G_{g}))}{\psi}.$

The local subsolution is given by

(6.20) $\displaystyle
h_{k,\epsilon,\delta}(\mathrm{d})=\log\frac{k}{k\mathrm{d}+1}+\frac{1}{2}\log\frac{(1-\epsilon)^{2}(n-1)(n-2)}{2(\sup_{\partial
M}\psi+\epsilon)}+\frac{1}{\mathrm{d}+\delta}-\frac{1}{\delta}.$

The $``\frac{1}{\mathrm{d}+\delta}-\frac{1}{\delta}"$ part in (6.20) is
inspired by [22]. By Lemma 6.2 in [4] and Lemma 2.1,

$\displaystyle
f(\lambda(\mathfrak{g}[h_{k,\epsilon,\delta}]))=F(\frac{G_{g}}{n-2}+U[h_{k,\epsilon,\delta}])\geq
F(U[h_{k,\epsilon,\delta}])=f(\lambda(U[h_{k,\epsilon,\delta}])).$

Straightforward computation tells

$\displaystyle
h_{k,\epsilon,\delta}^{\prime}=-\frac{k}{k\mathrm{d}+1}-\frac{1}{(\mathrm{d}+\delta)^{2}},\,$
$\displaystyle
h_{k,\epsilon,\delta}^{\prime\prime}=\frac{k^{2}}{(k\mathrm{d}+1)^{2}}+\frac{2}{(\mathrm{d}+\delta)^{3}},$
$\displaystyle h_{k,\epsilon,\delta}^{\prime
2}=\frac{k^{2}}{(k\mathrm{d}+1)^{2}}+\frac{1}{(\mathrm{d}+\delta)^{4}}+\frac{2k}{(k\mathrm{d}+1)(\mathrm{d}+\delta)^{2}},$

and

$h_{k,\epsilon,\delta}^{\prime\prime}+\frac{n-3}{2}h_{k,\epsilon,\delta}^{\prime
2}\geq\frac{n-1}{2}\frac{k^{2}}{(k\mathrm{d}+1)^{2}}+\frac{2}{(d+\delta)^{3}}.$

It is easy to see $h_{k,\epsilon,\delta}^{\prime
2}-h_{k,\epsilon,\delta}^{\prime\prime}\geq 0$ provided
$\mathrm{d}+\delta\leq\frac{1}{2}$ or $k\geq\frac{1}{\delta}$. Thus

(6.21) $\displaystyle U[h_{k,\epsilon,\delta}]=$
$\displaystyle\left(h_{k,\epsilon,\delta}^{\prime\prime}+\frac{n-3}{2}h_{k,\epsilon,\delta}^{\prime
2}\right)|\nabla\mathrm{d}|^{2}g+h^{\prime}(\Delta\mathrm{d}g-\nabla^{2}\mathrm{d})$
$\displaystyle+(h_{k,\epsilon,\delta}^{\prime
2}-h_{k,\epsilon,\delta}^{\prime\prime})d\mathrm{d}\otimes d\mathrm{d}$
$\displaystyle\geq$
$\displaystyle\left(\frac{n-1}{2}\frac{k^{2}}{(k\mathrm{d}+1)^{2}}+\frac{2}{(d+\delta)^{3}}\right)|\nabla\mathrm{d}|^{2}g$
$\displaystyle-\left(\frac{k}{k\mathrm{d}+1}+\frac{1}{(\mathrm{d}+\delta)^{2}}\right)(\Delta\mathrm{d}g-\nabla^{2}\mathrm{d}).$

Note that $|\nabla\mathrm{d}|^{2}=1$ on ${\partial M}$,
$\Delta\mathrm{d}g-\nabla^{2}\mathrm{d}$ is bounded in $\bar{M}$,
$\frac{k}{k\mathrm{d}+1}=\frac{1}{\mathrm{d}+\frac{1}{k}}\geq\frac{1}{\mathrm{d}+\delta}$
for $k\geq\frac{1}{\delta}$, and $\frac{1}{\mathrm{d}+\delta}\gg 1$ in
$\Omega_{\delta}$ if $0<\delta\ll 1$. For the $\epsilon$ fixed before, there
is a $\delta_{\epsilon}$ depending on $\epsilon$ and other known data such
that for $0<\delta<\delta_{\epsilon}$, in $\Omega_{\delta}$

  * •

$\mathrm{d}$ is smooth, and $|\nabla\mathrm{d}|^{2}\geq 1-\epsilon$.

  * •

$\frac{2}{(d+\delta)^{3}}|\nabla\mathrm{d}|^{2}g-\frac{1}{(\mathrm{d}+\delta)^{2}}(\Delta\mathrm{d}g-\nabla^{2}\mathrm{d})\geq
0$.

  * •

$\frac{\epsilon(n-1)}{2}\frac{k^{2}}{(k\mathrm{d}+1)^{2}}|\nabla\mathrm{d}|^{2}g-\frac{k}{k\mathrm{d}+1}(\Delta\mathrm{d}g-\nabla^{2}\mathrm{d})\geq
0$.

  * •

$\sup_{\partial M}\psi+\epsilon\geq\sup_{\Omega_{\delta}}{\psi}$.

And on $\partial\Omega_{\delta}$

  * •

$u_{k,\epsilon,\delta}=h_{k,\epsilon,\delta}$ on $\partial M$, and
$u_{k,\epsilon,\delta}|_{\mathrm{d}=\delta}\geq h_{k,\epsilon,\delta}(\delta)$
(according to (6.19)). Here we also use
$\frac{\log\frac{k}{k\delta+1}}{\frac{1}{2\delta}}\leq\frac{\log\frac{1}{\delta}}{\frac{1}{2\delta}}\rightarrow
0$ as $\delta\rightarrow 0^{+}$.

From now on we fix $0<\delta<\delta_{\epsilon}$ (for example
$\delta=\frac{\delta_{\epsilon}}{2}$) and $k\geq\frac{1}{\delta}$. Putting the
discussion above together we have in $\Omega_{\delta}$

$\displaystyle f(\lambda(\mathfrak{g}[h_{k,\epsilon,\delta}]))\geq$
$\displaystyle F(U[h_{k,\epsilon,\delta}])$ $\displaystyle\geq$ $\displaystyle
F\left(\frac{(n-1)(1-\epsilon)}{2}\cdot\frac{k^{2}}{(k\mathrm{d}+1)^{2}}|\nabla\mathrm{d}|^{2}g\right)$
$\displaystyle\geq$
$\displaystyle\frac{(n-1)(1-\epsilon)}{2}\cdot\frac{2(\sup_{\partial
M}\psi+\epsilon)}{(1-\epsilon)^{2}(n-1)(n-2)}e^{2h_{k,\epsilon,\delta}(\mathrm{d})}\cdot(1-\epsilon)$
$\displaystyle\geq$
$\displaystyle\frac{\sup_{\Omega_{\delta}}\psi}{n-2}e^{2h_{k,\epsilon,\delta}(\mathrm{d})}$
$\displaystyle\geq$
$\displaystyle\frac{\psi}{n-2}e^{2h_{k,\epsilon,\delta}(\mathrm{d})}.$

Here we also use $\frac{k^{2}}{(k\mathrm{d}+1)^{2}}\geq\frac{2(\sup_{\partial
M}\psi+\epsilon)}{(1-\epsilon)^{2}(n-1)(n-2)}e^{2h_{k,\epsilon,\delta}}.$ The
maximum principle again yields on $\Omega_{\delta}$

$\displaystyle u_{k,\epsilon,\delta}\geq
h_{k,\epsilon,\delta}=\log\frac{k}{k\mathrm{d}+\delta}+\frac{1}{2}\log\frac{(1-\epsilon)^{2}(n-1)(n-2)}{2((\sup_{\partial
M}\psi+\epsilon)}+\frac{1}{\mathrm{d}+\delta}-\frac{1}{\delta}.$

Since $u_{\infty}\geq u_{k,\epsilon,\delta}$, let $k\rightarrow+\infty$, we
obtain in $\Omega_{\delta}$

$\displaystyle
u_{\infty}(x)+\log\mathrm{d}(x)\geq\frac{1}{2}\log\frac{(n-1)(n-2)(1-\epsilon)^{2}}{2(\sup_{\partial
M}\psi+\epsilon)}+\frac{1}{\mathrm{d}+\delta}-\frac{1}{\delta}.$

Thus (6.16) follows.

Finally, if $\inf_{\partial M}\psi=\sup_{\partial M}\psi$, (6.17) follows from
(6.15) and (6.16). Furthermore, follows the line of [22] the uniqueness of
complete conformal metric can be proved by using (6.17).

∎

###  6.3. Remark on the existence of admissible conformal metrics

We will show that the condition (1.5) automatically holds if
$\mathcal{A}(\bar{M},[g],\Gamma)\neq\emptyset.$

######  Proposition 6.12.

Let $\mathcal{A}(\bar{M},[g],\Gamma)$ be as defined in (1.7). If $(M,g)$
supposes a $C^{2}$-function in $\mathcal{A}(\bar{M},[g],\Gamma)$, then there
exists an admissible conformal metric on $\bar{M}$.

###### Proof.

For $v\in\mathcal{A}(\bar{M},[g],\Gamma)$ (assume $v\geq 0$, otherwise
replaced by $v-\inf_{M}v$), we set

$\displaystyle\underline{u}=e^{Av},\,$
$\displaystyle\tilde{g}=e^{2\underline{u}}g,$

then

$\displaystyle U[\underline{u}]=$ $\displaystyle
A^{2}e^{Av}(1+\frac{n-3}{2}e^{Av})|\nabla v|^{2}g+Ae^{Av}(\Delta
vg-\nabla^{2}v)+A^{2}e^{Av}(e^{Av}-1)dv\otimes dv$ $\displaystyle\geq$
$\displaystyle Ae^{Av}\left\\{A|\nabla v|^{2}g+(\Delta
vg-\nabla^{2}v)\right\\}.$

We denote the critical set of $v$ by

$\displaystyle\mathcal{C}(v):=\left\\{x\in\bar{M}:\nabla v(x)=0\right\\}.$

For $x\in\mathcal{C}(v)$, there is $r_{x}>0$ ($r_{x}$ is small) such that

$\lambda(\Delta vg-\nabla^{2}v)\in\Gamma\mbox{ in
}\mathrm{V}_{x}:=\\{y\in\bar{M}:\mathrm{dist}(x,y)<r_{x}\\}.$

Fix $r_{x}$ as above. Set
$U=\cup_{x\in\mathcal{C}(v)}\\{y\in\bar{M}:\mathrm{dist}(x,y)<\frac{r_{x}}{2}\\}.$
Then $U$ is an open subset. The subset $\bar{M}\setminus{U}$ as well as the
closure of $U$, denoted by $\overline{U}$, are both compact. From above we see
$\lambda(\Delta vg-\nabla^{2}v)\in\Gamma$ in $\overline{U}$ and there is a
uniformly positive constant $a_{0}$ such that $|\nabla v|^{2}\geq a_{0}$ in
$\bar{M}\setminus{U}$.

If $x\in\bar{M}\setminus U$, then
$U[\underline{u}]\geq\frac{1}{2}a_{0}A^{2}e^{Av}g\mbox{ for }A\gg 1.$

For $x\in\overline{U}$,
$\lambda(\frac{1}{n-2}G_{\tilde{g}})=\lambda(\frac{1}{n-2}G_{g}+U[\underline{u}])\in\Gamma$
in $\overline{U}$ when $A\gg 1$, since $U[\underline{u}]\geq Ae^{Av}(\Delta
vg-\nabla^{2}v)$ and $\lambda(\Delta vg-\nabla^{2}v)\in\Gamma\mbox{ in
}\overline{U}$.

∎

###### Proof of Theorem 1.3.

Obviously, there are many smooth functions $v$ on $\bar{\Omega}$ with $\nabla
v\neq 0$ everywhere. According to Proposition 6.12 admissible conformal
metrics exist.

∎

######  Remark 6.13.

If $\bar{M}$ admits a $C^{1}$-gradient field without any zero, then
$\mathcal{A}(\bar{M},[g],\Gamma)\neq\emptyset$ for each conformal class $[g]$
on $\bar{M}$. Furthermore, if a manifold $Y$ supposes a function $v$ with
$\nabla^{Y}v\neq 0$ everywhere, then so does products $Y\times X$.

##  7\. Conformal deformation of general modified Schouten tensors

In this section, for modified Schouten tensors $S_{g}^{\tau}$ with
$\tau>1+(n-2)(1-\kappa_{\Gamma}\vartheta)$, we are going to find conformal
metrics $\tilde{g}=e^{2u}g$ to solve (1.16). Also, we assume throughout this
section $\psi\in C^{\infty}(\bar{M})$, $\psi>0\mbox{ in }\bar{M}$, and $f$
satisfies (1.2), (1.3), (1.4).

Recall that for $S^{\tau}_{\tilde{g}}$, $\tilde{g}=e^{2u}g$,

$S_{\tilde{g}}^{\tau}=S_{g}^{\tau}+\frac{\tau-1}{n-2}\Delta
ug-\nabla^{2}u+\frac{\tau-2}{2}|\nabla u|^{2}g+du\otimes du.$

Let $h_{k}=h_{k}(\mathrm{d})=\log\frac{k\delta^{2}}{k\mathrm{d}+\delta^{2}}$.
The straightforward computation gives $h_{k}^{\prime\prime}=h_{k}^{\prime 2}$
and

(7.1) $\displaystyle\frac{\tau-1}{n-2}\Delta
h_{k}g-\nabla^{2}h_{k}+\frac{\tau-2}{2}|\nabla h_{k}|^{2}g+dh_{k}\otimes
dh_{k}$ $\displaystyle=$
$\displaystyle\left(\frac{\tau+1-n}{n-2}+\frac{\tau}{2}\right)(h_{k}^{\prime})^{2}|\nabla\mathrm{d}|^{2}g+h^{\prime}_{k}\left(\frac{\tau-1}{n-2}\Delta\mathrm{d}g-\nabla^{2}\mathrm{d}\right).$

It is easy to see $0<\vartheta\leq\frac{1}{n}$ and $0\leq\kappa_{\Gamma}\leq
n-1$; furthermore $\vartheta=\frac{1}{n}$ and $\kappa_{\Gamma}=n-1$ cannot
occur simultaneity (otherwise
$f_{i}(\lambda)=\frac{1}{n}\sum_{j=1}^{n}f_{j}(\lambda)$,
$\forall\lambda\in\Gamma$, $\forall i$). Henceforce,

$\displaystyle\tau>1+(n-2)(1-\kappa_{\Gamma}\vartheta)>1+(n-2)(1-\frac{n-1}{n})=\frac{2n-2}{n}.$

As a result, we verify the positive sign of coefficient of quadratic gradient
term, $(h_{k}^{\prime})^{2}|\nabla\mathrm{d}|^{2}g$, playing a key role, that
is

(7.2)
$\displaystyle\frac{\tau+1-n}{n-2}+\frac{\tau}{2}=\frac{n\tau+2-2n}{2(n-2)}>0.$

######  Remark 7.1.

The importance of assumption (1.15):

  * •

It assures (1.14) (corresponding to
$0<\varrho<\frac{1}{1-\kappa_{\Gamma}\vartheta}$ in (1.13)).

  * •

It implies the positive sign as shown in (7.2).

Following the outline of proof in Section 6, we can prove Theorems 1.7.
Similarly, we also define

$\displaystyle\mathcal{A}^{\tau}(\bar{M},[g],\Gamma)=\left\\{u\in
C^{2}(\bar{M}):\lambda_{\tilde{g}}(\frac{\tau-1}{n-2}\Delta_{\tilde{g}}u\tilde{g}-\tilde{\nabla}^{2}u)(x_{0})\in\Gamma\mbox{
at }x_{0}\in\mathcal{C}(u)\mbox{ for }\tilde{g}\in[g]\right\\}$

as above $\mathcal{C}(u)$ is the critical set of $u$. The conformal metric
satisfying (1.17) exists if
$\mathcal{A}^{\tau}(\bar{M},[g],\Gamma)\neq\emptyset$. As a consequence, we
obtain Theorem 1.8. Furthermore, we obtain existence result for conformal
metrics with prescribed boundary metric.

######  Theorem 7.2.

Suppose (1.17) holds. For a Riemannian metric $h$ on $\partial M$ which is
conformal to $g|_{\partial M}$, there is a smooth conformal metric $\tilde{g}$
with prescribed boundary condition $\tilde{g}|_{\partial M}=h$ to satisfy
(1.16). Moreover, $\lambda_{\tilde{g}}(S^{\tau}_{\tilde{g}})\in\Gamma\mbox{ in
}\bar{M}$.

For complete conformal metric satisfying (1.16) in Theorem 1.7, we have the
following asymptotic property.

######  Proposition 7.3.

Let $g_{\infty}=e^{2u_{\infty}}g$ be the complete conformal metric claimed in
Theorem 1.7, then

(7.3) $\displaystyle\frac{1}{2}\log\frac{n\tau+2-2n}{2\sup_{\partial
M}\psi}\leq\lim_{x\rightarrow\partial
M}(u_{\infty}(x)+\log\mathrm{d}(x))\leq\frac{1}{2}\log\frac{n\tau+2-2n}{2\inf_{\partial
M}\psi}.$

Furthermore, if $\inf_{\partial M}\psi=\sup_{\partial M}\psi$ then

(7.4) $\displaystyle\lim_{x\rightarrow\partial
M}(u_{\infty}(x)+\log\mathrm{d}(x))=\frac{1}{2}\log\frac{n\tau+2-2n}{2\sup_{\partial
M}\psi}$

and the complete conformal metric is unique.

##  8\. Hessian and Weingarten equations

Finally, we present in this section another application of partial uniform
ellipticity. The following interesting assertion confirms a key assumption
used by Li [34] and Urbas [39] who studied Weingarten and Hessian equations,
respectively.

######  Proposition 8.1.

If $f$ satisfies (1.2), (1.3) and (1.12), then for each $\lambda\in\Gamma$ we
have

(8.1) $\displaystyle
f_{i}(\lambda)\geq\vartheta\sum_{j=1}^{n}f_{j}(\lambda)\mbox{ if
}\lambda_{i}\leq 0.$

Here $\vartheta$ is the constant as in Theorem 1.5. Moreover, (1.12) can be
removed if $n=2$.

###### Proof.

Fix $\lambda=(\lambda_{1},\cdots,\lambda_{n})\in\Gamma.$ Without loss of
generality, we assume $\lambda_{n}\geq\cdots\geq\lambda_{1}.$ By the
definition of $\kappa_{\Gamma}$,

$\lambda_{\kappa_{\Gamma}+1}>0.$

Therefore, $i\leq\kappa_{\Gamma}\mbox{ if }\lambda_{i}\leq 0.$ So (8.1)
immediately follows from Theorem 1.5. ∎

###  8.1. Hessian equations

Let’s consider Hessian equations

(8.2) $\displaystyle f(\lambda(\nabla^{2}u+A))=\psi,$

where $\psi\in C^{\infty}(\bar{M})$ and $A$ is a smooth symmetric $(0,2)$-type
tensor.

The gradient estimate is important for studying Hessian equations. However, it
is rather hard to derive on curved manifolds. In [34] it was obtained under
assumptions that the Riemannian manifold admits nonnegative sectional
curvature and $f$ satisfies
$\lim_{|\lambda|\rightarrow+\infty}\sum_{i=1}^{n}f_{i}(\lambda)=+\infty$, and
later generalized by [39] with replacing such two restrictions by (8.1) and
$\sum_{i=1}^{n}f_{i}\geq\delta>0$ in $\\{\lambda\in\Gamma:\inf_{M}\infty\leq
f(\lambda)\leq\sup_{M}\psi\\}$.

Combining Proposition 8.1 and (5.4), one can follow [39] to obtain the
gradient bound for solutions to (8.2) with existence of
$\mathcal{C}$-subsolution ([38]) satisfying

$\displaystyle\lim_{t\rightarrow+\infty}f(\lambda(\nabla^{2}\underline{u}+A)+te_{i})>\psi,\mbox{
in }\bar{M}\mbox{ for each }i=1,\cdots,n,$

where $e_{i}$ is the $i$-$\mathrm{th}$ standard basis vector.

######  Theorem 8.2.

Let $u$ be an admissible solution $u\in C^{3}(M)\cap C^{1}(\bar{M})$ to
equation (8.2) with nondegenerate right-hand side
$\inf_{M}\psi>\sup_{\partial\Gamma}f$. In addition to (1.2), (1.3) and (1.12),
there exists a $C^{2}$-smooth $\mathcal{C}$-subsolution $\underline{u}$. Then

$\displaystyle\sup_{M}|\nabla u|\leq C(1+\sup_{\partial M}|\nabla u|),$

where $C$ depends on $|\psi|_{C^{1}(\bar{M})}$,
$|\underline{u}|_{C^{2}(\bar{M})}$ and other known data. Moreover, $C$ is
independent of $(\delta_{\psi,f})^{-1}$,
$\delta_{\psi,f}:=\inf_{M}\psi-\sup_{\partial\Gamma}f$.

For Dirichlet problem, with a subsolution assumption

$\displaystyle f(\lambda(\nabla^{2}\underline{u}+A))\geq\psi,\mbox{
}\lambda(\nabla^{2}\underline{u}+A)\in\Gamma\mbox{ in }\bar{M},\mbox{
}\underline{u}=\varphi\mbox{ on }\partial M,$

the gradient estimate was proved by [16] for more general Hessian equations
(8.2) with a different method; while if the background space is a closed
Riemannian manifold, Theorem 8.2 was stated in [38] where the proof is based
on second estimate

(8.3) $\displaystyle\sup_{M}\Delta u\leq C(1+\sup_{M}|\nabla u|^{2})$

which extends Hou-Ma-Wu’s [27] second estimate for complex $k$-Hessian
equations on closed Kähler manifolds. The estimate (8.3) was extended by the
author [43] to compact Riemannian manifolds with concave boundary by
establishing quantitative boundary estimate of the form

(8.4) $\displaystyle\sup_{\partial M}\Delta u\leq C(1+\sup_{M}|\nabla
u|^{2}).$

Indeed, by using a strategy developed further in [44], (8.4) can be extended
to more general case that the principal curvatures of $\partial M$,
$\kappa_{1},\cdots,\kappa_{n-1}$, satisfy

(8.5)
$\displaystyle(-\kappa_{1},\cdots,-\kappa_{n-1})\in\overline{\Gamma}^{\infty}_{\mathbb{R}^{n-1}}\mbox{
in }\partial M,$

where $\overline{\Gamma}^{\infty}_{\mathbb{R}^{n-1}}$ is the closure of
$\Gamma^{\infty}_{\mathbb{R}^{n-1}}$. (In [43, 44] the author primarily deals
with Dirichlet problem on Hermitian manifolds, in which (8.4) was established
for complex fully nonlinear equations. In fact the gradient estimate in
complex variables is much more hard to handle). It is a remarkable fact that,
besides with proposing a new and effective approach to gradient estimate, the
quantitative boundary estimate indeed applies to Dirichlet problem for
equation (8.2) with degenerate right-hand side
$\inf_{M}\psi=\sup_{\partial\Gamma}f$. To solve degenerate equations, the
admissible subsolution shall be replaced by a strictly admissible subsolution
with $f(\lambda(\nabla^{2}\underline{u}+A))>\psi$ in $\bar{M}$.

As a result, using arguments and results in [39, 15] (nondegenerate case) and
[44] (degenerate case) one solves Dirichlet problem with assuming admissible
subsolutions.

######  Theorem 8.3.

Let $(M,g)$ be a Riemannian manifold with smooth boundary. Let $\psi,\varphi$
be smooth functions, $A$ a smoothly symmetric $(0,2)$-type tensor. In addition
to (1.2), (1.3) and (1.12), we assume that
$\inf_{M}\psi>\sup_{\partial\Gamma}f$ and there is a $C^{3,1}$-smooth
admissible subsolution $\underline{u}$. Then there exists a unique smooth
admissible function $u$ with $u|_{\partial M}=\varphi$ to solve the Hessian
equation (8.2).

Furthermore, if $\partial M$ satisfies (8.5) and $\underline{u}$ is a strict
subsolution, then the equation with degenerate right-hand side admits a weak
solution $u$ with $u|_{\partial M}=\varphi$, $u\in C^{1,1}(\bar{M})$, $\Delta
u\in L^{\infty}(\bar{M})$ and $\lambda(\nabla^{2}u+A)\in\overline{\Gamma}$.

###  8.2. Weingarten equation

Let
$S_{u}:=\\{(x^{\prime},x_{n+1})\subset\mathbb{R}^{n+1}:x^{\prime}=(x_{1},\cdots,x_{n}),x_{n+1}=u(x^{\prime})\\}$
be the graph of $u$. We denote $\kappa=(\kappa_{1},\cdots,\kappa_{n})$ the
principal curvatures of the graph $S_{u}$. The Weingarten equation is given by

(8.6) $\displaystyle f(\kappa)=\psi(x,u,\nu)$

where $\nu=(\nu_{1},\cdots,\nu_{n+1})$ is the downward normal vector of
$S_{u}$, and $\psi$ is in the class of $C^{1}$, which satisfies that there are
some positive constants $A_{0}$, $A_{1}$, $A_{2}$ such that

(8.7) $\displaystyle\psi_{u}\geq 0,\mbox{ }|\psi|\leq A_{0},\mbox{
}|\nabla\psi|\leq A_{1},\mbox{ }\psi\geq A_{2}>\sup_{\partial\Gamma}f.$

Interior gradient estimates for admissible solutions of certain Weingarten
equation with $f=(\sigma_{k})^{\frac{1}{k}}$ have been obtained by Korevaar
[29], and it was later generalized by Li [33] to more general Weingarten
equations satisfying

(8.8) $\displaystyle\liminf_{\lambda\rightarrow 0}f(\lambda)>-\infty,$ (8.9)
$\displaystyle
f_{i}(\lambda)\geq\theta+\theta\sum_{j=1}^{n}f_{j}(\lambda)\mbox{ for some
$\theta>0$},\mbox{ if }\lambda_{i}\leq 0$

which is satisfied if $f$ obeys (1.2), (1.3), (1.12), according to Proposition
8.1 and (5.4).

######  Theorem 8.4.

Suppose (1.2), (1.3), (1.12), (8.7) and (8.8) hold. Let
$\overline{B_{1}(0)}=\\{x^{\prime}\in\mathbb{R}^{n}:|x^{\prime}|\leq 1\\}$,
and $u\in C^{3}(\overline{B_{1}(0)})$ be a solution to (8.6) with
$\kappa\in\Gamma$ in graph $S_{u}$. In addition $u|_{\overline{B_{1}(0)}}<0$,
$u(0)=-A_{3}$. Then there is a positive constant $C$ depending only on $f$,
$n$, $A_{0}$, $A_{1}$, $A_{2}$ and $A_{3}$, such that $|\nabla u(0)|\leq C.$

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