# The partial uniform ellipticity and prescribed problems on the conformal
classes of complete metrics

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

We clarify how close a second order fully nonlinear equation can come to
uniform ellipticity, through counting large eigenvalues of the linearized
operator. This suggests an effective and novel way to understand the structure
of fully nonlinear equations of elliptic and parabolic type. As applications,
we solve a fully nonlinear version of the Loewner-Nirenberg problem and a
noncompact complete version of fully nonlinear Yamabe problem. Our method is
delicate as shown by a topological obstruction.

###### Key words and phrases:

Fully nonlinear equation; conformal transformation; fully nonlinear Loewner-
Nirenberg problem; complete noncompact version of fully nonlinear Yamabe
problem; symmetric concave function; partial uniform ellipticity

##  1\. Introduction

###  1.1. Partial uniform ellipticity

In their pioneering paper [8], Caffarelli-Nirenberg-Spruck initiated the study
of a fully nonlinear equation

(1.1) $\displaystyle F(D^{2}u)=\psi\mbox{ }\mbox{ in
}\Omega\subset\mathbb{R}^{n}.$

This equation has the form $F(D^{2}u)=f(\lambda(D^{2}u))$, where
$\lambda(D^{2}u)=(\lambda_{1},\cdots,\lambda_{n})$ are the eigenvalues of
$D^{2}u$. As suggested by Caffarelli-Nirenberg-Spruck, $f$ is a smooth
symmetric function of $n$ real variables defined in an open symmetric convex
cone $\Gamma\subset\mathbb{R}^{n}$, with vertex at the origin and boundary
$\partial\Gamma\neq\emptyset$, containing the positive cone $\Gamma_{n}$. In
addition, $f$ satisfies some fundamental structural conditions including

(1.2) $f\mbox{ is concave in }\Gamma,$ (1.3) $\displaystyle
f_{i}(\lambda):=\frac{\partial f}{\partial\lambda_{i}}(\lambda)>0\mbox{ in
}\Gamma,\mbox{ }\forall 1\leqslant i\leqslant n.$

In some cases, one may replace (1.3) by the following weaker condition

(1.4) $\displaystyle f_{i}(\lambda)\geqslant 0,\mbox{ }\forall 1\leqslant
i\leqslant n,\mbox{ }\sum_{i=1}^{n}f_{i}(\lambda)>0,\mbox{ in }\Gamma.$

This includes among others Poisson equation and Monge-Ampère equation, on
which there are substantial literature. Other typical examples are as follows:

$\displaystyle f=(\sigma_{k}/\sigma_{l})^{1/(k-l)},\mbox{
}\Gamma=\Gamma_{k},\,\mbox{ or }\,f=(\sigma_{k}/\sigma_{l})^{1/(k-l)}\circ
P_{n-1},\mbox{ }\Gamma=\mathcal{P}_{k}$

for $0\leqslant l<k\leqslant n$, where $\sigma_{k}$ is the $k$-th elementary
symmetric function, $\Gamma_{k}$ is the $k$-th Gårding cone, and
$P_{n-1}(\lambda)=\sum_{i=1}^{n}\lambda_{i}\vec{\bf 1}-\lambda,$ $\vec{\bf
1}=(1,\cdots,1)\in\mathbb{R}^{n},$
$\mathcal{P}_{k}=\\{\lambda:P_{n-1}(\lambda)\in\Gamma_{k}\\}$.

We first look at the structure of equation (1.1). The eigenvalues of
coefficient matrix of the linearlized operator at $u$ are precisely given by

$f_{1}(\lambda),\cdots,f_{n}(\lambda),\mbox{ for }\lambda=\lambda(D^{2}u).$

As a consequence, (1.3) yields that (1.1) is elliptic at $u\in
C^{2}(\bar{\Omega})$ satisfying $\lambda(D^{2}u)\in\Gamma$, while (1.2)
implies that the operator $F(D^{2}u)=f(\lambda(D^{2}u))$ is concave with
respect to $D^{2}u$ when $\lambda(D^{2}u)\in\Gamma$, thereby Evans-Krylov
theorem [15, 36] applies once the bound of real Hessian has been proved.

As is well known, the uniform ellipticity plays central roles in the study of
second order elliptic equations, especially in the theory of a priori
estimates, see e.g. [19]. However, the uniform ellipticity of (1.1) breaks
down in general, thereby causing various serious difficulties. Consequently,
it is of importance to determine when a fully nonlinear equation becomes
uniformly elliptic.

This leads to the concept of partial uniform ellipticity.

######  Definition 1.1.

We say that $f$ is of $\mathrm{m}$-uniform ellipticity in $\Gamma$, if $f$
satisfies (1.4) and there is a uniform positive constant $\vartheta$ such that
for any $\lambda\in\Gamma$ with
$\lambda_{1}\leqslant\cdots\leqslant\lambda_{n}$,

(1.5) $\displaystyle
f_{i}(\lambda)\geqslant\vartheta\sum_{j=1}^{n}f_{j}(\lambda),\mbox{ }\forall
1\leqslant i\leqslant\mathrm{m}.$

In particular, $n$-uniform ellipticity is also called fully uniform
ellipticity.

Accordingly, we have a similar notion of partial uniform ellipticity for a
second order equation, if its linearized operator satisfies a similar
condition.

There naturally arises a problem:

######  Problem 1.2.

To determine the integer $\mathrm{m}$ from (1.5) in Definition 1.1.

The first part of this paper is devoted to this problem. For any $f$
satisfying

(1.6) $\displaystyle\lim_{t\rightarrow+\infty}f(t\lambda)>f(\mu)\mbox{ for any
}\lambda,\mbox{ }\mu\in\Gamma,$

we observe that a criterion, extending the one proposed initially in [62]
leads to bridging partial uniform ellipticity and the maximum count of zero
components of vectors contained in $\Gamma$. To this end, we introduce an
integer for $\Gamma$.

######  Definition 1.3.

For the cone $\Gamma$, we define

$\displaystyle{\kappa}_{\Gamma}=\max\left\\{k:({\overbrace{0,\cdots,0}^{k}},{\overbrace{1,\cdots,1}^{n-k}})\in\Gamma\right\\}.$

Below we state our main result on the partial uniform ellipticity.

######  Theorem 1.4.

Suppose $(f,\Gamma)$ satisfies (1.2) and (1.6). Then we have (1.4) and that
there exists a universal positive constant $\vartheta_{\Gamma}$ depending only
on $\Gamma$, such that for any $\lambda\in\Gamma$ with
$\lambda_{1}\leqslant\cdots\leqslant\lambda_{n}$,

$\displaystyle
f_{{i}}(\lambda)\geqslant\vartheta_{\Gamma}\sum_{j=1}^{n}f_{j}(\lambda),\mbox{
}\forall 1\leqslant i\leqslant\kappa_{\Gamma}+1.$

In particular, $f$ is of fully uniform ellipticity in the type 2 cone.

######  Remark 1.5.

A choice of $\vartheta_{\Gamma}$ can be found in Remark 2.7. In Proposition
2.5 we prove $\kappa_{\Gamma}$ is in effect the maximum count of negative
components of vectors in $\Gamma$.

This theorem asserts that $f$ is of $(\kappa_{\Gamma}+1)$-uniform ellipticity
in $\Gamma$, which is in effect sharp as shown in Corollary 2.10 below. This
is one of the main contributions we make in this paper. The theorem applied to
the case $(f,\Gamma)=(\sigma_{k}^{1/k},\Gamma_{k})$ gives back [40, Theorem
1.1] of Lin-Trudinger with a different method, essentially using specific
properties of $\sigma_{k}^{1/k}$ which cannot be adapted to general functions.

Theorem 1.4 suggests an effective and novel way to understand the structure of
fully nonlinear equations of elliptic and parabolic type. As a by-product, we
can confirm the following inequality in general context

$\displaystyle
f_{i}(\lambda)\geqslant\vartheta_{\Gamma}\sum_{j=1}^{n}f_{j}(\lambda)\mbox{ if
}\lambda_{i}\leqslant 0.$

This inequality was imposed as a key assumption by Li [39], subsequently by
Trudinger [58], Guan-Spruck [25] and Sheng-Urbas-Wang [53] to study Weingarten
equations. Also, it can be applied to various geometric PDEs from conformal
geometry and real Hessian type fully nonlinear equations, see e.g. [10, 11,
59, 23].

Another remarkable consequence is that a type 2 cone (see Definition 3.1)
guarantees the uniform ellipticity of the corresponding equations. This allows
us to study a class of fully nonlinear equations, with applications to a fully
nonlinear version of Loewner-Nirenberg problem and a complete noncompact
version of fully nonlinear Yamabe problem.

###  1.2. Fully nonlinear equations on Riemannian manifolds

Let $(M,g)$ be a connected Riemannian manifold of dimension $n$ with Levi-
Civita connection $\nabla$. Let $\partial M$ denote the boundary, and
$\bar{M}=M\cup\partial M$. Notice $\bar{M}=M$ if $\partial M=\emptyset$.

We denote $\Delta u$, $\nabla^{2}u$ and $\nabla u$ the Laplacian, Hessian and
gradient of $u$ with respect to $g$, respectively. For $\mathrm{U}$ a
symmetric $(0,2)$-tensor, $\lambda(g^{-1}\mathrm{U})$ denote the eigenvalues
of $\mathrm{U}$ with respect to $g$, and the meaning of
$\lambda(\tilde{g}^{-1}\mathrm{U})$ is obvious.

We apply partial uniform ellipticity to study a fully nonlinear equation

(1.7) $\displaystyle f(\lambda(g^{-1}(\Delta u\cdot
g-\varrho\nabla^{2}u+A(x,\nabla u))))=\psi(x,u),$

with a constant $\varrho$ subject to

(1.8)
$\displaystyle\varrho<\frac{1}{1-\kappa_{\Gamma}\vartheta_{\Gamma}}\mbox{ and
}\varrho\neq 0,$

where the right-hand side is a smooth positive function such that

(1.9) $\displaystyle\psi_{z}(x,z)>0,\mbox{
}\lim_{z\to-\infty}\psi(x,z)=0,\mbox{ }\psi(x,z)\geqslant
h(x)e^{\alpha(x)z},\mbox{ }(x,z)\in\bar{M}\times\mathbb{R}$

holds with positive continuous functions $h$ and $\alpha$. In addition,

(1.10) $\sup_{\Gamma}f=+\infty,$ (1.11) $\displaystyle f>0\mbox{ in
}\Gamma,\quad f=0\mbox{ on }\partial\Gamma,$

and $A(x,p)$ is a smooth symmetric $(0,2)$-tensor satisfying for any $(x,p)\in
T\bar{M},$

(1.12)
$\displaystyle\lim_{|p|\to+\infty}\left(\frac{|A(x,p)|}{|p|^{2}}+\frac{|D_{p}A(x,p)|}{|p|}\right)\leqslant\eta(x),\mbox{
}\lim_{|p|\to+\infty}\frac{|\nabla^{\prime}A(x,p)|}{|p|^{3}}=0,$

where $\eta$ is a positive continuous function, $\nabla^{\prime}A$ denotes
partial covariant derivative of $A$, the meaning of $D_{p}A$ is obvious when
viewed as depending on $x\in\bar{M}$.

The equations of this form closely connect with prescribed curvature equations
from conformal geometry. An analogue of equation (1.7) with $\varrho=1$ is
also called a $(n-1)$ type fully nonlinear equation. In recent years, $(n-1)$
type Monge-Ampère equation has been studied extensively in complex geometry,
due to its close connection to the Calabi-Yau theorem for Gauduchon and
balanced metrics [17, 56].

First, we recall the concept of admissible and maximal solution.

######  Definition 1.6.

We say $u$ is an admissible function for the equation (1.7) if

$\lambda(g^{-1}(\Delta u\cdot g-\varrho\nabla^{2}u+A(x,\nabla u)))\in\Gamma$

pointwisely. Meanwhile, we say $u$ is a maximal solution of an equation, if
$u\geqslant w$ in $M$ for any admissible solution $w$ to the same equation.

Similarly, we have notions of admissible and maximal conformal metrics.

Below, we solve the Dirichlet problem with infinite boundary value condition.

######  Theorem 1.7.

Let $(M,g)$ be a compact connected Riemannian manifold of dimension
$n\geqslant 3$ with smooth boundary. Suppose, in addition to (1.2), (1.8),
(1.9), (1.10), (1.11) and (1.12), that there is a $C^{2}$-admissible function
on $\bar{M}$. Then the equation (1.7) admits a smooth admissible solution $u$
with $\underset{x\to\partial M}{\lim}\,u(x)=+\infty.$

When the background manifold is complete and noncompact, we can solve the
equation, given an asymptotic condition at infinity which is indeed sharp

(1.13) $\displaystyle f(\lambda(g^{-1}(\Delta\underline{u}\cdot
g-\varrho\nabla^{2}\underline{u}+A(x,\nabla\underline{u}))))\geqslant\psi(x,\underline{u}),\mbox{
}\forall x\in M.$

######  Theorem 1.8.

Assume $(M,g)$ is a complete noncompact Riemannian manifold of dimension
$n\geqslant 3$ and suppose a $C^{2}$-admissible function satisfying (1.13). In
addition to (1.8), (1.9) and (1.12), we assume $(f,\Gamma)$ satisfies (1.2),
(1.10) and (1.11). Then the equation (1.7) possesses a unique smooth maximal
admissible solution $u$. Moreover, $u\geqslant\underline{u}$ in $M$.

To prove the above results, the primary problem is to derive local estimates
for first and second derivatives. The key ingredient is that, in the presence
of (1.2), (1.8), (1.10) and (1.11), we can apply Theorem 1.4 to confirm the
fully uniform ellipticity of equation (1.7) in Proposition 3.2. This is the
most important step for our approach. With this crucial procedure at hand, we
can prove the above theorems. To be brief, we apply singularity solutions of a
class of semi-linear equations obtained by McOwen [46] to construct
supersolutions. Then we complete the proof of Theorem 1.7. In an attempt to
show Theorem 1.8, we set up appropriate approximate problems on an exhaustion
series of domains, building on Theorem 1.7. The approximation is based on a
maximum principle: the approximate Dirichlet problems with infinite boundary
value condition produce a decreasing sequence of approximate solutions (see
Proposition 4.4).

###  1.3. Prescribed problems for complete conformal metrics

Let $\mathrm{Sec}_{g}$, ${Ric}_{g}$ and ${R}_{g}$ denote the sectional, Ricci
and scalar curvature of $g$, respectively. When $n\geqslant 3$ we denote for
$g$

$\displaystyle
A_{{g}}^{\tau,\alpha}=\frac{\alpha}{n-2}\left({Ric}_{g}-\frac{\tau}{2(n-1)}{R}_{g}\cdot{g}\right),\,$
$\displaystyle\alpha=\pm 1,\mbox{ }\tau\in\mathbb{R}.$

When $\alpha=1$, it is the modified Schouten tensor
$A_{g}^{\tau}=\frac{1}{n-2}({Ric}_{g}-\frac{\tau}{2(n-1)}{R_{g}}\cdot g).$
When $\tau=\alpha=1$, it is the Schouten tensor
$A_{g}=\frac{1}{n-2}({Ric}_{g}-\frac{1}{2(n-1)}{R_{g}}\cdot g).$ For
$\tau=n-1$ and $\alpha=1$, it corresponds to the Einstein tensor
$G_{g}={Ric}_{g}-\frac{1}{2}{R}_{g}\cdot g.$

One natural question to ask is: given a smooth positive function $\psi$, is
there a smooth complete metric $\tilde{g}=e^{2{u}}g$ satisfying

(1.14) $\displaystyle
f(\lambda(\tilde{g}^{-1}A_{\tilde{g}}^{\tau,\alpha}))=\psi\mbox{ }\mbox{ in
}M.$

The equations of this type include many important equations as special cases.
When $f=\sigma_{1}$, $\tau=0$ and $\psi$ is a proper constant, it is closely
related to the well-known Yamabe problem, proved by Schoen [50] combining the
important work of Aubin [1] and Trudinger [57]. For $f=\sigma_{k}^{1/k}$,
$\psi=1$ and $\tau=\alpha=1$, the equation on closed Riemannian manifolds was
proposed by Viaclovsky [60], known as $k$-Yamabe problem. From then on it has
received much attention in [6, 9, 18, 27, 37, 31, 52]. When $M$ has boundary,
the prescribed scalar curvature problem with certain boundary properties was
studied by [12, 13, 14], and further complemented in [32, 7, 44, 45], where
the obtained metrics are not complete. The case is fairly different when the
metric is complete. A deep result of Aviles-McOwen [4], extending the
celebrated theorem of Loewner-Nirenberg [41], yields that every compact
Riemannian manifold of dimension $n\geqslant 3$ with smooth boundary admits a
complete conformal metric with negative constant scalar curvature. Since then
Loewner-Nirenberg and Aviles-McOwen’s results were extended by many experts to
prescribed $\sigma_{k}$ curvature equation

(1.15)
$\displaystyle\sigma_{k}^{1/k}(\lambda(\tilde{g}^{-1}A_{\tilde{g}}^{\tau,\alpha}))=\psi\mbox{
}\mbox{ in }M,$

when imposing restrictions to $(\alpha,\tau)$

(1.16) $\begin{cases}\tau<1\,&\mbox{ if }\alpha=-1,\\\ \tau>n-1\,&\mbox{ if
}\alpha=1.\end{cases}$

Under this assumption, the equation is automatically of fully uniform
ellipticity according to the formula (3.3) below. Below we list part of
related literature. On compact manifolds with boundary, the existence of
solutions to prescribed $\sigma_{k}$-curvature equation for $-{Ric}_{g}$ was
obtained by Guan [22] and Gursky-Streets-Warren [29], and the case $\tau>n-1$,
$\alpha=1$ was considered by Li-Sheng [38]. On the other hand, the prescribed
$\sigma_{k}$-curvature equation on a closed manifold was studied by Gursky-
Viaclovsky [30] with $\tau<1$, $\alpha=-1$, and by Sheng-Zhang [54] for
$\tau>n-1$, $\alpha=1$ based on a flow approach.

Nevertheless, most known results on fully nonlinear Loewner-Nirenberg problem
for modified Schouten tensors require condition (1.16), and it is rarely known
under a broader assumption. To close this gap, one challenge we face is the
critical case $\tau=n-1$, in which the uniform ellipticity has to break down
when $\Gamma=\Gamma_{n}$. This critical case coincides with prescribed
curvature equation for the Einstein tensor

(1.17) $\displaystyle f(\lambda(\tilde{g}^{-1}G_{\tilde{g}}))=\psi\mbox{
}\mbox{ in }M.$

Unfortunately, the topological obstruction as described in Remark 1.13 below
indicates that when $\Gamma=\Gamma_{n}$, at least for $n=3$, the fully
nonlinear Loewner-Nirenberg problem for positive Einstein tensor is unsolvable
in general. This manifests that the obstruction to the solvability of (1.14)
in the conformal class of complete metrics arises primarily from the lack of
fully uniform ellipticity.

####  1.3.1. Prescribed problem I: compact manifolds with boundary

When restricted to the equation (1.14), the assumption (1.8) reads as follows

(1.18) $\begin{cases}\tau<1\,&\mbox{ if }\alpha=-1,\\\
\tau>1+(n-2)(1-\kappa_{\Gamma}\vartheta_{\Gamma})\,&\mbox{ if
}\alpha=1.\end{cases}$

It would be worthwhile to note that the assumption (1.18) is much more broader
than (1.16), and it allows the critical case $\tau=n-1$ whenever
$\Gamma\neq\Gamma_{n}$. As a corollary of Theorem 1.7, we solve (1.14) in the
conformal class of complete metrics.

######  Theorem 1.9.

In addition to (1.2), (1.11) and (1.18), we assume

(1.19) $\displaystyle f(t\lambda)=t^{\mathrm{\varsigma}}f(\lambda),\mbox{
}\forall\lambda\in\Gamma,\mbox{}\forall t>0,\mbox{ for some constant
}0<\mathrm{\varsigma}\leqslant 1.$

Assume that $(M,g)$ is a compact connected Riemannian manifold of dimension
$n\geq 3$ with smooth boundary and support a $C^{2}$ compact conformal
admissible metric. Then for any $0<\psi\in C^{\infty}(\bar{M})$, there exists
at least one smooth complete metric $\tilde{g}=e^{2u}g$ satisfying (1.14).

We construct $C^{2}$ compact conformal admissible metrics, using Morse
functions.

######  Theorem 1.10.

In Theorem 1.9 the assumption on the existence of a compact conformal
admissible metric can be dropped if $(\alpha,\tau)$ further satisfies

(1.20) $\begin{cases}\tau\leqslant 0\,&\mbox{ if }\alpha=-1,\\\ \tau\geqslant
2\,&\mbox{ if }\alpha=1.\end{cases}$

We apply this construction to draw geometric conclusions. For the Ricci tensor
case, together with Theorem 7.1, Theorem 1.10 deduces the following theorem.

######  Theorem 1.11.

Suppose $(M,g)$ is a compact connected Riemannian manifold of dimension
$n\geqslant 3$ with smooth boundary. Then there is a unique complete conformal
metric $\tilde{g}$ with $Ric_{\tilde{g}}<-1$ and
$\sigma_{n}/\sigma_{n-1}(\lambda(-Ric_{\tilde{g}}))=1$.

This gives a new proof of [43, Theorem 1] when $n\geq 3$. Notice also that the
complete metric is obtained in each conformal class. This is in contrast with
[42, Theorems A and C] of Lohkamp.

In addition, we can deform the Einstein tensor.

######  Theorem 1.12.

Let $(M,g)$ be a compact connected Riemannian manifold of dimension
$n\geqslant 3$ with smooth boundary. Let $(f,\Gamma)$ satisfy (1.2), (1.11)
and (1.19). Suppose in addition that $\Gamma\neq\Gamma_{n}$. Then for each
$0<\psi\in C^{\infty}(\bar{M})$, there exists a smooth admissible complete
metric $\tilde{g}=e^{2u}g$ satisfying (1.17).

Note that this theorem is significantly interesting in dimension three, since
the Einstein tensor is closely connected with the sectional curvature. Based
on this, we present a topological obstruction to reveal that the condition
$\Gamma\neq\Gamma_{n}$ imposed in Theorem 1.12 is crucial and cannot be
further dropped.

######  Remark 1.13 (Topological obstruction).

Denote $B_{r}(a)=\left\\{x\in\mathbb{R}^{3}:|x-a|^{2}<r^{2}\right\\}$. Let
$\bar{B}_{r_{1}}(a_{1}),\cdots,\bar{B}_{r_{m}}(a_{m})$ be pairwise disjoint.
For $r\gg 1$ with $\cup_{i=1}^{m}B_{r_{i}}(a_{i})\subset B_{r}(0)$, we denote
$\Omega=B_{r+1}(0)\setminus(\cup_{i=1}^{m}\bar{B}_{r_{i}}(a_{i}))$, $g$ a
Riemannian metric on $\Omega$. Fix $x\in\Omega$, let $\Sigma\subset
T_{x}\Omega$ be a tangent $2$-plane, $\vec{\bf n}\in T_{x}\Omega$ the unit
normal vector to $\Sigma$, then

$\displaystyle G_{g}(\vec{\bf n},\vec{\bf n})=-\mathrm{Sec}_{g}(\Sigma),$

see [28, Section 2] or (3.2) below. If Theorem 1.12 holds in the positive cone
case, then the solution on $\Omega$ is a complete conformal metric with
negative sectional curvature. This contradicts to the Cartan-Hadamard theorem.

####  1.3.2. Prescribed problem II: complete noncompact manifolds

In contrast with the resolution of Yamabe problem on closed Riemannian
manifolds, the complete noncompact version of Yamabe problem is not always
solvable as shown by Jin [34]. Consequently, one could only expect the
solvability of prescribed curvature problem in the conformal class of complete
noncompact metrics under proper additional assumptions. Indeed prior to Jin’s
work, Ni [48, 49] obtained the existence and nonexistence of solutions to
prescribed scalar curvature equation on standard Euclidean spaces, which was
partially extended by Sui [55] to the prescribed $\sigma_{k}$-curvature
equation for negative Ricci curvature. When imposed fairly strong restrictions
to the asymptotic ratio of prescribed functions and curvature, or even to the
topology of background manifolds, Aviles-McOwen [2, 3] and Jin [35]
investigated prescribed scalar curvature equation on negatively curved
complete noncompact Riemannian manifolds, followed by Fu-Sheng-Yuan [16]
recently, who studied prescribed $\sigma_{k}$-curvature equation (1.15) for
$\alpha=-1$ and $\tau<1$. Unfortunately, the restrictions imposed there are
not optimal, even for conformal prescribed scalar curvature equation.

It still remains widely open to determine under which conditions the conformal
prescribed curvature equation is solvable on complete noncompact manifolds. We
close this gap as a consequence of Theorem 1.8. That is, we can solve fully
nonlinear version of Yamabe problem on a complete noncompact Riemannian
manifold, assuming existence of a $C^{2}$ complete admissible metric
$\underline{g}=e^{2\underline{u}}g$ with

(1.21)
$\displaystyle{f(\lambda(\underline{g}^{-1}A_{\underline{g}}^{\tau,\alpha}))}\geqslant\Lambda_{0}{\psi}\mbox{
holds uniformly in $M\setminus K_{0}$.}$

Here $\Lambda_{0}$ is a uniform positive constant, and $K_{0}$ is a compact
subset of $M$.

######  Theorem 1.14.

Let $(M,g)$ be a complete noncompact Riemannian manifold of dimension
$n\geqslant 3$. Suppose $(f,\Gamma)$ satisfies (1.2), (1.11) and (1.19). Given
a smooth positive function $\psi$ and $(\alpha,\tau)$ obeying (1.18), we
assume that $(M,g)$ carries a $C^{2}$ complete admissible conformal metric
subject to (1.21). Then there exists a unique smooth complete maximal
conformal admissible metric satisfying (1.14).

######  Remark 1.15.

Theorems 1.8 and 1.14 reveal that all of geometric and analytic obstructions
to solvability of the corresponding equations are embodied in the assumption
of asymptotic condition at infinity, which is sufficient and necessary:

  * •

The solution we expect to obtain tautologically satisfies the asymptotic
condition at infinity, which is therefore necessary for the existence.

  * •

Prescribed curvature problem (1.14) is not always solvable. Please refer to
[35, 49, 55] for some nonexistence results on conformal prescribed scalar and
Ricci curvature equations.

In conclusion, except the asymptotic assumption at infinity, it is not
required to impose further restrictions to curvatures, prescribed functions
and the topology of background manifolds. This is in contrast to related works
cited above. This is new even in Euclidean spaces. An analogue of Theorem 1.14
also holds for conformal prescribed scalar curvature equation, see Theorem 4.9
below.

######  Remark 1.16.

A somewhat surprising fact to us is that we impose neither (1.3) nor (1.4) in
Theorems 1.4, 1.7, 1.8, 1.9, 1.10, 1.12 and 1.14. This is in contrast with
huge literature on fully nonlinear elliptic equations.

In conclusion, in this paper we first prove partial uniform ellipticity for
fully nonlinear equations and then use it to confirm the uniform ellipticity
of a special class of fully nonlinear equations. As an application, we solve a
fully nonlinear Loewner-Nirenberg problem, given a compact conformal
admissible metric. This assumption is partially confirmed, utilizing Morse
theory. As a result, we can deduce various geometric conclusions. Furthermore,
we examine asymptotic behavior and uniqueness of solutions to fully nonlinear
Loewner-Nirenberg problem. Building on these solutions, we prove the existence
and uniqueness of complete maximal conformal metric solving a complete
noncompact version of fully nonlinear Yamabe problem, under an asymptotic
condition at infinity which is in effect sharp. Our approach also works for
conformal scalar curvature equation. Moreover, the topological obstruction
indicates that our strategy is delicate.

The paper is organized as follows. In Section 2 we investigate the partial
uniform ellipticity. This is one of the most important parts of the paper. In
Section 3 we confirm the uniform ellipticity of (1.7). This is crucial for our
approach. In Section 4 we solve a class of equations of fully uniform
ellipticity on complete noncompact Riemannian manifolds, using an approximate
argument. The key ingredient is the solution of the Dirichlet problem with
infinite boundary value condition. The proof of existence of such solutions is
left to Section 5. Moreover, in Section 7 the approximation method is also
used to analyze asymptotic behavior and uniqueness of complete conformal
metrics. In Section 6 we construct conformal admissible metrics, based on a
result on Morse functions. In the final section, we derive local and boundary
estimates for equations of fully uniform ellipticity.

##  2\. The partial uniform ellipticity

We start with the following lemma.

######  Lemma 2.1.

For $(f,\Gamma)$ satisfying (1.2), the following statements are equivalent.

  1. $(1)$

$(f,\Gamma)$ satisfies (1.6).

  2. $(2)$

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

  3. $(3)$

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

###### Proof.

It follows from the concavity of $f$ that

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

$\mathrm{(1)}\Rightarrow\mathrm{(2)}$: Fix $\lambda\in\Gamma$. The condition
(1.6) implies that for any $\mu\in\Gamma$, there is $T\geqslant 1$ (may depend
on $\mu$) such that for each $t>T$, $f(t\mu)>f(\lambda)$. Together with (2.1),
one gets $\sum_{i=1}^{n}f_{i}(\lambda)(t\mu_{i}-\lambda_{i})>0$. Thus,
$\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}>0$ (if one takes $\mu=\lambda$) and
moreover $\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}>0$.

$\mathrm{(2)}\Rightarrow\mathrm{(3)}$: The proof uses (2.1).

$\mathrm{(3)}\Rightarrow\mathrm{(1)}$: For any $\lambda,\mbox{ }\mu\in\Gamma$,
$t\lambda-\mu\in\Gamma$ for $t>t_{\lambda,\mu}$, depending only on $\lambda$
and $\mu$. So $f(t\lambda)>f(\mu)$ for such $t$. ∎

######  Remark 2.2.

In the presence of (1.2) and (1.3), this lemma was initially proposed by the
author in [62, Lemma 3.2] to set up quantitative boundary estimate of the form

$\displaystyle\sup_{\partial M}|\partial\overline{\partial}u|\leqslant
C(1+\sup_{M}|\partial u|^{2})$

for the Dirichlet problem of fully nonlinear elliptic equations on complex
manifolds. In Lemma 2.1, the assumption (1.3) has been removed.

######  Corollary 2.3.

If $f$ satisfies (1.2) and (1.6), then (1.4) holds.

###### Proof.

Fix $\lambda\in\Gamma.$ According to Lemma 2.1,
$\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}>0,\mbox{ }\forall\mu\in\Gamma$. Note that
$\Gamma_{n}\subseteq\Gamma$, then $f_{i}(\lambda)\geqslant 0,$ $\forall
1\leqslant i\leqslant n.$ We have $\sum_{i=1}^{n}f_{i}(\lambda)>0$ by setting
$\mu=\vec{\bf 1}$. ∎

In the following proposition, we relate $\kappa_{\Gamma}$ to the maximal count
of negative components of vectors in $\Gamma$. Let’s denote

######  Definition 2.4.

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

######  Proposition 2.5.

$\kappa_{\Gamma}=\widetilde{\kappa}_{\Gamma}$.

###### Proof.

The case $\Gamma=\Gamma_{n}$ is true since $\kappa_{\Gamma_{n}}=0$ and
$\widetilde{\kappa}_{\Gamma_{n}}=0$. Next, we consider the case
$\Gamma\neq\Gamma_{n}$. Assume
$(-\alpha_{1},\cdots,-\alpha_{\widetilde{\kappa}_{\Gamma}},\alpha_{\widetilde{\kappa}_{\Gamma}+1},\cdots,\alpha_{n})\in\Gamma$,
for $\alpha_{i}>0$, $\forall 1\leqslant i\leqslant n$. Then
$(0,\cdots,0,\alpha_{\widetilde{\kappa}_{\Gamma}+1},\cdots,\alpha_{n})\in\Gamma$,
which implies ${\kappa}_{\Gamma}\geqslant\widetilde{\kappa}_{\Gamma}$.
Conversely, if

$({\overbrace{0,\cdots,0}^{{\kappa}_{\Gamma}}},{\overbrace{1,\cdots,1}^{n-{\kappa}_{\Gamma}}})\in\Gamma,$

then by the openness of $\Gamma$, we have for some $0<\epsilon\ll 1$,

$({\overbrace{-\epsilon,\cdots,-\epsilon}^{{\kappa}_{\Gamma}}},{\overbrace{1,\cdots,1}^{n-{\kappa}_{\Gamma}}})\in\Gamma.$

Consequently, $\widetilde{\kappa}_{\Gamma}\geqslant{\kappa}_{\Gamma}$. ∎

Corollary 2.3 is a part of Theorem 1.4. Below we complete the proof. Fix
$\lambda\in\Gamma$. It follows from the concavity and symmetry of $f$ that

$f_{i}(\lambda)\geqslant f_{j}(\lambda)\mbox{ for
}\lambda_{i}\leqslant\lambda_{j}.$

In particular

$f_{1}(\lambda)\geqslant\frac{1}{n}\sum_{i=1}^{n}f_{i}(\lambda)\mbox{ if
}\lambda_{1}\leqslant\cdots\leqslant\lambda_{n}.$

Therefore, for the case $\Gamma=\Gamma_{n}$ $(\mbox{i.e. }\kappa_{\Gamma}=0)$,
we immediately obtain Theorem 1.4. For general $\Gamma$, Theorem 1.4 is a
consequence of Proposition 2.5 and the following proposition.

######  Proposition 2.6.

Assume $\Gamma\neq\Gamma_{n}$ and $f$ satisfies (1.2) and (1.6) in $\Gamma$.
For the $\kappa_{\Gamma}$ as defined in Definition 1.3, let
$\alpha_{1},\cdots,\alpha_{n}$ 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
$\alpha_{1}\geqslant\cdots\geqslant\alpha_{\kappa_{\Gamma}}$. Then for each
$\lambda\in\Gamma$ with order
$\lambda_{1}\leqslant\cdots\leqslant\lambda_{n}$,

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

###### Proof.

Fix $\lambda\in\Gamma$ and we assume
$\lambda_{1}\leqslant\lambda_{2}\leqslant\cdots\leqslant\lambda_{n}$. From
Corollary 2.3, $f_{i}(\lambda)\geqslant 0$ and
$\sum_{i=1}^{n}f_{i}(\lambda)>0$. By Lemma 2.1,

(2.3)
$\displaystyle-\sum_{i=1}^{\kappa_{\Gamma}}\alpha_{i}f_{i}(\lambda)+\sum_{i=\kappa_{\Gamma}+1}^{n}\alpha_{i}f_{i}(\lambda)>0$

which simply yields
$f_{\kappa_{\Gamma}+1}(\lambda)>\frac{\alpha_{1}}{\sum_{i=\kappa_{\Gamma}+1}^{n}\alpha_{i}}f_{1}(\lambda)$.
In addition, one can derive (2.2) by using the iteration.

∎

######  Remark 2.7.

When $\Gamma\neq\Gamma_{n}$ the constant $\vartheta_{\Gamma}$ in Theorem 1.4
can be achieved as follows:

$\vartheta_{\Gamma}=\sup_{(-\alpha_{1},\cdots,-\alpha_{\kappa_{\Gamma}},\alpha_{\kappa_{\Gamma}+1},\cdots,\alpha_{n})\in\Gamma,\,\alpha_{i}>0}\,\frac{\alpha_{1}/n}{\sum_{i=\kappa_{\Gamma}+1}^{n}\alpha_{i}-\sum_{i=2}^{\kappa_{\Gamma}}\alpha_{i}}.$

A somewhat remarkable fact to us is that in the description of partial uniform
ellipticity, $\kappa_{\Gamma}$ and $\vartheta_{\Gamma}$ depend only on
$\Gamma$ instead specifically on $f$. Consequently, it has great advantages in
applications to PDEs and geometry. For instance, Theorem 1.4 allows us to
compute the partial uniform ellipticity of fully nonlinear equations in
connection with the notions of $p$-convexity [51, 61],
$\mathbb{F}$-subharmonic and $\mathbb{G}$-plurisubharmonic functions in the
sense of Harvey-Lawson [33].

The following proposition precisely reveals the interaction between
$\kappa_{\Gamma}$ and partial uniform ellipticity. Denote

(2.4)
$\Gamma^{\infty}_{\mathbb{R}^{k}}:=\left\\{\lambda^{\prime}\in\mathbb{R}^{k}:(\lambda^{\prime},c,\cdots,c)\in\Gamma\mbox{
for some }c>0\right\\}.$

######  Proposition 2.8.

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

###### Proof.

Let $\vartheta$ be as in (1.5). As above, $\vec{\bf 1}=(1,\cdots,1)$. For the
$(f,\Gamma)$, we denote

$\displaystyle\Gamma^{\sigma}=\\{\lambda\in\Gamma:f(\lambda)>\sigma\\}.$

Let $c_{0}$ be the positive constant with $f(c_{0}\vec{\bf
1})>\sup_{\partial\Gamma}f$, and then we set $a=1+c_{0}$. Thus $f(a\vec{\bf
1})>f(c_{0}\vec{\bf 1})$ by $\sum_{i=1}^{n}f_{i}(\lambda)>0$. For
$0<\epsilon<a$ and $R=\frac{a}{\vartheta}$, we denote
$\lambda_{\epsilon,R}=({\overbrace{\epsilon,\cdots,\epsilon}^{k}},{\overbrace{R,\cdots,R}^{n-k}}).$
We can deduce from (2.1) that

$\displaystyle f(\lambda_{\epsilon,R})\geqslant$ $\displaystyle f(a\vec{\bf
1})+\epsilon\sum_{i=1}^{k}f_{i}(\lambda_{\epsilon,R})+R\sum_{i=k+1}^{n}f_{i}(\lambda_{\epsilon,R})-a\sum_{i=1}^{n}f_{i}(\lambda_{\epsilon,R})$
$\displaystyle\geqslant$ $\displaystyle f(a\vec{\bf
1})+(R\vartheta-a)\sum_{i=1}^{n}f_{i}(\lambda_{\epsilon,R})\mbox{ (using
$(k+1)$-uniform ellipticity)}$ $\displaystyle=$ $\displaystyle f(a\vec{\bf
1})\mbox{ (noticing }R=\frac{a}{\vartheta})$ $\displaystyle>$ $\displaystyle
f(c_{0}\vec{\bf 1}).$

So
$\lambda_{\epsilon,R}=(\epsilon,\cdots,\epsilon,R,\cdots,R)\in\overline{\Gamma^{f(a\vec{\bf
1})}}$. Notice that $R=\frac{a}{\vartheta}$ does not depend on $\epsilon$.
Taking $\epsilon\rightarrow 0^{+}$, we get
$(0,\cdots,0,R,\cdots,R)\in\overline{\Gamma^{f(a\vec{\bf
1})}}\subset\Gamma^{f(c_{0}\vec{\bf 1})}\subset\Gamma.$ Thus
$\kappa_{\Gamma}\geqslant k$. ∎

######  Remark 2.9.

The proof uses the simple fact $\Gamma_{n}\subseteq\Gamma$. This is different
from the proof of Proposition 3.3 below.

######  Corollary 2.10.

The $(\kappa_{\Gamma}+1)$-uniform ellipticity asserted in Theorem 1.4 cannot
be improved.

The following two corollaries are key ingredients in the construction of local
barriers in Section 5.

######  Corollary 2.11.

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

######  Corollary 2.12.

If $\Gamma$ carries a smooth symmetric concave $f$ satisfying (1.2) and (1.6),
then the following statements are equivalent to each other.

  1. $\mathrm{(1)}$

$f$ is of fully uniform ellipticity in $\Gamma$.

  2. $\mathrm{(2)}$

$\Gamma$ is of type 2. That is
$\Gamma^{\infty}_{\mathbb{R}^{n-1}}=\mathbb{R}^{n-1}$, where
$\Gamma^{\infty}_{\mathbb{R}^{n-1}}$ is as we denoted in (2.4).

######  Lemma 2.13.

Let $\kappa_{\Gamma}$ be as in Definition 1.3, $\vartheta_{\Gamma}$ be as in
Theorem 1.4, then

  1. $\mathrm{(1)}$

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

  2. $\mathrm{(2)}$

For $\Gamma=\Gamma_{k}$, $\kappa_{\Gamma}=n-k$.

  3. $\mathrm{(3)}$

$\kappa_{\Gamma}=0$ if and only if $\Gamma=\Gamma_{n}$.

  4. $\mathrm{(4)}$

$\kappa_{\Gamma}=n-1$ if and only if $\Gamma$ is of type 2.

  5. $\mathrm{(5)}$

$0<\vartheta_{\Gamma}\leqslant\frac{1}{n}$.

Moreover, if $\vartheta_{\Gamma}=\frac{1}{n}$ and $\kappa_{\Gamma}=n-1$ occur
simultaneity then

$f_{i}(\lambda)=\frac{1}{n}\sum_{j=1}^{n}f_{j}(\lambda),\quad\forall\lambda\in\Gamma,\mbox{
}\forall 1\leqslant i\leqslant n.$

###### Proof.

Note that $\Gamma\subseteq\Gamma_{1}$, we know $\kappa_{\Gamma}\leqslant n-1$.
Next we prove the last statement. Set $\lambda=\vec{\bf 1}$, then
$f_{i}(\vec{\bf 1})=\frac{1}{n}\sum_{j=1}^{n}f_{j}(\vec{\bf 1}),\mbox{
}\forall 1\leqslant i\leqslant n$. This simply yields
$\vartheta_{\Gamma}\leqslant\frac{1}{n}$. The proofs of the other three
statements are obvious, and we omit them here. ∎

As consequences of Lemma 2.13, we obtain the following lemmas.

######  Lemma 2.14.

Suppose $\varrho$ obeys (1.8). Then $\varrho<n$ and $\varrho\neq 0.$

######  Lemma 2.15.

Suppose $(\alpha,\tau)$ satisfies (1.18). Then

$\displaystyle\alpha(n\tau+2-2n)>0,\quad(\tau-1)(n\tau+2-2n)>0.$

It seems very interesting to answer whether Gårding’s cone is the largest one
to share same $\kappa_{\Gamma}$.

######  Problem 2.16.

Fix $1<k<n$. For the cone $\Gamma$ with $\kappa_{\Gamma}=n-k$, is
$\Gamma\subseteq\Gamma_{k}$ true?

######  Problem 2.17.

For any $\Gamma$, $\hat{\Gamma}$ with
$\kappa_{\Gamma}\geqslant\kappa_{\hat{\Gamma}}+1$, is
$\hat{\Gamma}\subset\Gamma$ true?

Finally, we confirm (1.6) under the assumptions (1.10) and (1.11).

######  Lemma 2.18.

Assume, in addition to (1.2), that $\sup_{\Gamma}f=+\infty$ and

(2.5) $\displaystyle\lim_{t\rightarrow+\infty}f(t\lambda)>-\infty,\mbox{
}\forall\lambda\in\Gamma.$

Then $(f,\Gamma)$ satisfies (1.6).

###### Proof.

We have by (2.1) the following inequality

$\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}\geqslant\limsup_{t\rightarrow+\infty}f(t\mu)/t,\quad\forall\lambda,\mbox{
}\mu\in\Gamma.$

Thus

(2.6) $\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}\geqslant
0,\quad\forall\lambda,\mbox{ }\mu\in\Gamma.$

So $f_{i}(\lambda)\geqslant 0$, $\forall\lambda\in\Gamma,1\leqslant i\leqslant
n$. Together with $\sup_{\Gamma}f=+\infty$ and (2.1), we derive

(2.7) $\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)>0\mbox{ in }\Gamma.$

By (2.6), (2.7) and the openness of $\Gamma$, we get $(2)$ from Lemma 2.1, as
required.

∎

##  3\. Problem reduction

###  3.1. 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 $M$. From now on we denote

$\langle X,Y\rangle=g(X,Y),\quad g_{ij}=\langle
e_{i},e_{j}\rangle,\quad\\{g^{ij}\\}=\\{g_{ij}\\}^{-1}.$

Under Levi-Civita connection of $(M,g)$,
$\nabla_{e_{i}}e_{j}=\Gamma_{ij}^{k}e_{k}$, and $\Gamma_{ij}^{k}$ denote the
Christoffel symbols. The curvature coefficients are given by $R_{ijkl}=\langle
e_{i},R(e_{k},e_{l})e_{j}\rangle$. For simplicity we write

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

Under the local unit orthogonal frame

(3.1) $\displaystyle{Ric}_{g}(e_{i},e_{i})=\sum_{j=1}^{n}\langle
R(e_{i},e_{j})e_{i},e_{j}\rangle,\,$
$\displaystyle{R}_{g}=\sum_{i=1}^{n}{Ric}_{g}(e_{i},e_{i}),$ (3.2)
$\displaystyle{Ric}_{g}(e_{i},e_{i})=\frac{{R}_{g}}{2}-\frac{1}{2}\sum_{k,l\neq
i}\langle R(e_{k},e_{l})e_{k},e_{l}\rangle,\mbox{ }\forall 1\leqslant
i\leqslant n.$

Under the conformal change $\tilde{g}=e^{2u}g$, one has (see e.g. [5])

$\displaystyle{Ric}_{\tilde{g}}=$ $\displaystyle{Ric}_{g}-\Delta
ug-(n-2)\nabla^{2}u-(n-2)|\nabla u|^{2}g+(n-2)du\otimes du.$

Thus

(3.3) $\displaystyle
A_{\tilde{g}}^{\tau,\alpha}=A_{g}^{\tau,\alpha}+\frac{\alpha(\tau-1)}{n-2}\Delta
ug-\alpha\nabla^{2}u+\frac{\alpha(\tau-2)}{2}|\nabla u|^{2}g+\alpha du\otimes
du.$

In particular, the Schouten tensor obeys

(3.4) $\displaystyle A_{\tilde{g}}=A_{g}-\nabla^{2}u-\frac{1}{2}|\nabla
u|^{2}g+du\otimes du.$

###  3.2. Fully uniform ellipticity and construction of type 2 cones

First, we shall summarize the definition of type 2 cones.

######  Definition 3.1 ([8]).

$\Gamma$ is said to be of type 1 if the positive $\lambda_{i}$ axes belong to
$\partial\Gamma$; otherwise it is called of type 2.

Except $\Gamma_{1}$, it seems not easy to find general type 2 cones. Based on
Theorem 1.4, we can construct some examples of type 2 cones.

Given a cone $\Gamma$, we take a constant $\varrho$ satisfying (1.8). Note
that $\varrho<n$ and $\varrho\neq 0$ by Lemma 2.14. We let

(3.5)
$\displaystyle\mu_{i}=\frac{1}{n-\varrho}\left(\sum_{j=1}^{n}\lambda_{j}-\varrho\lambda_{i}\right),\mbox{
i.e.
}\lambda_{i}=\frac{1}{\varrho}\left(\sum_{j=1}^{n}\mu_{j}-(n-\varrho)\mu_{i}\right),$
(3.6)
$\displaystyle\tilde{\Gamma}=\left\\{(\lambda_{1},\cdots,\lambda_{n}):\lambda_{i}=\frac{1}{\varrho}\left(\sum_{j=1}^{n}\mu_{j}-(n-\varrho)\mu_{i}\right),\mbox{
}(\mu_{1},\cdots,\mu_{n})\in\Gamma\right\\}.$

So $\tilde{\Gamma}$ is also an open symmetric convex cone of $\mathbb{R}^{n}$
with $\tilde{\Gamma}\subseteq\Gamma_{1}$. One has a symmetric concave function
$\tilde{f}$ on $\tilde{\Gamma}$ as follows:

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

######  Proposition 3.2 (Fully uniform ellipticity).

Let $(f,\Gamma)$ satisfy (1.2) and (1.6). Let $(\tilde{f},\tilde{\Gamma})$ be
as above. Suppose in addition that $\varrho$ satisfies (1.8). Then there is a
uniform positive constant $\theta$ such that

(3.8)
$\displaystyle\frac{\partial\tilde{f}}{\partial\lambda_{i}}(\lambda)\geqslant\theta\sum_{j=1}^{n}\frac{\partial\tilde{f}}{\partial\lambda_{j}}(\lambda)>0\mbox{
in }\tilde{\Gamma},\quad\forall 1\leqslant i\leqslant n.$

Moreover, $(\tilde{f},\tilde{\Gamma})$ also satisfies (1.2) and (1.6).

###### Proof.

Case 1: $\varrho<0$. A straightforward computation shows

$\frac{\partial\tilde{f}}{\partial\lambda_{i}}=\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}\frac{\partial\mu_{j}}{\partial\lambda_{i}}=\frac{1}{n-\varrho}\left(\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}-\varrho\frac{\partial
f}{\partial\mu_{i}}\right)\geqslant\frac{1}{n-\varrho}\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}=\frac{1}{n-\varrho}\sum_{j=1}^{n}\frac{\partial\tilde{f}}{\partial\lambda_{j}}.$

Case 2: $0<\varrho<\frac{1}{1-\kappa_{\Gamma}\vartheta_{\Gamma}}$. By Theorem
1.4, $(1-\kappa_{\Gamma}\vartheta_{\Gamma})\sum_{j=1}^{n}\frac{\partial
f}{\partial\mu_{j}}\geqslant\frac{\partial f}{\partial\mu_{i}},$ $\forall
1\leqslant i\leqslant n$. Thus

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

∎

######  Proposition 3.3 (Construction of type 2 cones).

Let $\varrho$ satisfy (1.8), and let $\tilde{\Gamma}$ be the cone as defined
in (3.6). Then $\Gamma_{n}\subset\tilde{\Gamma}$ and $\tilde{\Gamma}$ is a
type 2 cone.

###### Proof.

Given $(f,\Gamma)$, as in (3.6) and (3.7), we obtain
$(\tilde{f},\tilde{\Gamma})$. It only requires to verify

$(0\cdots,0,1)\in\tilde{\Gamma}.$

Denote $\lambda_{t}=(t\cdots,t,1)$ and
$I=\\{t\in[0,1]:\lambda_{t}\in\tilde{\Gamma}\\}.$ Clearly, $1\in I$, and $I$
is an open subset of $I$ by the openness of $\tilde{\Gamma}$. It is sufficient
to prove the closeness. Let $\\{t_{i}\\}\subset I$, $0<t_{i}\leqslant 1$ and

$t_{0}=\lim_{i\to+\infty}t_{i}.$

It only requires to prove $t_{0}\in I$. We denote
$\tilde{\Gamma}^{\sigma}=\\{\lambda\in\tilde{\Gamma}:\tilde{f}(\mu)>\sigma\\}.$
Let $\theta$ be as in (3.8), and let $\tilde{f}(a\vec{\bf
1})>\sup_{\partial\tilde{\Gamma}}\tilde{f}$. We deduce from (3.8) and the
concavity of $\tilde{f}$ that

$\displaystyle\tilde{f}\left(\frac{a}{\theta}\lambda_{t_{i}}\right)\geqslant\tilde{f}(a\vec{\bf
1})+\frac{at_{i}}{\theta}\sum_{j=1}^{n-1}\frac{\partial\tilde{f}}{\partial\lambda_{j}}\left(\frac{a}{\theta}\lambda_{t_{i}}\right)+\frac{a}{\theta}\cdot\frac{\partial\tilde{f}}{\partial\lambda_{n}}\left(\frac{a}{\theta}\lambda_{t_{i}}\right)-a\sum_{j=1}^{n}\frac{\partial\tilde{f}}{\partial\lambda_{j}}\left(\frac{a}{\theta}\lambda_{t_{i}}\right)>\tilde{f}(a\vec{\bf
1}).$

Thus
$\frac{a}{\theta}\lambda_{t_{0}}\in\overline{\tilde{\Gamma}^{\tilde{f}(a\vec{\bf
1})}}$. As required, $\lambda_{t_{0}}\in\tilde{\Gamma}.$ This gives $I=[0,1]$.
∎

######  Corollary 3.4.

For $\varrho$ satisfying (1.8), $(1,\cdots,1,1-\varrho)\in\Gamma.$

######  Remark 3.5.

From the construction of $\tilde{\Gamma}$, it is not easy to see
$\Gamma_{n}\subset\tilde{\Gamma}$.

###  3.3. Reduction to Schouten tensor case

We reduce the prescribed curvature equation (1.14) to a uniform elliptic
equation for conformal deformation of Schouten tensor. To do this, we can
check

$\displaystyle\mathrm{tr}\left(g^{-1}(-A_{g})\right)g-\varrho\left(-A_{g}\right)=\frac{n-2}{\alpha(\tau-1)}A_{g}^{\tau,\alpha},$

where we take

$\displaystyle\varrho=\frac{n-2}{\tau-1}.$

######  Lemma 3.6.

Let $(\alpha,\tau)$ satisfy (1.18), let $\varrho=\frac{n-2}{\tau-1}$. Then

$\displaystyle\varrho<\frac{1}{1-\kappa_{\Gamma}\vartheta_{\Gamma}}\mbox{ and
}\varrho\neq 0.$

Therefore, when $f$ is homogeneous of degree $\mathrm{\varsigma}$, the
equation (1.14) is equivalent to

(3.9)
$\displaystyle\tilde{f}(\lambda(-\tilde{g}^{-1}A_{\tilde{g}}))=\left(\frac{n-2}{\alpha(n\tau+2-2n)}\right)^{\mathrm{\varsigma}}\psi,$

in which $\tilde{f}$ is of fully uniform ellipticity in $\tilde{\Gamma}$
according to Proposition 3.2. Here $(\tilde{f},\tilde{\Gamma})$ is as in
(3.6)-(3.7). To study this equation, according to the formula (3.4), it
requires to consider

(3.10) $\displaystyle\tilde{f}(\lambda(g^{-1}(\nabla^{2}u+\frac{1}{2}|\nabla
u|^{2}g-du\otimes du-
A_{g})))=\left(\frac{n-2}{\alpha(n\tau+2-2n)}\right)^{\mathrm{\varsigma}}\psi
e^{2\mathrm{\varsigma}u}.$

######  Remark 3.7.

In what follows, we replace $(\tilde{f},\tilde{\Gamma})$ by $(f,\Gamma)$ if
necessary.

##  4\. The equations on complete noncompact manifolds and applications to
prescribed curvature problem

From Proposition 3.2 and Lemma 2.18, (1.7) falls into the equation of the form

(4.1) $\displaystyle f\left(\lambda(g^{-1}(\nabla^{2}u+A(x,\nabla
u)))\right)=\psi(x,u),$

in which ${f}$ is of fully uniform ellipticity in ${\Gamma}$, i.e., there is a
uniform constant $\theta$ such that

(4.2) $\displaystyle
f_{i}(\lambda)\geqslant\theta\sum_{j=1}^{n}f_{j}(\lambda)>0\mbox{ in
}\Gamma,\quad\forall 1\leqslant i\leqslant n.$

In addition, $\psi(x,t)$ is a smooth positive function on $M\times\mathbb{R}$
satisfying (1.9), and $A(x,p)$ is a smooth symmetric $(0,2)$-tensor satisfying
(1.12).

In this section we solve the equation (4.1) on a complete noncompact
Riemannian manifold, given an asymptotic condition at infinity

(4.3) $\displaystyle
f(\lambda(g^{-1}(\nabla^{2}\underline{u}+A(x,\nabla\underline{u}))))\geqslant\psi(x,\underline{u}),\mbox{
}\forall x\in M.$

Also, for the equation (4.1), we say a $C^{2}$ function $w$ is an admissible
function, if

$\displaystyle\lambda(g^{-1}(\nabla^{2}w+A(x,\nabla w)))\in\Gamma.$

######  Theorem 4.1.

Assume $(M,g)$ is a complete noncompact Riemannian manifold of dimension
$n\geqslant 3$ and suppose a $C^{2}$-admissible function $\underline{u}$
satisfying the asymptotic condition (4.3). Suppose in addition that (1.2),
(1.10), (1.11), (1.9), (1.12) and (4.2) hold. Then the equation (4.1) has a
unique smooth maximal admissible solution $u$. Moreover,
$u\geqslant\underline{u}$ in $M.$

As a consequence, we obtain Theorem 1.8.

######  Remark 4.2.

Suppose for any positive constant $\Lambda$ that

$\displaystyle\Lambda\psi(x,t)\geqslant\psi(x,t+C_{\Lambda}),\mbox{
}\forall(x,t)\in M\times\mathbb{R}$

holds for a constant $C_{\Lambda}$. Then the asymptotic condition (4.3) can be
replaced by

$\displaystyle
f(\lambda(g^{-1}(\nabla^{2}\underline{u}+A(x,\nabla\underline{u}))))\geqslant\Lambda\psi(x,\underline{u}).$

### Comparison principle

First of all, we deduce a comparison principle.

######  Lemma 4.3.

Let $(\Omega,g)$ be a compact Riemannian manifold with boundary
$\partial\Omega$. Suppose (1.2), (1.10), (1.11) and (1.9) hold. Let $w,v\in
C^{2}(\Omega)\cap C(\bar{\Omega})$ be admissible functions subject to

$f(\lambda(g^{-1}(\nabla^{2}w+A(x,\nabla w))))\geqslant\psi(x,w),\mbox{
}f(\lambda(g^{-1}(\nabla^{2}v+A(x,\nabla v))))\leqslant\psi(x,v)\mbox{ in
}\Omega,$ $w-v\leqslant 0\mbox{ on }\partial\Omega.$

Then

$\displaystyle w-v\leqslant 0\mbox{ in }\Omega.$

###### Proof.

Assume by contradiction that there is an interior point $x_{0}\in\Omega$ such
that $0<(w-v)(x_{0})=\sup_{\Omega}(w-v)$. Therefore at $x_{0}$,

$\displaystyle\nabla^{2}v\geqslant\nabla^{2}w,\,$ $\displaystyle\nabla
v=\nabla w.$

By Lemma 2.18 and Corollary 2.3, one has (1.4). Thus at $x_{0}$

$\displaystyle f(\lambda(g^{-1}(\nabla^{2}v+A(x,\nabla v))))\geqslant
f(\lambda(g^{-1}(\nabla^{2}w+A(x,\nabla w)))),$

which further yields $v(x_{0})\geqslant w(x_{0})$ by (1.9). This contradicts
to $w(x_{0})>v(x_{0})$. ∎

### Approximation and the proof of Theorem 4.1

We employ an approximate method, based on Theorem 5.3 below. Let
$\\{M_{k}\\}_{k=1}^{+\infty}$ be an exhaustion series of domains with

  * •

$M=\cup_{k=1}^{\infty}M_{k},\mbox{ }\bar{M}_{k}=M_{k}\cup\partial M_{k},\mbox{
}\bar{M}_{k}\subset\subset M_{k+1}$.

  * •

$\bar{M}_{k}$ is a compact $n$-manifold with smooth boundary $\partial M_{k}$.

According to Theorem 5.3, for each $k\geqslant 1$, there is an admissible
function $u_{k}\in C^{\infty}(M_{k})$ satisfying

(4.4) $\displaystyle f\left(\lambda(g^{-1}(\nabla^{2}u_{k}+A(x,\nabla
u_{k})))\right)=\psi(x,u_{k})\mbox{ in }M_{k},$ (4.5)
$\displaystyle\lim_{x\rightarrow\partial M_{k}}u_{k}(x)=+\infty.$

By the comparison principle (Lemma 4.3), we immediately derive

######  Proposition 4.4.

Let $u_{k}$ be the admissible solution to problem (4.4)-(4.5), then

$\displaystyle u_{k+1}\leqslant u_{k}\mbox{ in }M_{k},\quad\forall k\geqslant
1.$

######  Proposition 4.5.

For any admissible solution $u_{k}$ to (4.4)-(4.5), we have

$u_{k}\geqslant\underline{u}\mbox{ in }M_{k},\quad\forall k\geqslant 1.$

Proposition 4.4 states that $\\{u_{k}\\}_{k=m}^{\infty}$ is a decreasing
sequence of functions on $M_{m}$; while such a decreasing sequence is
uniformly bounded from below according to Proposition 4.5. Thus we can take
the limit

(4.6) $u_{\infty}=\lim_{k\to+\infty}u_{k}.$

Moreover, $u_{\infty}\geqslant\underline{u}$ in $M.$ In Propositions 4.4-4.5,
and Theorems 8.1-8.2 below, we establish the local estimates up to second
order derivatives for each $u_{k}$. Combining with Evans-Krylov theorem [15,
36] and classical Schauder theory, we know that $u_{\infty}$ is in fact a
smooth admissible solution to the equation (4.1).

Let $w_{\infty}$ be another admissible solution to the equation (4.1), then by
the comparison principle we have $u_{k}\geqslant w_{\infty}$ in $M_{k}$ for
all $k\geqslant 1$. Thus $u_{\infty}(x)\geqslant w_{\infty}(x),$ $\forall x\in
M.$ This means that the solution given by (4.6) is the maximal solution. The
uniqueness of maximal solution is obvious.

### Prescribed curvature problem on complete noncompact manifolds

Note that the equation (3.10) is a special case of (4.1). As a consequence of
Theorem 4.1, we obtain the following result and Theorem 1.14.

######  Theorem 4.6.

Suppose $(f,\Gamma)$ satisfies (1.2), (1.11), (1.19) and (4.2). Let $(M,g)$ be
a complete noncompact Riemannian manifold of dimension $n\geqslant 3$ and with
a $C^{2}$ complete conformal metric $\underline{g}$ subject to

(4.7) $\displaystyle\lambda(-g^{-1}A_{\underline{g}})\in\Gamma\mbox{ }\mbox{
in }M$

and

(4.8) $\displaystyle
f(\lambda(-\underline{g}^{-1}A_{\underline{g}}))\geqslant\Lambda_{1}\psi\mbox{
}\mbox{ in }M$

where $0<\psi\in C^{\infty}(M)$ and $\Lambda_{1}$ is a uniform positive
constant. Then there exists a unique smooth maximal complete metric
$\tilde{g}=e^{2u}g$ satisfying $f(\lambda(-\tilde{g}^{-1}A_{\tilde{g}}))=\psi$
and $f(\lambda(-{g}^{-1}A_{\tilde{g}}))\in\Gamma.$

######  Remark 4.7.

When $M\subset\mathbb{R}^{n}$ is a bounded domain, the continuous viscosity
solution for $f(\lambda(-\tilde{g}^{-1}A_{\tilde{g}}))=1$ was studied by
González-Li-Nguyen [20] using Perron’s method. The Dirichlet problem for
$\sigma_{k}^{1/k}(\tilde{g}^{-1}A_{\tilde{g}})=\psi$ was studied by Guan [21],
while the obtained metric is not complete.

Our method is fairly different, which is based on partial uniform ellipticity.
In addition, the metrics obtained in this paper are smooth and complete.

As a consequence of Theorem 1.14, together with the Maclaurin inequalities

$\displaystyle\left({\sigma_{k}(\lambda)}/{C_{n}^{k}}\right)^{1/k}\leqslant\left({\sigma_{l}(\lambda)}/{C_{n}^{l}}\right)^{1/l},\quad\forall\lambda\in\Gamma_{k},\mbox{
}1\leqslant l<k\leqslant n$

we obtain the existence of solutions to prescribed $\sigma_{k}$-curvature
equations.

######  Theorem 4.8.

Fix $\Gamma=\Gamma_{k}$ and $(\alpha,\tau)$ satisfying (1.18), we assume
$(M,g)$ is a complete noncompact Riemannian manifold of dimension $n\geqslant
3$ with $\lambda(g^{-1}A_{g}^{\tau,\alpha})\in\Gamma_{k}$ and

$\displaystyle\sigma_{k}^{1/k}(\lambda(g^{-1}A_{g}^{\tau,\alpha}))\geqslant\delta$

for some positive constant $\delta.$ Then there is a unique
$(\tilde{g}_{1},\tilde{g}_{2},\cdots,\tilde{g}_{k})$, which consists of smooth
maximal complete conformal metrics, to satisfy

$\begin{cases}-{R}_{\tilde{g}_{1}}=\frac{n(n-2)}{\alpha(n\tau+2-2n)},\\\
\sigma_{2}(\lambda(\tilde{g}_{2}^{-1}A^{\tau,\alpha}_{\tilde{g}_{2}}))=\frac{n(n-1)}{2},\mbox{
}\lambda({g}^{-1}A^{\tau,\alpha}_{\tilde{g}_{2}})\in\Gamma_{2},\\\
\quad\quad\quad\quad\vdots\\\
\sigma_{k}(\lambda(\tilde{g}_{k}^{-1}A^{\tau,\alpha}_{\tilde{g}_{k}}))=\frac{n!}{k!(n-k)!},\mbox{
}\lambda({g}^{-1}A^{\tau,\alpha}_{\tilde{g}_{k}})\in\Gamma_{k}.\end{cases}$

The approximate method also works for conformal scalar curvature equation.

######  Theorem 4.9.

Let $(M,g)$ be a complete noncompact Riemannian manifold of dimension
$n\geqslant 3$ with negative scalar curvature. Let $\psi$ be a smooth positive
function which is bounded from above in terms of $-{R}_{g}$, i.e. there is a
positive constant $\delta$ such that $0<\psi\leqslant-\delta{R}_{g}.$ Then
there exists a unique smooth maximal complete conformal metric $\tilde{g}$
with prescribed scalar curvature $-{R}_{\tilde{g}}=\psi.$

######  Remark 4.10.

One can apply Theorem 4.9 to improve [3, Theorem A] of Aviles-McOwen. More
precisely, let $(M,g)$ be a complete noncompact Riemannian manifold with
nonpositive scalar curvature subject to ${R}_{g}(x)<-\delta<0$ outside a
compact subset $K$, then for any smooth positive function $\psi$ with
$\sup_{M}\psi<+\infty$, there is a unique smooth maximal complete conformal
metric $\tilde{g}$ with $-R_{\tilde{g}}=\psi$.

##  5\. The Dirichlet problem

As described in Section 4, it only requires to solve the equation (4.4) on
$M_{k}$ with infinite boundary value condition (4.5). Throughout this section,
we assume for simplicity that $(M,g)$ is a compact connected Riemannian
manifold with smooth boundary $\partial M$, and $\varphi\in
C^{\infty}(\partial M)$.

###  5.1. The Dirichlet problem with finite boundary value condition

First we solve the Dirichlet problem with finite boundary value condition.

######  Theorem 5.1.

Suppose, in addition to (1.2), (1.10), (1.11), (1.9), (1.12) and (4.2), that
there is a $C^{2}$ admissible function. Then the equation (4.1) admits a
unique smooth admissible solution $u$ with $u|_{\partial M}=\varphi$.

####  $C^{0}$-estimate

By the maximum principle, we obtain $C^{0}$-estimate.

######  Lemma 5.2.

In addition to (1.2), (1.4), (1.9), (1.10) and (1.11), we assume that there is
an admissible function $\underline{w}$. Let $u\in C^{2}(\bar{M})$ be an
admissible solution to the equation (4.1) with $u=\varphi$ on $\partial M$,
then

$\displaystyle\min\left\\{\inf_{\partial M}(\varphi-\underline{w}),\mbox{
}A_{1}-\sup_{M}\underline{w}\right\\}\leqslant
u-\underline{w}\leqslant\max\left\\{\sup_{\partial
M}(\varphi-\underline{w}),\mbox{ }A_{2}-\inf_{M}\underline{w}\right\\},$

where

$\displaystyle\sup_{x\in\bar{M}}\psi(x,A_{1})\leqslant\inf_{x\in\bar{M}}f(\lambda(g^{-1}(\nabla^{2}\underline{w}+A(x,\nabla\underline{w})))),$
$\displaystyle\inf_{x\in\bar{M}}\psi(x,A_{2})\geqslant\sup_{x\in\bar{M}}f(\lambda(g^{-1}(\nabla^{2}\underline{w}+A(x,\nabla\underline{w})))).$

#### Boundary gradient estimate

We denote the distance from $x$ to $\partial M$ with respect to $g$ by

(5.1) $\mathrm{\sigma}(x)=\mathrm{dist}_{g}(x,\partial M).$

Then $\mathrm{\sigma}(x)$ is smooth near the boundary. We use
$\mathrm{\sigma}$ to construct local barriers. Denote

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

The assumption (1.12) on $A(x,p)$ yields that there exists a positive uniform
constant $B$ such that

(5.3) $\displaystyle\big{|}\mathrm{tr}(g^{-1}A(x,p))\big{|}\leqslant
B(1+|p|^{2}).$

Local upper barrier. Take
$\bar{w}=\epsilon\log(1+\frac{\mathrm{\sigma}}{\delta^{2}})+\varphi,$ where
$\epsilon$ is a positive constant to be determined later. By (5.3), one can
see that there is a uniform constant $C_{B}^{\prime}$ such that

$\displaystyle\mathrm{tr}(g^{-1}A(x,\bar{w}))\leqslant
C_{B}^{\prime}\left(1+|\nabla\varphi|^{2}+\frac{\epsilon^{2}|\nabla\mathrm{\sigma}|^{2}}{(\delta^{2}+\mathrm{\sigma})^{2}}\right).$

Set $\epsilon=\frac{1}{2C_{B}^{\prime}}$. The computation shows that on
$\Omega_{\delta}$,

$\displaystyle\mathrm{tr}(g^{-1}A(x,\nabla\bar{w}))$
$\displaystyle+\Delta\bar{w}\leq\frac{2(\delta^{2}+\mathrm{\sigma})\Delta\mathrm{\sigma}-|\nabla\mathrm{\sigma}|^{2}}{4C_{B}^{\prime}(\delta^{2}+\mathrm{\sigma})^{2}}+C_{B}^{\prime}|\nabla\varphi|^{2}+C_{B}^{\prime}+\Delta\varphi\leqslant
0$

provided $0<\delta\ll 1$. Notice also that $\underset{\delta\to
0^{+}}{\lim}\,\bar{w}|_{\mathrm{\sigma}=\delta}=+\infty.$ Combining with the
$C^{0}$-estimate in Lemma 5.2, when $\delta$ is sufficiently small, we know

(5.4) $\displaystyle u\leqslant\bar{w}\mbox{ on }\Omega_{\delta}.$

Local lower barrier. Fix $k\geqslant 1$. Pick $\varepsilon$ a positive
constant to be determined later. Inspired by [22] we take

(5.5) $\displaystyle
w=\varepsilon\log\frac{\delta^{2}}{k\mathrm{\sigma}+\delta^{2}}+\varphi.$

The straightforward computation gives that for any $k\geqslant 1$,

$\displaystyle\nabla
w=-\frac{k\varepsilon\nabla\mathrm{\sigma}}{\delta^{2}+k\mathrm{\sigma}}+\nabla\varphi,\quad\nabla^{2}w=\frac{k^{2}\varepsilon
d\mathrm{\sigma}\otimes
d\mathrm{\sigma}}{(\delta^{2}+k\mathrm{\sigma})^{2}}-\frac{k\varepsilon\nabla^{2}\mathrm{\sigma}}{\delta^{2}+k\mathrm{\sigma}}+\nabla^{2}\varphi.$

Note that $|\nabla\mathrm{\sigma}|=1$ on $\partial M$. By (5.3) again, there
exist positive constants $C_{B}$ and $\delta^{\prime}$ such that

$\displaystyle\left|\nabla^{2}\varphi+A\left(x,\nabla\varphi-\frac{k\varepsilon\nabla\mathrm{\sigma}}{\delta^{2}+k\mathrm{\sigma}}\right)\right|\leqslant\frac{k^{2}\varepsilon^{2}C_{B}|\nabla\mathrm{\sigma}|^{2}}{(\delta^{2}+k\mathrm{\sigma})^{2}}\mbox{
in }\Omega_{\delta},\mbox{ }\forall 0<\delta<\delta^{\prime}.$

By Lemma 2.18, $f$ satisfies (1.6). According to Corollary 2.11 or 2.12,

$\Gamma^{\infty}_{\mathbb{R}^{n-1}}=\mathbb{R}^{n-1}\mbox{ and
}(0,\cdots,0,1)\in\Gamma.$

Notice $\lambda(g^{-1}\frac{\varepsilon k^{2}d\mathrm{\sigma}\otimes
d\mathrm{\sigma}}{(\delta^{2}+k\mathrm{\sigma})^{2}})=\frac{\varepsilon
k^{2}|\nabla\mathrm{\sigma}|^{2}}{(\delta^{2}+k\mathrm{\sigma})^{2}}(0,\cdots,0,1)$.
We take $\varepsilon$, $\delta$ small enough such that

$\displaystyle\lambda\left(g^{-1}\left(\frac{\varepsilon
k^{2}d\mathrm{\sigma}\otimes
d\mathrm{\sigma}}{2(\delta^{2}+k\mathrm{\sigma})^{2}}-\frac{\varepsilon
k\nabla^{2}\mathrm{\sigma}}{\delta^{2}+k\mathrm{\sigma}}+\nabla^{2}\varphi+A\left(x,\nabla\varphi-\frac{\varepsilon
k\nabla\mathrm{\sigma}}{\delta^{2}+k\mathrm{\sigma}}\right)\right)\right)\in\Gamma\mbox{
in }\Omega_{\delta}.$

Fix $\varepsilon$, $\delta$ as above. From Lemma 2.1, for $0<\delta\ll 1$, we
have on $\Omega_{\delta}$

$\displaystyle f\left(\lambda(g^{-1}(\nabla^{2}w+A(x,\nabla
w)))\right)\geqslant f\left(\frac{\varepsilon
k^{2}|\nabla\mathrm{\sigma}|^{2}}{2(\delta^{2}+k\mathrm{\sigma})^{2}}\left(0,\cdots,0,1\right)\right).$

Since $\psi_{z}(x,z)>0$ and $w\leqslant\varphi$, we further obtain for
$0<\delta\ll 1$

$\displaystyle
f\left(\frac{k^{2}\varepsilon|\nabla\mathrm{\sigma}|^{2}}{2(\delta^{2}+k\mathrm{\sigma})^{2}}\left(0,\cdots,0,1\right)\right)\geqslant\psi(x,w)\mbox{
on }\Omega_{\delta}.$

It follows from Lemmas 4.3, 5.2 and $\underset{\delta\to
0^{+}}{\lim}\,w|_{\mathrm{\sigma}=\delta}=-\infty$ that

(5.6) $\displaystyle u\geqslant
w=\varepsilon\log\frac{\delta^{2}}{k\mathrm{\sigma}+\delta^{2}}+\varphi\mbox{
on }\Omega_{\delta}$

for some positive constant $\delta$.

From (5.6) (setting $k=1$) and (5.4) we derive

$\displaystyle|\nabla u|\leqslant C\mbox{ on }\partial M.$

Combining with local estimates in Theorems 8.1-8.2, and boundary estimate in
Theorem 8.4, we obtain estimates up to second order

$\displaystyle|\nabla^{2}u|\leqslant C.$

Together with Evans-Krylov theorem and Schauder theory, one can derive higher
estimates. By the standard continuity method, we obtain Theorem 5.1.

###  5.2. The Dirichlet problem with infinite boundary value condition

######  Theorem 5.3.

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$ with
smooth boundary. Suppose in addition that (1.2), (1.9), (1.12), (1.10), (1.11)
and (4.2) hold. Then the equation (4.1) admits a smooth admissible solution
$u$ with $\underset{x\to\partial M}{\lim}\,u(x)=+\infty,$ provided that there
is a $C^{2}$ admissible function on $\bar{M}$.

The following lemma is useful. It is a consequence of the concavity.

######  Lemma 5.4.

Let $f$ satisfy (1.2) and $f(\vec{\bf 1})<\sup_{\Gamma}f$, then

(5.7) $\displaystyle\sum_{i=1}^{n}\lambda_{i}\geqslant
n+A_{f}\left(f(\lambda)-f(\vec{\bf 1})\right),\mbox{
}\forall\lambda\in\Gamma,$

where $A_{f}={n}\left(\sum_{i=1}^{n}f_{i}(\vec{\bf 1})\right)^{-1}.$ In
particular, if $f$ satisfies in addition

(5.8) $\displaystyle f(t\lambda)=tf(\lambda),\mbox{
}\forall\lambda\in\Gamma,\mbox{ }t>0,\mbox{ with normalization }f(\vec{\bf
1})=1,$

then

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

###### Proof of Theorem 5.3.

Let $u^{(k)}$ be an admissible solution to

(5.10) $\displaystyle f\left(\lambda(g^{-1}(\nabla^{2}u^{(k)}+A(x,\nabla
u^{(k)})))\right)=\psi(x,u^{(k)}),\,$ $\displaystyle u^{(k)}|_{\partial
M}=\varepsilon\log k,$

where the constant $\varepsilon$ comes from (5.5). The comparison principle
(Lemma 4.3) simply yields

(5.11) $\displaystyle u^{(k+1)}\geqslant u^{(k)},\quad\forall k\geqslant 1.$

Namely, $\\{u^{(k)}\\}$ is an increasing sequence. From Lemma 5.4 we get

$\displaystyle A_{f}\psi(x,u^{(k)})\leqslant\Delta
u^{(k)}+\mathrm{tr}(g^{-1}A(x,\nabla u^{(k)}))+A_{f}f(\vec{\bf 1})-n.$

On the other hand, from (5.3)

(5.12) $\displaystyle\mathrm{tr}(g^{-1}A(x,\nabla u^{(k)}))\leqslant
B(1+|\nabla u^{(k)}|^{2}).$

By the assumption (1.9) on $\psi(x,t)$ and the compactness of $\bar{M}$, there
are constants $\Lambda_{0}$ and $\Theta$ such that

$\displaystyle\psi(x,u^{(k)})\geqslant\Lambda_{0}e^{\Theta u^{(k)}}.$

Fix $\Theta$. Take $q$ a constant bigger than 1 but sufficiently close to $1$,
such that

$\displaystyle\frac{\Theta}{q-1}\geqslant B,$

where $B$ is as in (5.12). Fix such $q$. As a corollary of [46, Theorem 2.2],
there is $\tilde{u}\in C^{\infty}(M)$ such that

(5.13) $\displaystyle\Delta\tilde{u}+$
$\displaystyle\frac{\Theta}{q-1}|\nabla\tilde{u}|^{2}=\frac{q-1}{\Theta}e^{\Theta\tilde{u}}\mbox{
}\mbox{ in }M,\,$ $\displaystyle\lim_{x\to\partial M}\tilde{u}(x)\to+\infty.$

We know that $u^{(k)}-\tilde{u}$ achieves its maximum at an interior point
$x_{0}\in M$. At $x_{0}$,

$\displaystyle\nabla
u^{(k)}=\nabla\tilde{u},\quad\nabla^{2}u^{(k)}\leqslant\nabla^{2}\tilde{u}.$

Thus

$\displaystyle
A_{f}\Lambda_{0}e^{\Theta(u^{(k)}-\tilde{u})}\leqslant\frac{q-1}{\Theta}+\left(A_{f}f(\vec{\bf
1})+B\right)e^{-\Theta\tilde{u}}.$

Notice also that $\underset{M}{\inf}\,\tilde{u}>-\infty.$ We conclude

(5.14) $\displaystyle u^{(k)}\leqslant\tilde{u}+C$

for a uniform positive constant $C$, independent of $k$.

We have derived the (local) $C^{0}$-estimate in (5.11) and (5.14). We then
have

(5.15) $\displaystyle u(x)=\lim_{k\to+\infty}u^{(k)}(x),\mbox{ }\forall x\in
M.$

Combining with Evans-Krylov theorem, Schauder theory as well as the local
estimates up to second derivatives established in Theorems 8.1-8.2 below, we
can conclude that $u$ is a smooth admissible solution to (4.1) with
$\underset{x\to\partial M}{\lim}\,u(x)=+\infty.$

Moreover, the inequality (5.6) gives (note $\varphi=\varepsilon\log k$)

(5.16) $\displaystyle
u^{(k)}\geqslant\varepsilon\log\frac{k\delta^{2}}{k\mathrm{\sigma}+\delta^{2}}\mbox{
on }\Omega_{\delta},$

which implies $u\geqslant-\varepsilon\log\mathrm{\sigma}-C_{0}$ for some
$C_{0}$ near the boundary.

∎

######  Proposition 5.5.

Suppose that $A(x,du)$ in the equation (4.1) is of the form

(5.17) $\displaystyle A(x,\nabla u)=\mathrm{U}+\alpha(x)|\nabla
u|^{2}g+\beta(x)du\otimes du+R(x,du)$

where $\mathrm{U}$ is a smooth symmetric $(0,2)$ tensor, $R(x,p)$ is a smooth
symmetric $(0,2)$ tensor depending linearly on $p$. Assume in addition that
$\alpha(x)$ and $\beta(x)$ are smooth functions with

(5.18)
$\displaystyle(\alpha(x),\cdots,\alpha(x),\alpha(x)+\beta(x)+1)\in\Gamma\mbox{
in }\bar{M}.$

Then in (5.5) we can choose $\varepsilon=1$, hence the solution $u$ (defined
by (5.15)) in Theorem 5.3 satisfies

(5.19) $\displaystyle u\geqslant-\log\mathrm{\sigma}-C_{0}$

for some $C_{0}$ near the boundary. Moreover, the metric $\tilde{g}=e^{2u}g$
is complete.

###### Proof.

We take $\varepsilon=1$ in (5.5), i.e.
$w=\log\frac{k\delta^{2}}{k\mathrm{\sigma}+\delta^{2}}$. By (5.1),

$\displaystyle\nabla^{2}w+A(x,\nabla
w)=U+\frac{k^{2}\alpha|\nabla\mathrm{\sigma}|^{2}g+k^{2}(1+\beta)d\mathrm{\sigma}\otimes
d\mathrm{\sigma}}{(\delta^{2}+k\mathrm{\sigma})^{2}}-\frac{k\nabla^{2}\mathrm{\sigma}}{\delta^{2}+k\mathrm{\sigma}}+R\left(x,\frac{-kd\mathrm{\sigma}}{\delta^{2}+k\mathrm{\sigma}}\right)$

Note that

$\displaystyle\lambda\left(g^{-1}\left(\frac{k^{2}\alpha|\nabla\mathrm{\sigma}|^{2}g+k^{2}(1+\beta)d\mathrm{\sigma}\otimes
d\mathrm{\sigma}}{(\delta^{2}+k\mathrm{\sigma})^{2}}\right)\right)=\frac{k^{2}|\nabla\mathrm{\sigma}|^{2}}{(\delta^{2}+k\mathrm{\sigma})^{2}}(\alpha,\cdots,\alpha,\alpha+\beta+1).$

So we can prove by Lemma 2.1, that for $0<\delta\ll 1$,

$\displaystyle f\left(\lambda(g^{-1}(\nabla^{2}w+A(x,\nabla
w)))\right)\geqslant\psi(x,w)\mbox{ on }\Omega_{\delta}.$

Similarly, we obtain an inequality analogous to (5.6) (or (5.16)) near
boundary

$\displaystyle
u^{(k)}\geqslant\log\frac{k\delta^{2}}{k\mathrm{\sigma}+\delta^{2}}$

as required. ∎

######  Remark 5.6.

Since $\Gamma$ is of type 2 (by Corollary 2.12 or 2.11), the assumption (5.18)
is automatically satisfied, provided

$\displaystyle\alpha(x)\geqslant 0,\mbox{ }\alpha(x)+\beta(x)+1\geqslant
0,\mbox{ }n\alpha(x)+\beta(x)+1>0.$

As a corollary, we get the following result and Theorem 1.9.

######  Theorem 5.7.

Suppose $(f,\Gamma)$ satisfies (1.2), (1.11), (1.19) and (4.2). Let $(M,g)$ be
a compact connected Riemannian manifold of dimension $n\geqslant 3$ with
smooth boundary and support a $C^{2}$ conformal metric satisfying (4.7). Then
for any $0<\psi\in C^{\infty}(\bar{M})$, there exists at least one smooth
complete metric $\tilde{g}=e^{2u}g$ satisfying
${f}(\lambda(-\tilde{g}^{-1}A_{\tilde{g}}))=\psi$ and
$\lambda(-{g}^{-1}A_{\tilde{g}})\in\Gamma$ in $M$.

##  6\. Construction of admissible metrics

Now we focus on the equation (1.14). Throughout this section and Section 7,
the parameters in $A_{g}^{\tau,\alpha}$ obey (1.20), i.e.

$\begin{cases}\tau\leqslant 0\,&\mbox{ if }\alpha=-1,\\\ \tau\geqslant
2\,&\mbox{ if }\alpha=1.\end{cases}$

And we denote

$\displaystyle V[u]=\Delta ug-\varrho\nabla^{2}u+\gamma|\nabla u|^{2}g+\varrho
du\otimes du+A,$ $\displaystyle\varrho=\frac{n-2}{\tau-1},\mbox{
}\gamma=\frac{(\tau-2)(n-2)}{2(\tau-1)},\mbox{
}A=\frac{n-2}{\alpha(\tau-1)}A_{g}^{\tau,\alpha}.$

Here one can check
$V[u]=\frac{n-2}{\alpha(\tau-1)}A^{\tau,\alpha}_{\tilde{g}},\mbox{
}\tilde{g}=e^{2u}g$.

Under the assumption (1.20), we have the following lemma.

######  Lemma 6.1.

Under the assumption (1.20), we have

(6.1) $\gamma=\frac{(n-2)(\tau-2)}{2(\tau-1)}\geqslant
0,\quad\gamma+\varrho=\frac{\tau(n-2)}{2(\tau-1)}\geqslant 0.$

The following lemma asserts that a compact manifold with boundary carries a
Morse function without critical points. The construction of such a Morse
function is standard in differential topology.

######  Lemma 6.2.

Let $(M,g)$ be a compact connected Riemannian manifold of dimension $n\geq 2$
with smooth boundary. Then there is a smooth Morse function $v$ without
critical points, that is $dv\neq 0$ in $\bar{M}$.

###### Proof.

Let $X$ be the double of $M$. Let $w$ be a smooth Morse function on $X$ with
the critical set $\\{p_{i}\\}_{i=1}^{m+k}$, among which $p_{1},\cdots,p_{m}$
are all the critical points being in $\bar{M}$. Pick $q_{1},\cdots,q_{m}\in
X\setminus\bar{M}$ but not the critical point of $w$. By homogeneity lemma
(see e.g. [47]), one can find a diffeomorphism $h:X\to X$, which is smoothly
isotopic to the identity, such that

  * •

$h(p_{i})=q_{i}$, $1\leqslant i\leqslant m$.

  * •

$h(p_{i})=p_{i}$, $m+1\leqslant i\leqslant m+k$.

Then $v=w\circ h^{-1}\big{|}_{\bar{M}}$ is the desired Morse function.

∎

######  Proposition 6.3.

For $(\alpha,\tau)$ satisfying (1.20), there exists a smooth conformal
admissible metric on $\bar{M}$.

###### Proof.

According to Lemma 6.2, there exists a smooth function $v$ with

$|\nabla v|^{2}\geqslant a_{0}\mbox{ in }\bar{M}$

for some positive constant $a_{0}$. Without loss of generality, $v\geqslant
0$.

Set $\underline{u}=e^{Nv},$ $\underline{g}=e^{2\underline{u}}g,$ then

$\displaystyle V[\underline{u}]=N^{2}e^{Nv}\left((\Delta
vg-\varrho\nabla^{2}v)/N+(1+\gamma e^{Nv})|\nabla
v|^{2}g+\varrho(e^{Nv}-1)dv\otimes dv\right)+A.$

The discussion consists of two cases:

Case 1: $\varrho>0$. In this case

$(1+\gamma e^{Nv})|\nabla v|^{2}g+\varrho(e^{Nv}-1)dv\otimes
dv\geqslant|\nabla v|^{2}g.$

Case 2: $\varrho<0$. In this case

$(1+\gamma e^{Nv})|\nabla v|^{2}g+\varrho(e^{Nv}-1)dv\otimes
dv\geqslant\left((1-\varrho)+(\gamma+\varrho)e^{Nv}\right)|\nabla
v|^{2}g\geqslant|\nabla v|^{2}g.$

Here we use (6.1). Therefore, $\lambda(g^{-1}V[\underline{u}])\in\Gamma$ in
$\bar{M}$ for $N\gg 1$. That is, $\underline{g}=e^{2\underline{u}}g$ is an
admissible metric.

∎

######  Remark 6.4.

By Corollary 3.4, the construction also works if
$(\gamma,\cdots,\gamma,\gamma+\varrho)\in\overline{\Gamma}$.

##  7\. Asymptotic behavior and uniqueness of solutions

In this section we further assume $f$ satisfies (5.8), i.e.,

$\displaystyle f(t\lambda)=tf(\lambda),\mbox{ }\forall\lambda\in\Gamma,\mbox{
}t>0,\mbox{ with normalization }f(\vec{\bf 1})=1.$

Then the equation (1.14) reads as follows

(7.1) $\displaystyle
f(\lambda(g^{-1}V[u]))=\frac{(n-2)\psi}{\alpha(\tau-1)}e^{2u}.$

For the complete conformal metrics satisfying (1.14), we prove the following
asymptotic behavior and uniqueness of metrics, when the prescribed function is
a constant when restricted to boundary. Let $\sigma$ denote as in (5.1) the
distance function.

######  Theorem 7.1.

Let $(M,g)$ be a compact connected Riemannian manifold of dimension
$n\geqslant 3$ with smooth boundary. Suppose in addition that (1.2), (1.11),
(5.8), (1.18) and (1.20) hold. Then for any positive smooth function with
$\psi|_{\partial M}\equiv 1$, there is a unique smooth complete admissible
metric $\widetilde{g}=e^{2\widetilde{u}_{\infty}}g$ satisfying (1.14).
Moreover,

$\displaystyle\lim_{x\rightarrow\partial
M}(\widetilde{u}_{\infty}(x)+\log\mathrm{\sigma}(x))=\frac{1}{2}\log\frac{\alpha(n\tau+2-2n)}{2(n-2)}.$

We now summarize a theorem due to Aviles-McOwen, extending a famous result of
Loewner-Nirenberg to general Riemannian manifolds.

######  Theorem 7.2 ([4]).

Let $(M,g)$ be a compact connected Riemannian manifold of dimension
$n\geqslant 3$ with smooth boundary. There is a smooth complete conformal
metric with scalar curvature $-1$.

The theorem yields that there is a $\tilde{u}\in C^{\infty}(M)$ such that

(7.2) $\displaystyle
2(n-1)\Delta\tilde{u}+(n-1)(n-2)|\nabla\tilde{u}|^{2}-R_{g}=e^{2\tilde{u}}\mbox{
in }M,\,$ $\displaystyle\lim_{x\rightarrow\partial M}\tilde{u}(x)=+\infty.$

Geometrically, $e^{2\tilde{u}}g$ is a smooth complete metric with constant
scalar curvature $-1$.

For conformal scalar curvature equation (7.2) on $M=\Omega$ a smooth bounded
domain of Euclidean spaces, Loewner-Nirenberg [41] proved the asymptotic ratio

(7.3) $\displaystyle\lim_{x\to\partial
M}\left(\tilde{u}(x)+\log\mathrm{\sigma}(x)\right)=\frac{1}{2}\log{n(n-1)}.$

This asymptotic property can be extended to general Riemannian manifolds, as
pointed out by McOwen in [46]

Theorem 7.1 follows from the propositions below.

######  Proposition 7.3.

Let $\widetilde{g}_{\infty}=e^{2\widetilde{u}_{\infty}}g$ be a complete metric
obeying the equation (1.14) with (1.2), (1.11), (5.8), (1.18) and (1.20), then

(7.4) $\displaystyle\lim_{x\rightarrow\partial
M}(\widetilde{u}_{\infty}(x)+\log\mathrm{\sigma}(x))\leqslant\frac{1}{2}\log\frac{\alpha(n\tau+2-2n)}{2(n-2)\inf_{\partial
M}\psi}.$

###### Proof.

We utilize an approximation in analogy with that used in proof of Theorem 4.1.
Let $\Omega_{\delta}$ be as in (5.2), and let

$\Omega_{\delta,\delta^{\prime}}=\\{x\in
M:0<\delta^{\prime}<\mathrm{\sigma}(x)<\delta-\delta^{\prime}\\}.$

We choose $0<\delta^{\prime}<\frac{\delta}{2}\ll 1$ such that
$\Omega_{\delta}$ and $\Omega_{\delta,\delta^{\prime}}$ are both smooth.

We use Theorem 7.2 repeatedly. By Theorem 7.2, $\Omega_{\delta}$ admits a
smooth complete conformal metric
$g_{\infty}^{\delta}=e^{2w_{\infty}^{\delta}}g$ with constant scalar curvature
$-1$. Again, there is a smooth complete metric
$g_{\infty}^{\delta,\delta^{\prime}}=e^{2u_{\infty}^{\delta,\delta^{\prime}}}g$
on $\Omega_{\delta,\delta^{\prime}}$ with constant scalar curvature $-1$, that
is,

$\mathrm{tr}(g^{-1}V[{u_{\infty}^{\delta,\delta^{\prime}}}])=\frac{n\tau+2-2n}{2(n-1)(\tau-1)}e^{2u_{\infty}^{\delta,\delta^{\prime}}}\mbox{
in
}\Omega_{\delta,\delta^{\prime}},\quad{u_{\infty}^{\delta,\delta^{\prime}}}|_{\partial\Omega_{\delta,\delta^{\prime}}}=+\infty.$

By Lemma 2.15, $\frac{n\tau+2-2n}{2(n-1)(\tau-1)}>0$. By maximum principle

(7.5) $\displaystyle u_{\infty}^{\delta,\delta^{\prime}}\geqslant
w_{\infty}^{\delta}\mbox{ in }\Omega_{\delta,\delta^{\prime}}.$

Furthermore, for any $0<\delta^{\prime}<\delta_{0}^{\prime}<\frac{\delta}{2}$,
we have
$\Omega_{\delta,\delta_{0}^{\prime}}\subset\Omega_{\delta,\delta^{\prime}}$
and

(7.6) $\displaystyle u_{\infty}^{\delta,\delta^{\prime}}\leqslant
u_{\infty}^{\delta,\delta_{0}^{\prime}}\mbox{ in
}\Omega_{\delta,\delta_{0}^{\prime}}.$

According to (5.9), the solution $\widetilde{u}_{\infty}$ satisfies

$\mathrm{tr}(g^{-1}V[\widetilde{u}_{\infty}])\geqslant\frac{n(n-2)\psi}{\alpha(\tau-1)}e^{2\widetilde{u}_{\infty}}.$

The maximum principle implies

$\displaystyle
u_{\infty}^{\delta,\delta^{\prime}}\geqslant\widetilde{u}_{\infty}+\frac{1}{2}\log\left(\frac{2n(n-1)(n-2)}{\alpha(n\tau+2-2n)}\inf_{\Omega_{\delta,\delta^{\prime}}}\psi\right)\mbox{
in }\Omega_{\delta,\delta^{\prime}}.$

Take $u_{\infty}^{\delta}(x)=\underset{\delta^{\prime}\rightarrow
0}{\lim}\,u_{\infty}^{\delta,\delta^{\prime}}(x)$ (such a limit exists by
(7.5)-(7.6)), then

(7.7) $\displaystyle u_{\infty}^{\delta}\geqslant w_{\infty}^{\delta},\,$
$\displaystyle
u_{\infty}^{\delta}\geqslant\widetilde{u}_{\infty}+\frac{1}{2}\log\left(\frac{2n(n-1)(n-2)}{\alpha(n\tau+2-2n)}\inf_{\Omega_{\delta}}{\psi}\right)\mbox{
in }\Omega_{\delta}$

and $e^{2u_{\infty}^{\delta}}g$ is a smooth complete conformal metric on
$\Omega_{\delta}$ of scalar curvature $-1$. (We have local estimates since it
is of uniform ellipticity). Furthermore, (7.3) gives

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

Putting them together, we get (7.4). ∎

######  Proposition 7.4.

Assume (1.2), (1.11), (5.8), (1.18) and (1.20) hold. Then any admissible
complete metric $\widetilde{g}_{\infty}=e^{2\widetilde{u}_{\infty}}g$
satisfying (1.14) obeys

(7.8) $\displaystyle\lim_{x\rightarrow\partial
M}(\widetilde{u}_{\infty}(x)+\log\mathrm{\sigma}(x))\geqslant\frac{1}{2}\log\frac{\alpha(n\tau+2-2n)}{2(n-2)\sup_{\partial
M}\psi}.$

###### Proof.

The proof is different from that presented in [41, 29]. The key ingredients in
the proof are (5.9) and $|\nabla\mathrm{\sigma}|=1$ on $\partial M$. Fix a
constant $0<\epsilon<1$. We consider

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

The solvability is obtained by Theorem 5.1. The maximum principle implies

(7.10) $\displaystyle\widetilde{u}_{\infty}\geqslant u_{k,\epsilon}\mbox{ in
}M,$ (7.11)
$\displaystyle\inf_{M}u_{k,\epsilon}\geqslant\frac{1}{2}\min\left\\{\inf_{M}\log\frac{f(\lambda(g^{-1}A_{g}^{\tau,\alpha}))}{{\psi}},\log\frac{k^{2}(1-\epsilon)^{2}\alpha(n\tau+2-2n)}{2(n-2)(\sup_{\partial
M}{\psi}+\epsilon)}\right\\}.$

The local subsolution is given by

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

The part $``\frac{1}{\mathrm{\sigma}+\delta}-\frac{1}{\delta}"$ is inspired by
[29]. Straightforward computation tells

$\displaystyle
h_{k,\epsilon,\delta}^{\prime}=-\frac{k}{k\mathrm{\sigma}+1}-\frac{1}{(\mathrm{\sigma}+\delta)^{2}},\,$
$\displaystyle
h_{k,\epsilon,\delta}^{\prime\prime}=\frac{k^{2}}{(k\mathrm{\sigma}+1)^{2}}+\frac{2}{(\mathrm{\sigma}+\delta)^{3}},$
$\displaystyle h_{k,\epsilon,\delta}^{\prime
2}=\frac{k^{2}}{(k\mathrm{\sigma}+1)^{2}}+\frac{1}{(\mathrm{\sigma}+\delta)^{4}}+\frac{2k}{(k\mathrm{\sigma}+1)(\mathrm{\sigma}+\delta)^{2}},$
$h_{k,\epsilon,\delta}^{\prime\prime}+\gamma h_{k,\epsilon,\delta}^{\prime
2}=\frac{(1+\gamma)k^{2}}{(k\mathrm{\sigma}+1)^{2}}+\frac{2}{(\mathrm{\sigma}+\delta)^{3}}+\frac{\gamma}{(\mathrm{\sigma}+\delta)^{4}}+\frac{2k\gamma}{(k\mathrm{\sigma}+1)(\mathrm{\sigma}+\delta)^{2}}.$
$\displaystyle V[h_{k,\epsilon,\delta}]=$
$\displaystyle(h_{k,\epsilon,\delta}^{\prime\prime}+\gamma
h_{k,\epsilon,\delta}^{\prime
2})|\nabla\mathrm{\sigma}|^{2}g+\varrho(h_{k,\epsilon,\delta}^{\prime
2}-h_{k,\epsilon,\delta}^{\prime\prime})d\mathrm{\sigma}\otimes
d\mathrm{\sigma}$
$\displaystyle+h_{k,\epsilon,\delta}^{\prime}(\Delta\mathrm{\sigma}\cdot
g-\varrho\nabla^{2}\mathrm{\sigma})+A.$

By (6.1) we have $\varrho+\gamma\geqslant 0$, then

(7.12) $\displaystyle h_{k,\epsilon,\delta}^{\prime\prime}+\gamma
h_{k,\epsilon,\delta}^{\prime 2}+\varrho(h_{k,\epsilon,\delta}^{\prime
2}-h_{k,\epsilon,\delta}^{\prime\prime})\geqslant\frac{(1+\gamma)k^{2}}{(k\mathrm{\sigma}+1)^{2}}+\frac{2(1-\varrho)}{(\mathrm{\sigma}+\delta)^{3}}.$

Next we are going to verify that

######  Claim.

If $\delta\leqslant\frac{1}{4}$ or $k\geqslant\frac{1}{\delta}$ then

$\displaystyle
V[h_{k,\epsilon,\delta}]\geqslant\left(\frac{(1+\gamma)k^{2}}{(k\mathrm{\sigma}+1)^{2}}+\frac{2}{(\mathrm{\sigma}+\delta)^{3}}\right)|\nabla\mathrm{\sigma}|^{2}g-\left(\frac{k}{k\mathrm{\sigma}+1}+\frac{1}{(\mathrm{\sigma}+\delta)^{2}}\right)(\Delta\mathrm{\sigma}g-\varrho\nabla^{2}\mathrm{\sigma})+A.$

First, it is easy to see $h_{k,\epsilon,\delta}^{\prime
2}-h_{k,\epsilon,\delta}^{\prime\prime}\geqslant 0$ provided
$\delta\leqslant\frac{1}{4}$ or $k\geqslant\frac{1}{\delta}$. So the case
$\varrho>0$ is trivial. When $\varrho<0$, together with (7.12), we get

$\displaystyle(h_{k,\epsilon,\delta}^{\prime\prime}+\gamma
h_{k,\epsilon,\delta}^{\prime
2})|\nabla\mathrm{\sigma}|^{2}g+\varrho(h_{k,\epsilon,\delta}^{\prime
2}-h_{k,\epsilon,\delta}^{\prime\prime})d\mathrm{\sigma}\otimes
d\mathrm{\sigma}\geqslant\left(\frac{(1+\gamma)k^{2}}{(k\mathrm{\sigma}+1)^{2}}+\frac{2}{(\mathrm{\sigma}+\delta)^{3}}\right)|\nabla\mathrm{\sigma}|^{2}g.$

This completes the proof of the claim.

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

  1. (1)

$\mathrm{\sigma}(x)$ is smooth, and $|\nabla\mathrm{\sigma}|^{2}\geqslant
1-\epsilon$.

  2. (2)

$\frac{2}{(\mathrm{\sigma}+\delta)^{3}}|\nabla\mathrm{\sigma}|^{2}g-\frac{1}{(\mathrm{\sigma}+\delta)^{2}}(\Delta\mathrm{\sigma}\cdot
g-\varrho\nabla^{2}\mathrm{\sigma})+A\geqslant 0$.

  3. (3)

$\frac{k^{2}\epsilon(1+\gamma)}{(k\mathrm{\sigma}+1)^{2}}|\nabla\mathrm{\sigma}|^{2}g-\frac{k}{k\mathrm{\sigma}+1}(\Delta\mathrm{\sigma}\cdot
g-\varrho\nabla^{2}\mathrm{\sigma})\geqslant 0$.

  4. (4)

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

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

(7.13) $\displaystyle f(\lambda(g^{-1}V[h_{k,\epsilon,\delta}]))\geqslant
f\left(\frac{k^{2}{(1+\gamma)(1-\epsilon)}}{(k\mathrm{\sigma}+1)^{2}}\cdot|\nabla\mathrm{\sigma}|^{2}I_{n}\right)\geqslant\frac{(n-2)\psi}{\alpha(\tau-1)}e^{2h_{k,\epsilon,\delta}}.$

Note that $u_{k,\epsilon}=h_{k,\epsilon,\delta}$ on $\partial M$, and
$u_{k,\epsilon}|_{\mathrm{\sigma}=\delta}\geqslant
h_{k,\epsilon,\delta}(\delta)$ (according to (7.11)). Here we also use
$\frac{\log\frac{k}{k\delta+1}}{\frac{1}{2\delta}}\leqslant\frac{\log\frac{1}{\delta}}{\frac{1}{2\delta}}\rightarrow
0$ as $\delta\rightarrow 0^{+}$. Then the maximum principle yields

$\displaystyle u_{k,\epsilon}\geqslant
h_{k,\epsilon,\delta}=\log\frac{k}{k\mathrm{\sigma}+\delta}+\frac{1}{2}\log\frac{(1-\epsilon)^{2}\alpha(n\tau+2-2n)}{2(n-2)(\sup_{\partial
M}{\psi}+\epsilon)}+\frac{1}{\mathrm{\sigma}+\delta}-\frac{1}{\delta}$

on $\Omega_{\delta}$. From (7.10), let $k\rightarrow+\infty$, we know that on
$\Omega_{\delta}$,

$\displaystyle\widetilde{u}_{\infty}(x)+\log\mathrm{\sigma}(x)\geqslant\frac{1}{2}\log\frac{(1-\epsilon)^{2}\alpha(n\tau+2-2n)}{2(n-2)(\sup_{\partial
M}{\psi}+\epsilon)}+\frac{1}{\mathrm{\sigma}+\delta}-\frac{1}{\delta}.$

Thus (7.8) follows.

∎

######  Proposition 7.5.

Suppose (1.2), (1.11), (1.18), (5.8) and (1.20) hold. Let
$u_{\infty}(x)=\underset{k\to+\infty}{\lim}\,u^{(k)}(x),$ where

$\displaystyle
f(\lambda(g^{-1}V[u^{(k)}]))=\frac{(n-2)\psi}{\alpha(\tau-1)}e^{2u^{(k)}}\mbox{
in }M,\,$ $\displaystyle u^{(k)}=\log k\mbox{ on }\partial M.$

For any admissible solution $\widetilde{u}_{\infty}$ to the Dirichlet problem
for equation (7.1) with infinite boundary value condition, then

$u_{\infty}\leqslant\widetilde{u}_{\infty}\leqslant
u_{\infty}+\frac{1}{2}(\sup_{\partial M}\log\psi-\inf_{\partial
M}\log\psi)\mbox{ in }M.$

###### Proof.

The comparison principle yields $\widetilde{u}_{\infty}\geqslant u^{(k)}$
$(\forall k\geqslant 1)$. Then $\widetilde{u}_{\infty}\geqslant u_{\infty}$.

Next, we prove the second inequality. Note that (7.4) and (7.8) imply

$\lim_{x\rightarrow\partial
M}(\widetilde{u}_{\infty}(x)-u_{\infty}(x))\leqslant\frac{1}{2}(\sup_{\partial
M}\log\psi-\inf_{\partial M}\log\psi).$

Assume there is an interior point $x_{0}\in M$ such that

$(\widetilde{u}_{\infty}-u_{\infty})(x_{0})=\sup_{M}(\widetilde{u}_{\infty}-u_{\infty})>\frac{1}{2}(\sup_{\partial
M}\log\psi-\inf_{\partial M}\log\psi).$

This is a contradiction since the maximum principle yields
$\widetilde{u}_{\infty}(x_{0})\leqslant u_{\infty}(x_{0})$. ∎

##  8\. Local and boundary estimates for nonlinear uniformly elliptic
equations

This section is devote to deriving the local and boundary estimates for a more
general equation

(8.1) $\displaystyle F(\nabla^{2}u+A(x,u,\nabla
u)):=f\left(\lambda(g^{-1}(\nabla^{2}u+A(x,u,\nabla
u)))\right)=\psi(x,u,\nabla u)$

which is of fully uniform ellipticity, i.e.,

$\displaystyle
f_{i}(\lambda)\geqslant\theta\sum_{j=1}^{n}f_{j}(\lambda)>0\mbox{ in
}\Gamma,\quad\forall 1\leqslant i\leqslant n,$

where $\psi(x,z,p)$ and $A(x,z,p)$ denote respectively smooth function and
symmetric $(0,2)$-type tensor of variables $(x,z,p)$, $(x,p)\in TM$,
$z\in\mathbb{R}$.

Again, we say $u$ is an admissible function for (8.1) if

$\lambda(g^{-1}(\nabla^{2}u+A(x,u,\nabla u)))\in\Gamma.$

Throughout this section, for $u$ we denote

$\mathfrak{g}[u]=\nabla^{2}u+A(x,u,\nabla u),\quad\psi[u]=\psi(x,u,\nabla u).$

Moreover, for any admissible solution $u$ to (8.1),
$\mathfrak{g}=\mathfrak{g}[u]$, $\lambda=\lambda(g^{-1}\mathfrak{g})$,
$F^{ij}=\frac{\partial F}{\partial a_{ij}}(\mathfrak{g})$. Then the matrix
$\\{F^{ij}\\}$ (with respect to $\\{g^{ij}\\}$) has eigenvalues
$f_{1},\cdots,f_{n}$. Moreover,
$\sum_{i=1}^{n}f_{i}=F^{ij}g_{ij},\sum_{i=1}^{n}f_{i}\lambda_{i}=F^{ij}\mathfrak{g}_{ij}.$

The local estimates for second derivatives can be stated as follows.

######  Theorem 8.1.

Let $B_{r}\subset M$ be a geodesic ball of radius $r$. Let $u\in C^{4}(B_{r})$
be an admissible solution to equation (8.1) in $B_{r}$. Assume (1.2), (1.6)
and (4.2) hold. Then

$\displaystyle\sup_{B_{{r}/{2}}}|\nabla^{2}u|\leqslant C,$

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

We will prove local gradient estimate under asymptotic assumptions:

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

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

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

Here $\nabla^{\prime}\psi$, $\nabla^{\prime}A$ and $D_{p}A$ are as in
Introduction, and the meanings of $D_{z}A$, $D_{z}\psi$ and $D_{p}\psi$ are
similar.

######  Theorem 8.2.

Let $B_{r}\subset M$ as above and $u\in C^{3}(B_{r})$ be an admissible
solution to (8.1) in $B_{r}$. In addition to (1.2), we assume (4.2) and (8.2)
hold. Suppose there is a positive constant $\kappa_{0}$ depending not on
$\nabla u$ such that

(8.3) $\displaystyle F^{ij}g_{ij}\geqslant\kappa_{0}.$

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

$\sup_{B_{{r}/{2}}}|\nabla u|\leqslant{C}/{r}.$

######  Remark 8.3.

By Lemma 2.1 and (2.1), if (1.2) and (1.6) hold then

(8.4) $\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)>\kappa:={(f(R_{0}\vec{\bf
1})-\psi[u])}/{R_{0}}>0\mbox{ for some }R_{0}>0.$

Moreover, if $f$ is homogeneous of degree one, then
$\sum_{i=1}^{n}f_{i}(\lambda)\geqslant f(\vec{\bf 1})>0.$

As a result, the assumption (8.3) holds when $f$ is a homogeneous function of
degree one, or $f$ satisfies (1.6) and $\psi[u]=\psi(x,u)$.

The boundary estimates for second derivatives are also obtained.

######  Theorem 8.4.

Let $(M,g)$ be a compact Riemannian manifold with smooth boundary. Suppose
(1.2), (1.6) and (4.2) hold. Let $u\in C^{3}(M)\cap C^{2}(\bar{M})$ be an
admissible solution to (8.1) with $u=\varphi$ on $\partial M$ for $\varphi\in
C^{3}(\bar{M})$, then

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

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

###  8.1. Useful computation

Let $e_{1},...,e_{n}$ be the local frame as above. We denote
$A_{ij}(x,u,\nabla u)=A(x,u,\nabla u)(e_{i},e_{j})$, and
$\mathfrak{g}_{ij}=\nabla_{ij}u+A_{ij}(x,u,\nabla u)$. The linearized operator
of (8.1) is given by

(8.5)
$\displaystyle\mathcal{L}v=F^{ij}\nabla_{ij}v+(F^{ij}A_{ij,p_{l}}-\psi_{p_{l}})\nabla_{l}v\mbox{
for }v\in C^{2}(\bar{M}).$

Similar computation can be found in [24]. We have

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

By differentiating equation (8.1),

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

Using (8.6), $\nabla_{ij}u=\nabla_{ji}u$ and
$\nabla_{ijk}u=\nabla_{kij}u-R_{ljik}\nabla_{l}u$,

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

Let $w=|\nabla u|^{2}$. The straightforward computation yields

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

###  8.2. Local gradient estimate

Note that the equation (8.1) is of fully uniform ellipticity, the proof of
local estimates is standard. For convenience, we present the details below. As
above we denote $w=|\nabla u|^{2}$. We consider the quantity

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

where $\phi$ is determined later. Following [26] let $\eta$ be a smooth
function with compact support in $B_{r}\subset M$ and

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

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

(8.12)
$\displaystyle\frac{\nabla_{i}\eta}{\eta}+\frac{\nabla_{i}w}{w}+\nabla_{i}\phi=0,\,$
$\displaystyle\mathcal{L}(\log\eta+\log w+\phi)\leqslant 0.$

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

Step 1. Computation and estimation for $\mathcal{L}(\log w)$. By (8.9),

(8.13) $\displaystyle F^{ij}\nabla_{i}w\nabla_{j}w\leqslant 4|\nabla
u|^{2}F^{ij}\nabla_{ik}u\nabla_{jk}u.$

Using Cauchy-Schwarz inequality, one derives by (8.12)

$\displaystyle
F^{ij}\frac{\nabla_{i}w\nabla_{j}w}{w^{2}}\leqslant(1+\epsilon)\left(\frac{1}{\epsilon}F^{ij}\frac{\nabla_{i}\eta\nabla_{j}\eta}{\eta^{2}}+F^{ij}\nabla_{i}\phi\nabla_{j}\phi\right),$

which, together with (8.13), yields

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

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

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

Step 2. Construction and computation of $\phi$. As in [22], let $\phi=v^{-N}$
where $v=u-\inf_{B_{r}}u+2$ ($v\geqslant 2$ in $B_{r}$) and $N\geqslant 1$ to
be chosen later. By direct computation,

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

#### Step 3. Completion of the proof.

By (4.2),

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

By Cauchy-Schwarz inequality again, one derives

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

We choose $N\gg 1$ so that

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

Suppose furthermore that $\frac{1}{8}Nv^{-N-2}\theta w\geqslant C_{0}$ where
$C_{0}$ is as in (8.14) (otherwise we are done). Together with (8.12), (8.14),
(8.15) and (8.16), we derive

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

Using (8.11) and the asymptotic assumption (8.2), we obtain

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

which gives $\eta w\leqslant\frac{C}{r^{2}}$, as required.

###  8.3. Local estimate for second derivatives

Let’s consider

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

where $\eta$ is the cutoff function as given by (8.11) 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 $\mathfrak{g}_{11}(x_{0})\geqslant 1$. At $x_{0}$ one has

(8.17)
$\displaystyle\nabla_{i}\varphi+\frac{\nabla_{i}\mathfrak{g}_{11}}{\mathfrak{g}_{11}}+\frac{\nabla_{i}\eta}{\eta},\,$
$\displaystyle\mathcal{L}(\varphi+\log\eta+\log\mathfrak{g}_{11})\leqslant 0.$

Step 1. Estimation for $\mathcal{L}(\mathfrak{g}_{11})$. From (8.7), we get

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

The straightforward computation shows

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

Here we use (8.6). Combing (8.18) and (8.19),

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

Using (8.17) and Cauchy-Schwarz inequality,

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

Hence

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

Step 2. Construction and estimate for $\mathcal{L}\varphi$. Following [22] 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 $\varphi^{\prime}=\frac{\varphi^{3}}{4N},$
$\varphi^{\prime\prime}=\frac{3\varphi^{5}}{16N^{2}}$ and
$1\leqslant\varphi\leqslant\sqrt{2}$. By (8.10), (4.2) and Cauchy-Schwarz
inequality, we obtain

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

Consequently,

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

Step 3. Completion of the proof. From (8.11),

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

Next,

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

Finally, at $x_{0}$

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

Combining with (8.4) we have $\eta\mathfrak{g}_{11}\leqslant\frac{C}{r^{2}}$
at $x_{0}$. Therefore the proof is complete.

###  8.4. Boundary estimate

For a point $x_{0}\in\partial M$ 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 $\partial M$. We denote
$\rho(x):=\mathrm{dist}_{g}(x,x_{0})$ the distance function from $x$ to
$x_{0}$ with respect to $g$, and $M_{\delta}:=\\{x\in M:\rho(x)<\delta\\}.$ As
in (5.1), $\mathrm{\sigma}(x)$ is the distance from $x$ to $\partial M$. We
know that $\mathrm{\sigma}(x)$ is smooth and
$|\nabla\mathrm{\sigma}|\geqslant\frac{1}{2}$ in $M_{\delta}$ for small
$\delta$.

#### Case 1. Pure tangential derivatives.

From the boundary value condition,

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

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

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

#### Case 2. Mixed derivatives.

For $1\leqslant\alpha<n$, the local barrier function is given by

$\displaystyle\Psi=\pm\nabla_{\alpha}(u-\varphi)+A_{1}\left(\frac{N\mathrm{\sigma}^{2}}{2}-t\mathrm{\sigma}\right)-A_{2}\rho^{2}+A_{3}\sum_{k=1}^{n-1}|\nabla_{k}(u-\varphi)|^{2}\mbox{
in }M_{\delta},$

where $A_{1}$, $A_{2}$, $A_{3}$, $N$, $t$ are all positive constants to be
determined, and $N\delta-2t\leqslant 0.$ The construction of local barrier is
standard.

To deal with local barrier functions, we need the following formula

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

see e.g. [23]. Combining with (8.8) one derives

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

As in (8.10), we can prove

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

By [23, Proposition 2.19], there exists an index $1\leqslant r\leqslant n$
with

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

It follows from (2.1) and Lemma 2.1 that

$0<\sum_{i=1}^{n}f_{i}\lambda_{i}\leqslant\psi(x,u,\nabla u)-f(\vec{\bf
1})+\sum_{i=1}^{n}f_{i}.$

Combining with Cauchy-Schwarz inequality and

$\sum_{i=1}^{n}f_{i}|\lambda_{i}|=2\sum_{\lambda_{i}\geqslant
0}f_{i}\lambda_{i}-\sum_{i=1}^{n}f_{i}\lambda_{i}=\sum_{i=1}^{n}f_{i}\lambda_{i}-2\sum_{\lambda_{i}<0}f_{i}\lambda_{i},$

we have

$\displaystyle\sum_{i=1}^{n}f_{i}|\lambda_{i}|\leqslant\epsilon\sum_{i\neq
r}f_{i}\lambda_{i}^{2}+\frac{C}{\epsilon}\sum_{i=1}^{n}f_{i}+C.$

Let $\kappa$ be as in (8.4). If $\delta$ and $t$ are chosen small enough such
that

$\displaystyle\mathcal{L}\left(\frac{N\mathrm{\sigma}^{2}}{2}-t\mathrm{\sigma}\right)\geqslant\frac{N\kappa\theta}{8(1+\kappa)}\left(1+\sum_{i=1}^{n}f_{i}\right).$

Putting those inequalities together, if $A_{1}\gg A_{2}$, $A_{1}\gg A_{3}>1$
then

$\displaystyle\mathcal{L}(\Psi)>0\mbox{ in }M_{\delta}.$

On the other hand $\Psi|_{\partial M}=-A_{2}\rho^{2}\leqslant 0$; while on
$\partial M_{\delta}\setminus\partial M$, $\Psi\leqslant 0$ if
$N\delta-2t\leqslant 0$, $A_{2}\gg A_{1}$. Note that $\Psi(x_{0})=0$. Thus

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

#### Case 3. Double normal derivatives.

Fix $x_{0}\in\partial M$. Since $\mathrm{tr}(g^{-1}\mathfrak{g})>0$, (8.20)
and (8.21), we have $\mathfrak{g}_{nn}\geqslant-C$. We assume
$\mathfrak{g}_{nn}(x_{0})\geqslant 1$ (otherwise we are done). According to
(4.2), $F^{nn}\geqslant\theta\sum_{i=1}^{n}f_{i}.$ By the concavity of
equation, (8.20) and (8.21),

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

This gives $\mathfrak{g}_{nn}(x_{0})\leqslant C.$ Here we also use (8.4).

#### Acknowledgements

The author wishes to express his gratitude to Professor Robert McOwen for
sending his beautiful paper [46]. The author would like to thank Professor Yi
Liu for answering questions related to the proof of Lemma 6.2. The author also
wishes to thank Dr. Ze Zhou for useful discussions on the homogeneity lemma.
The author is supported by the National Natural Science Foundation of China
through grant 11801587.

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