The Gursky-Streets equation and its application to the $\sigma_k$ Yamabe problem

TL;DR

This paper introduces solutions to the Gursky-Streets equation, a key geometric PDE in conformal geometry, establishing uniform estimates, concavity, and uniqueness results, with applications to the $\sigma_k$ Yamabe problem on conformally flat manifolds.

Contribution

It proves uniform $C^{1,1}$ estimates and concavity of the Gursky-Streets operator for all $k \,\leq\, n$, and establishes the first uniqueness results for degenerate solutions.

Findings

Established uniform $C^{1,1}$ estimates for solutions.

Proved the concavity of the Gursky-Streets operator for all $k\leq n$.

Proved uniqueness of solutions to degenerate equations.

Abstract

The Gursky-Streets equation are introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of $\sigma_2$ Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equations with uniform $C^{1, 1}$ estimates for $2k\leq n$. An important new ingredient is to show the concavity of the operator which holds for all $k\leq n$. Our proof of the concavity heavily relies on Garding's theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform $C^{1, 1}$ \emph{a priori estimates} for all the cases $n\geq 2k$. Moreover, we establish the uniqueness of the solution to the degenerate equations for the first time. As an application, we prove that if $k\geq 3$ and $M^{2k}$ is conformally flat, any solution solution of $\sigma_k$ Yamabe problem is conformal diffeomorphic to the round sphere $S^{2k}$.

Equations

$$A:=\frac{1}{n-2}\left(Ric-\frac{1}{2(n-1)}Rg\right).$$

$${\mathcal{C}}_k^{+}=\left\{u|g_u\in [g],A_u\in {\Gamma }_k^{+}\right\}.$$

$$u_{tt}{\sigma }_k(A_u)-⟨T_{k-1}(A_u),\nabla u_t\otimes \nabla u_t⟩=0.$$

$$⟨\psi ,\varphi {⟩}_u={\int }_M\varphi \psi {\sigma }_2(g_u^{-1}{A}_u)d{V}_u.$$

$$u_{tt}{\sigma }_2(A_u)-⟨T_1(A_u),\nabla u_t\otimes \nabla u_t⟩=f$$

$$u_{tt}{\sigma }_2(A_u)-⟨T_1(A_u),\nabla u_t\otimes \nabla u_t⟩=0$$

$${\sigma }_k(\Lambda )=\sum _{i_1<...<i_k}{\lambda }_{i_1}\cdots {\lambda }_{i_k}.$$

$$u_{tt}{\sigma }_k(A_u)-⟨T_{k-1}(A_u),\nabla u_t\otimes \nabla u_t⟩=f,$$

$$|u{|}_{C^0}+|u_t|\le C=C(C_2,\sup f),\max \left\{|\nabla u|,u_{tt},|{\nabla }^{2}u|,|\nabla u_t|\right\}\le C_3.$$

$$u_{tt}{\sigma }_k(A_u)-⟨T_{k-1}(A_u),\nabla u_t\otimes \nabla u_t⟩=0$$

$$|u{|}_{C^0}+|u_t|+|\nabla u|+u_{tt}+|{\nabla }^{2}u|+|\nabla u_t|\le C_2.$$

$$\{\sup f+f^{-1}\left(|\nabla f|+|f_t|+|f_{tt}|+|\Delta f|\right)\}.$$

$$u_{tt}{\sigma }_k(A_u)-⟨T_{k-1}(A_u),\nabla u_t\otimes \nabla u_t⟩=0,$$

$$F_k(R)=r_{00}{\sigma }_k(r)-⟨T_{k-1}(r),x\otimes x⟩$$

$$\mathcal{S}=\{R:r\in {\Gamma }_k^{+},F_k(R)>0\}$$

$$H_k(r,x)=\frac{x{T}_{k-1}(r)x^T}{{\sigma }_k(r)}=x\left(\frac{\partial \log {\sigma }_k(r)}{\partial r_{ij}}\right)x^T$$

$$p_{k+1}(t)=F_k(R+tI)=(r_{00}+t){\sigma }_k(r+tI)-x{T}_{k-1}(r+tI)x^T$$

$$T_k(S)={\sigma }_k(S)I-{\sigma }_{k-1}(S)S+\cdots +(-1{)}^k{S}^k.$$

$$\begin{array}{rl}& ⟨T_k(A-X\otimes X),X\otimes X⟩=⟨T_k(A),X\otimes X⟩\\ & {\sigma }_k(A-X\otimes X)={\sigma }_k(A)-⟨T_{k-1}(A),X\otimes X⟩\end{array}$$

$$\left\{\begin{array}[]{ll}\sigma_{k}(r+tI)=\sum\limits_{1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n}(t+r_{i_{1}})(t+r_{i_{2}})\cdots(t+r_{i_{k}})\\ \\ xT_{k-1}(r+tI)x^{T}=\sum\limits_{1\leq i\leq n}q_{i,k}(t)x_{i}^{2}\end{array}\right.$$

$$p_{k+1}(t)=(r_{00}+t){\sigma }_k(r+tI)-\sum _{1\le i\le n}{q}_{i,k}(t)x_i^2.$$

$${\sigma }_k(r+tI)=\frac{{\pi }_n^{(n-k)}}{(n-k)!}.$$

$$\frac{{\pi }_n^{(n-k)}}{(n-k)!}$$

$$q_{i,k}$$

$$q_{i,k}^{(1)}$$

$$q_{i,n}$$

$$q_{i,k}=\frac{q_{i,k+1}^{(1)}}{n-k}=\frac{q_{i,k+2}^{(2)}}{(n-k)(n-k-1)}=\cdots =\frac{q_{i,n}^{(n-k)}}{(n-k)!}=\frac{{\pi }_{i,n}^{(n-k)}}{(n-k)!}.$$

$$v_n\le u_n\le v_{n-1}\le \cdots \le v_2\le u_2\le v_1\le u_1;$$

$$u_n\le v_{n-1}\le \cdots \le v_2\le u_2\le v_1\le u_1.$$

$$F(t)=p_{k+1}(t),a(t)=t+r_0,f(t)={\sigma }_k(r+tI),g_i(t)=q_{i,k}(t),b_i=-x_i^2.$$

$$Ric(g_u)=Ric+(n-2)\left({\nabla }^{2}u+\nabla u\otimes \nabla u-\frac{1}{2}|\nabla u{|}^{2}g\right)+\left(\Delta u-\frac{n-2}{2}|\nabla u{|}^2\right)g$$

Full text

The Gursky-Streets equation and its application to the $\sigma_{k}$ Yamabe problem

Weiyong He

Department of Mathematics, University of Oregon, Eugene, OR 97403.

whe@uoregon.edu

, 

Lu Xu

Institute of Mathematics, Hunan University, Changsha 410082 China.

xulu@hnu.edu.cn

 and 

Mingbo Zhang

Department of Mathematics, University of Science and Technology of China, Hefei 230026 China.

mbzhang@ustc.edu.cn

The first author is supported in part by an NSF grant, No. 1611797. The second author is supported by NSFC No. 11771132 and Hunan Science and Technology Project No. 2018JJ1004. The third author is supported by Youth Innovation Fund 2016 under grant No. WK0010460002.

2010 Mathematics Subject Classification: Primary 35B45; Secondary 35A02, 35J70, 35B50

Abstract

The Gursky-Streets equation are introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of $\sigma_{2}$ Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equations with uniform $C^{1,1}$ estimates for $2k\leq n$. An important new ingredient is to show the concavity of the operator which holds for all $k\leq n$. Our proof of the concavity heavily relies on Garding’s theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform $C^{1,1}$ a priori estimates for all the cases $n\geq 2k$. Moreover, we establish the uniqueness of the solution to the degenerate equations for the first time.

As an application, we prove that if $k\geq 3$ and $M^{2k}$ is conformally flat, any solution solution of $\sigma_{k}$ Yamabe problem is conformal diffeomorphic to the round sphere $S^{2k}$.

Keywords: A priori estimate, Uniqueness, Degenerate equation, Maximum principle

1. Introduction

Let $(M^{n},g)$ be a compact Riemannian manifold of dimension $n$, $n\geq 3$, with a fixed conformal class $[g]$. Write $Ric$ for the Ricci tensor of $g$. The Schouten tensor is defined as

[TABLE]

The well-known $\sigma_{k}$-curvature is the $k$-th elementary symmetric function of the eigenvalues of $g^{-1}A$. For $1\leq k\leq n$, we say $A\in\Gamma^{+}_{k}$ if $\sigma_{j}(g^{-1}A)>0$ for all $1\leq j\leq k$. Suppose $(M^{n},g)$ satisfies that $A\in\Gamma^{+}_{k}$. Let $g_{u}=e^{-2u}g$ be a conformal metric. We say $u$ is admissible, if $A_{u}\in\Gamma^{+}_{k}$, where $A_{u}$ is the Schouton tensor of $g_{u}$. Denote

[TABLE]

Given $u_{0},u_{1}\in{\mathcal{C}}^{+}_{k}$, we study the following equations.

[TABLE]

This equation is introduced by Gursky and Streets [18]. When $k=2,n=4$, Gursky-Streets [18] studied the metric structure extensively on the space of ${\mathcal{C}}^{+}_{2}$ defined by

[TABLE]

and explored its geometry. This geometric structure plays a substantial role in the proof of the uniqueness of $\sigma_{2}$-Yamabe problem on a compact four manifold.

In this article, we aim to further study the general case of the Gursky-Streets equation. When $M$ is conformally flat and $n=2k\geq 6$, the geometric structure is parallel to the case $n=4=2k$, see [18] and [19].

It is noteworthy that the geometry of Gursky-Streets’ metric on the space of conformal metrics has a parallel theory with the geometry of the space of Kähler metrics. It was originally inspired by the Mabuchi-Semmes-Donaldson metric [28, 29, 33, 14] of Kähler geometry. In 1990s, Donaldson [13] set up a program to study the geometry of the space of Kähler metrics and its various applications to the well-known problems in Kähler geometry, notably the existence and uniqueness of Calabi’s extremal Kähler metrics [6] (constant scalar curvature metrics). Donaldson’s program and related problems have great impacts on the Kähler geometry. A key ingredient is the geodesic equation, which can be written as a homogeneous complex Monge-Ampere equation by the work of Semmes [33] and Donaldson [13]. X.X. Chen [8] proved the existence of $C^{1,\bar{1}}$ geodesic given two boundary datum and solved a conjecture of Donaldson. Chen’s solution has played an important role in Donaldson’s program.

Inspired by the theory of the space of Kähler metrics, Gursky-Streets introduced the fully nonlinear degenerate elliptic equation (1), arising as the geodesic equation of the Gursky-Streets metric. Gursky and Streets proved a remarkable geometric consequence: solutions of the $\sigma_{2}$-Yamabe problem–whose existence follows from their positivity assumption and Chang, Gursky and Yang [12]-are unique, unless the manifold is conformally equivalent to the round sphere. This is a surprising departure from the classical Yamabe problem, where explicit examples of nonuniqueness are known (see e.g. [30, 2, 3, 32, 36] ). Therefore, the geodesic equation has played an essential and important role in the uniqueness of the solutions of the $\sigma_{2}$-Yamabe problem.

In [22] the first author solved the following equation, for a smooth $f>0$ and any $n\geq 4$,

[TABLE]

with the uniform $C^{1,1}$ estimates, and then gave a strong solution to the degenerate equation

[TABLE]

by letting $f=s\rightarrow 0$. Using the uniform $C^{1,1}$ estimates, the first author was able to carry out the approach introduced in [18] to give a simplified and more straightforward proof of the uniqueness of the $\sigma_{2}$-Yamabe problem [22].

Recall that for $\Lambda=(\lambda_{1},...,\lambda_{n})\in\mathbb{R}^{n}$, the $k$-th elementary symmetric function is defined as

[TABLE]

The operator $\sigma_{k}$ has many useful and important properties, and itself is very significant. The $\sigma_{k}$-Hessian and curvature equations are developed by Caffarealli-Nirenberg-Spruck in [5]. It appears in the classical $\sigma_{k}$-Yamabe problem in conformal geometry, see e.g. [24]. It also appears in the celebrated Christoffel-Minkowski problem in convex geometry, see e.g. [17].

In this paper we consider all the cases $n\geq 2k$, which generalizes substantially [22] for $k=2$. At first, we establish the a priori estimates of the following equations for $u:M\times[0,1]\rightarrow{\mathcal{C}}^{+}_{k}$

[TABLE]

where $f>0$ is a given smooth function. Specifically, we prove the following main result.

Theorem 1**.**

Let $2k\leq n$. Given $u_{0},u_{1}\in{\mathcal{C}}^{+}_{k}$ and a smooth function $f>0$, there exists a unique smooth solution $u$ of (3) such that $u(0,\cdot)=u_{0}$, $u(1,\cdot)=u_{1}$. Moreover, we have the following uniform $C^{1,1}$ a priori estimates,

[TABLE]

Taking $f=s$ for a positive constant $s$ and let $s\rightarrow 0^{+}$, we obtain the existence of solutions to degenerate equations. For the uniqueness results, we use maximum principle by constructing approximate solutions. At last, we conclude by a overall result as following.

Theorem 2**.**

Let $2k\leq n$. Given $u_{0},u_{1}\in{\mathcal{C}}^{+}_{k}$, there exists a unique function $u\in C^{1,1}\cap\overline{{\mathcal{C}}^{+}_{k}}$ with $u(0,\cdot)=u_{0}$, $u(1,\cdot)=u_{1}$ such that

[TABLE]

in $L^{\infty}$ sense. Moreover, we have the following uniform $C^{1,1}$ control of $u$,

[TABLE]

Note that in the above theorems the following conventions of dependence of the constants are used. We use $C_{1}$ to denote a uniformly bounded positive constant depending only on $(M^{n},g)$; $C_{2}$ to denote a uniformly bounded constant depending in addition on the boundary values $u_{0},u_{1}$; and $C_{3}$ to denote a uniformly bounded constant depending in addition on $f$. An important feature is that $C_{3}$ does not depend on $\inf f$, but rather on

[TABLE]

We use the notation $C=C(a_{1},a_{2},\cdots)$ to denote a uniform constant which depends on parameters $a_{1},a_{2},\cdots$. The precise dependence of constants on the boundary value $u_{0},u_{1}$ and $f$ can be easily traced in the proofs.

It is worth mentioning that there are several different key points compared with [22] when $k=2$ is assumed. To solve the Gursky-Streets equations with uniform $C^{1,1}$ estimates for $2k\leq n$, our technique is to first show the concavity of the operator. The approach in [22] for $k=2$ does not seem to work for the general case. The proof in this article relies heavily on the Garding theory of hyperbolic polynomials and the results in the theory of real roots for (interlacing) polynomials. This concavity certainly plays a substantial role in the a priori estimates.

Comparing to the case $k=2$, the general case is more involved technically and the computations are certainly more complicated. We should emphasize that a main contribution is the interior $C^{2}$ estimates. In the case $n=2k$, this interior $C^{2}$ estimate is extremely delicate and technically very involved. Moreover, we establish the uniqueness of solution to the degenerate equation which are not obvious at all, even for $k=2$. To achieve the uniqueness of the degenerate equation

[TABLE]

we use the notion of viscosity solution together with our $C^{1,1}$ estimates. The uniqueness does not follow directly from general uniqueness results of viscosity solution in literature, at least to our knowledge. In additional, the nonlinear structure of $A_{u}$ and $2k\leq n$ play important roles in our arguments and indeed the proof is rather technical.

In the Appendix, we include briefly the geometric structure in ${\mathcal{C}}_{k}^{+}$ and the Gursky-Streets equation as a geodesic equation when $M^{2k}$ is conformally flat. Moreover the functional introduced by Brendle and Viaclovsky [4] is geodesically convex. As an application, we prove that if $M^{2k}$ is locally conformally flat and ${\mathcal{C}}_{k}^{+}$ is not empty, then any solution of the $\sigma_{k}$ Yamabe problem is conformally diffeomorphic to the round sphere $S^{2k}$. This gives a totally different proof of a classical result of A.B. Li and Y. Y. Li [24, 25].

2. Concavity

In this section we establish the convexity of the Gursky-Streets equations. Denote the symmetric matrix $R=\left(r_{ij}\right)$ for $0\leq i,j\leq n$. We write $r=\left(r_{ij}\right)$ for $1\leq i,j\leq n$ and $x=\left(r_{01},\cdots,r_{0n}\right)$. For $r\in\Gamma_{k}^{+}$, define the operator

[TABLE]

Denote the set ${\mathcal{S}}$ to be the set of symmetric $(n+1)\times(n+1)$ matrices satisfying the following,

[TABLE]

Theorem 2.1**.**

The set ${\mathcal{S}}$ is a convex cone and $F_{k}^{\frac{1}{k+1}}(R)$ (and hence $\log F_{k}(R))$ is concave on ${\mathcal{S}}$.

This proves the concavity of Gursky-Streets equations and it confirms a conjecture of the first named author [22, Conjecture 4.3]. When $k=n$, the operator $F_{n}=\det$ and the concavity is a well-known result. When $k=1$ the operator $F_{1}$ is the Donaldson operator and the concavity was proved by S. Donaldson and Chen-He. The concavity of $\log F_{k}$ is equivalent to the convexity of $H_{k}(r,x)$ with

[TABLE]

It is well-known that $\log\sigma_{k}(r)$ is concave (over $\Gamma_{k}^{+}$). The convexity of $H_{k}$ (with $x$ fixed), implies that $\left(\frac{\partial\log\sigma_{k}(r)}{\partial r_{ij}}\right)$ as a function on $r$ is convex, which is about the positivity of “third derivatives ” of $\log\sigma_{k}$. In [22], the first named author proved the convexity of $H_{2}$ by very involved computations. The method is elementary but consists of delicate and complicated computations. For $3\leq k\leq n-1$, the approach used in [22] seems to be too involved to prove the convexity of $H_{k}$.

In this paper we adopt an approach relying on Garding’s theory of hyperbolic polynomials [20] to prove the concavity of $F_{k}^{1/{k+1}}$ (and hence $\log F_{k}$). Using the theory of hyperbolic polynomials (see [21, Proposition 2.1.31]), this is equivalent to proving the following

Theorem 2.2**.**

For each $1\leq k\leq n$, the polynomial

[TABLE]

has only real roots for any real $n$-dimention vector $x$ and $n\times n$ real symmetric matrix $r$. Furthermore, if $\alpha_{1}\leq\alpha_{2}\leq\cdots\leq\alpha_{k+1}$ are the all real roots of $p_{k+1}$ , then they are separated by the $k$ real roots of $\sigma_{k}(r+tI)$. We also have $\alpha_{i}\in[\min\limits_{1\leq j\leq n}\{-r_{j}\},\max\limits_{1\leq j\leq n}\{-r_{j}\}]$, for $2\leq i\leq k-1$, where $(r_{1},\cdots,r_{n})$ are real eigenvalues of $r$.

We need some facts about the convex cone $\Gamma^{+}_{k}$ and the Newton transformation $T_{k}(A)$. With the standard Euclidean metric, the $k$-th Newton transformation associated with a symmetric matrix $S$ (on $\mathbb{R}^{n}$) is given by

[TABLE]

Proposition 2.1**.**

We have,

1. (1)

Each $\Gamma^{+}_{k}$ is an open convex cone. 2. (2)

If $A\in\Gamma^{+}_{k}$, then $T_{k-1}(A)$ is positive definite. 3. (3)

$\log\sigma_{k}$* and $\sigma_{k}^{1/k}$ are concave on $\Gamma^{+}_{k}$.*

We also need the following fact (see [18]),

Proposition 2.2**.**

Given $A$ a symmetric matrix and $X$ a vector, then

[TABLE]

It is generally a hard problem to check whether a polynomial has all real roots. For example, when $k=2$, it is not easy to check that the cubic polynomial $F_{2}(R+tI)$ has all real roots by direct computations, even though the conditions on cubic polynomial having all real roots are well-known. Instead we exhibit the structure of real roots of $F_{k}(R+tI)$ and its relation with the polynomial $\sigma_{k}(r+tI)$. This allows us to use the theory of interlacing to assert that $F_{k}(R+tI)$ has all real roots. Since $F_{k}(R)$ involves $r_{00}$, $x$ and only eigenvalues of $r$, we can diagonalize $r$ such that both of $r+tI$ and $T_{k-1}(r+tI)$ are diagonal matrices. We denote the eigenvalues of $r$ by $(r_{1},\dots,r_{n})$ and write

[TABLE]

where $q_{i,k}(t)\triangleq\sum\limits_{0\leq j\leq k-1}(-1)^{j}\sigma_{k-1-j}(r+tI)(t+r_{i})^{j}$ and $\sigma_{0}(r+tI)$ is defined to be 1. Thus we have

[TABLE]

We introduce some notations for the convenience in the following discussion.

1. (1)

RZ$\triangleq$ the set of univariate polynomials with all real zeros. 2. (2)

$\pi_{n}(t)\triangleq(t+r_{1})(t+r_{2})\cdots(t+r_{n})$; 3. (3)

$\pi_{i,n}(t)\triangleq\frac{\pi_{n}(t)}{t+r_{i}}$.

Firstly we give two formulas about $\sigma_{k}(r+tI)$ and $q_{i,k}$, which will be used later.

Proposition 2.3**.**

Let $\pi_{n}^{(i)}$ denote the $i$th derivative of $\pi_{n}$ with respect to $t$. Then

[TABLE]

Proof.

This is straightforward computation,

[TABLE]

∎

Proposition 2.4**.**

$q_{i,k}=\dfrac{\pi^{(n-k)}_{i,n}}{(n-k)!}$. The superscript $(n-k)$ still stands for the $(n-k)$-th derivative.

Proof.

We have the following,

[TABLE]

while

[TABLE]

The last equality holds because $\pi_{i,n}$ is a polynomial of degree $n-1$. So we have

[TABLE]

∎

By the above-mentioned formulas for $\sigma_{k}(r+tI)$ and $q_{i,k}$ , we have:

Lemma 2.5**.**

$\sigma_{k}(r+tI)$* and $q_{i,k}(t)$ are real rooted for each $1\leq i,k\leq n$.*

Proof.

Obviously, $\pi_{n}(t)=(t+r_{1})(t+r_{2})\cdots(t+r_{n})$ has $n$ real roots $\{-r_{1},-r_{2},\cdots,-r_{n}\}$ by counting multiplicity. By Rolle’s Mediate Value Theorem, the $i$th derivative of $f$ has exactly $n-i$ real roots, for any $0\leq i\leq n$. Thus $\sigma_{k}(r+tI)$ has $n-(n-k)=k$ real roots by Proposition 2.3. While it is a polynomial of $t$ of degree $k$. Therefore $\sigma_{k}(r+tI)$ is real rooted.

Similarly, $\pi_{i,n}$ has $n-1$ real roots $\{-r_{1},-r_{2},\cdots,-\hat{r_{i}},-\cdots,-r_{n}\}$. Then $q_{i,k}$ has exactly $n-1-(n-k)=k-1$ real roots by Proposition 2.4 and it is of degree $k-1$. Then all the roots are real. ∎

Assume $f,g$ are two polynomials in RZ and let $\{u_{i}\},\{v_{j}\}$ be all roots of $f,g$ in nonincreasing order respectively. We say that $g$ interlaces $f$ , denoted by $g\preccurlyeq f$, if either deg$f=$deg $g=n$ and

[TABLE]

or deg $f=$ deg $g+1=n$ and

[TABLE]

If all inequalities are strict, then we say that $g$ strictly interlaces $f$, denoted by $g\prec f$. Interlacing describes the relative positions of the real roots of a pair of polynomials. It has shown its power in studying real rootedness of polynomials. We have the following important relation between $\sigma_{k}(r+tI)$ and $q_{i,k}$ :

Lemma 2.6**.**

$q_{i,k}$* interlaces $\sigma_{k}(r+tI)$ for any $1\leq i,k\leq n$.*

Proof.

By Propositions 2.3 and 2.4, it’s equivalent to prove $\pi^{(n-k)}_{i,n}(t)$ interlaces $\pi^{(n-k)}_{n}(t)$. This holds definitely because for each $1\leq i\leq n$, $\pi_{i,n}(t)$ interlaces $\pi_{n}(t)$ and the differential operator $\frac{d}{dt}$ preserves interlacing [34, Theorem 1.47, page 38/779]. ∎

Now we come to prove the real rootedness of $p_{k+1}$. The main idea of the proof depends on the following theorem [27, Theorem 2.3, page 5/19], which is very useful to prove the real rootedness of some specific categories of real polynomials.

Theorem 2.3**.**

[27]**: Let $F,f,g_{1},\cdots,g_{n}$ be real polynomials satisfying the following conditions:

1. (1)

$F(t)=a(t)f(t)+b_{1}(t)g_{1}(t)+\cdots+b_{n}(t)g_{n}(t)$* , where $a,b_{i}$ are real polynomials such that $\text{deg}\,F=degf$ or $\text{deg}f+1$.* 2. (2)

$f,g_{i}\in RZ$* and $g_{i}$ interlaces $f$ for each $i$.* 3. (3)

$F$* and $g_{i}$ have leading coefficients of the same sign.* 4. (4)

$b_{i}(z)\leq 0$* for each $i$ and each zero $z$ of $f$ .*

Then $F\in RZ$ and $f$ interlaces $F$. In particular, if for each zero $z$ of $f$ , there is an index $i$ such that $g_{i}$ strictly interlace $f$ and $b_{i}(z)<0$, then $f$ strictly interlaces $F$.

By setting

[TABLE]

in the above theorem, we can see :

**(1): **

deg($p_{k+1}$)$=k+1=$ deg\bigg{(}\sigma_{k}(r+tI)\bigg{)}+1

**(2): **

$\sigma_{k}(r+tI),q_{i,k}\in RZ$ and $q_{i,k}$ interlaces $\sigma_{k}(r+tI)$ for each $i$ and $k$ (Lemma 2.5 and 2.6).

**(3): **

$p_{k+1}(t)$ and $q_{i,k}(t)$ have both positive leading coefficients. In fact, the leading coefficient of $p_{k+1}$ equals $\frac{n!}{(n-k)!^{2}}$ and that of $q_{i,k}$ equals $\frac{(n-1)!}{(n-k)!^{2}}$.

**(4): **

$-x_{i}^{2}\leq 0$ for each $i$ and each zero of $\sigma_{k}(r+tI)$

Thus, all the four required conditions are all satisfied and then we get the real rootedness of $p_{k+1}(t)=F_{k}(R+tI)$. Hence by [21, Proposition 2.1.31], $F_{k}^{1/(k+1)}$ is concave on the component of $I$ in the set of $\{R:F_{k}(R)\neq 0\}$, which is a connected convex cone. We denote this convex cone as $\tilde{\mathcal{S}}$. Since $I$ is in ${\mathcal{S}}$ and ${\mathcal{S}}$ is connected such that $F_{k}(R)>0$ on ${\mathcal{S}}$, hence ${\mathcal{S}}$ is contained in $\tilde{\mathcal{S}}$. The concavity of $F_{k}^{1/(k+1)}$ on $\tilde{\mathcal{S}}$ (and hence on ${\mathcal{S}}$) implies that ${\mathcal{S}}$ is indeed a convex cone, which proves Theorem 2.2. Even though we do not really need this fact, it is a standard practice to show that ${\mathcal{S}}=\tilde{\mathcal{S}}$.

3. Existence and uniqueness

In this section we establish the a priori estimates to solve the equation. First we recall Gursky-Streets’ equations and related notations briefly. Let $(M^{n},g)$ be a compact Riemannian manifold with the conformal class $[g]$, $n\geq 3$. The metrics in $[g]$ can be parametrized by metrics of the form $g_{u}=e^{-2u}g$. The Ricci curvature is given by

[TABLE]

and the scalar curvature is given by

[TABLE]

Under the conformal change, the Schouten tensor is given by

[TABLE]

In this section we derive the a priori estimates to solve the equation. Denote the operator

[TABLE]

Given $u_{0},u_{1}\in C^{\infty}$ such that $A_{u_{0}},A_{u_{1}}\in\Gamma^{+}_{k}$, we assume that $u\in C^{\infty}$ such that $A_{u}\in\Gamma^{+}_{k}$, and solves the equation

[TABLE]

for a positive function $f\in C^{\infty}$, with the boundary condition $u(0,x)=u_{0}(x)$, $u_{1}(1,x)=u_{1}(x).$

We need the following standard formulas of elementary symmetric functions, see e.g. [26].

Proposition 3.1**.**

Let $\lambda=(\lambda_{1},...,\lambda_{n})\in\mathbb{R}^{n}$ and $k=0,1,...,n$, then

[TABLE]

Here we denote by $\sigma_{k}(\lambda|i)$ the symmetric function with $\lambda_{i}=0$.

Proposition 3.2**.**

(Newton’s inequality). For any $k\in\{1,2,...,n-1\},$ and $\lambda\in\mathbb{R}^{n}$, we have

[TABLE]

In particular, we have

[TABLE]

Proposition 3.3**.**

(MacLaurin inequality). For $\lambda\in\Gamma^{+}_{k}$ and $k\geq l\geq 1$, we have

[TABLE]

Proposition 3.4**.**

(Generalized Newton-MacLaurin inequality). For $\lambda\in\Gamma_{k}^{+}$ and $k>l\geq 0$, $r>s\geq 0$, $k\geq r$, $l\geq s$, we have

[TABLE]

and the equality holds if and only if $\lambda_{1}=\lambda_{2}=\cdots=\lambda_{n}>0$.

We set

[TABLE]

Then by Proposition 2.2, the operator in equation (3.1) is transformed into

[TABLE]

Proposition 3.5**.**

Suppose $A_{u}\in\Gamma_{k}^{+}$ and $u_{tt}>0$, $\sigma_{k}(E_{u})>0$, then $E_{u}\in\Gamma^{+}_{k}$. In particular, we have

[TABLE]

with $f_{i}=u_{tt}^{1-i}\sigma_{i}(E_{u})$, $f_{k}=f$.

Proof.

Since $\sigma_{k}(E_{u})>0$, it is sufficient to prove if $\sigma_{i}(E_{u})>0$, then $\sigma_{i-1}(E_{u})>0$ for all $2\leq i\leq\ k.$

Note

[TABLE]

It yields

[TABLE]

We compute

[TABLE]

So we only need to show

[TABLE]

is positive definite.

Diagonalize $A_{u}$ with eigenvalues $\tilde{\lambda}=(\tilde{\lambda_{1}},...,\tilde{\lambda_{n}})$, and denote by $\sigma_{k}(\tilde{\lambda}|l)$ the symmetric function with $\tilde{\lambda}_{l}=0$. Then we need to verify $\forall\;l\;\in\{1,...n\}$

[TABLE]

It is true by

[TABLE]

The last inequality holds from Newton’s inequality (3.8).

∎

In the following for simplicity we denote the symmetric tensor product as follows,

[TABLE]

Proposition 3.6**.**

Given $u\in C^{2}$ such that $A_{u}\in\Gamma^{+}_{k}$, then the equation (3.2) is strictly elliptic when $f>0$. The linearized operator is given by

[TABLE]

where ${\mathcal{L}}_{A_{u}}(v)$ is the linearization of $A_{u}$, given by

[TABLE]

Proof.

First note that when $f>0$, by the assumption $A_{u}\in\Gamma^{+}_{k}$, we have $u_{tt}>0$. Suppose $\delta u=v$, and we use the variation of $\sigma_{k}$, $\delta\sigma_{k}(E_{u})=\langle T_{k-1}(E_{u}),\delta E_{u}\rangle$. By direct computation we have

[TABLE]

To show the ellipticity, we only need to take care of second order derivatives of $v$. The leading terms reads,

[TABLE]

Replacing the derivatives of $(v_{t},\nabla v)$ by a vector $(\xi,X)\in T([0,1]\times M)=\mathbb{R}\times\mathbb{R}^{n}$, we need to show that the following quadratic form is positive definite,

[TABLE]

We compute

[TABLE]

where $Y=\sqrt{u_{tt}}X-\xi\nabla u_{t}$. It follows that

[TABLE]

Since $E_{u}\in\Gamma^{+}_{k}$ from Proposition 3.5, $T_{k-1}(E_{u})$ is positive definite. A direct computation by Proposition 3.1 gives

[TABLE]

It then follows that, for $(\xi,X)\neq 0$, $Q(\xi,X)>0$. To show the second identity in (3.20), we compute

[TABLE]

Applying (3.21) again we get the result. This completes the proof. ∎

We will need the comparison function as follows. Denote $U_{a}=at(1-t)+(1-t)u_{0}+tu_{1}$ for any number $a$. Note that $U_{0}=u_{0}$ at $t=0$, $U_{1}=u_{1}$ at $t=1$ for any $t$. In particular $U_{a}$ has the same boundary value with $u$. If $u_{0},u_{1}$ are admissible ($A_{u_{i}}\in\Gamma^{+}_{k}$ for $i=0,1$), $U_{0}=(1-t)u_{0}+tu_{1}$ is admissible [35] and hence $U_{a}$ are all admissible, since we have $A_{U_{0}}=A_{U_{a}},(\nabla U_{0})_{t}=(\nabla U_{a})_{t}$ for any $a$. Gursky-Streets [18] proved a uniform $C^{1}$ estimate for the equation

[TABLE]

where $E_{u}^{\epsilon}=(1+\epsilon)u_{tt}A_{u}-\nabla u_{t}\otimes\nabla u_{t}$, for any $k\geq 1$. They introduced an extra $\epsilon$-parameter for the purpose of $C^{1,1}$ estimates, which does not play any essential role in $C^{1}$ estimates. Hence their results clearly apply in our setting to obtain uniform $C^{1}$ estimates and most computations required in $C^{1}$ estimates can be found in [18]. Nevertheless we will include details of $C^{1}$ estimates for completeness (these computations will be needed for uniform $C^{1,1}$ estimates). Our arguments below are a variant of the case $k=2,n\geq 4$, explored in [22].

3.1. $C^{0}$ estimates

In this section we derive the $C^{0}$ estimates.

Proposition 3.7**.**

There exists $a=a(u_{0},u_{1},\sup f)>0$ sufficiently large, such that

[TABLE]

Proof.

First by $u_{tt}>0$, we have

[TABLE]

That gives the upper bound,

[TABLE]

We claim $u-U_{-a}\geq 0$ for $a>0$ sufficiently large. We argue by contradiction. Since $u-U_{-a}=0$ for $t=0$ and $t=1$, there exists an interior point $p=(t,x)\in(0,1)\times M$, such that $u-U_{-a}$ obtains its minimum at $p$. Denote $u^{s}=su+(1-s)U_{-a}$ and $v=\partial_{s}u^{s}(s=1)=u-U_{-a}$. Then $D^{2}v\geq 0$ and $\nabla v=0$ at $p$. By the concavity of $\log F_{k}$, it follows that for $s\in[0,1]$,

[TABLE]

At $s=1$, we get (at $p$),

[TABLE]

where ${F_{k}}^{-1}{\mathcal{L}}_{F_{k}}$ takes values at $u\;(s=1)$. We can choose $a$ large enough such that

[TABLE]

is sufficiently large. Then the right hand side of (3.22) is negative (at $p$) since $F_{k}(u_{tt},A_{u},\nabla u_{t})=f$. This is a contradiction given the claim that ${\mathcal{L}}_{F_{k}}(v)\geq 0$ at $p$. Note that $D^{2}v\geq 0,\nabla v=0$ at $p$. We compute, using $\nabla v=0$ at $p$,

[TABLE]

We can assume $v_{tt}>0$. Otherwise we have $v_{tt}=0$, then $u_{tt}=2a>0$. Notice that $\nabla v_{t}=0$ and $\nabla^{2}v\geq 0$ (since $D^{2}v\geq 0$ at $p$). In this case the claim follows trivially. If $v_{tt}>0$, the argument follows similarly as in Proposition 3.6. Indeed we write

[TABLE]

where $Y=\sqrt{v_{tt}/u_{tt}}\nabla u_{t}-\sqrt{u_{tt}/v_{tt}}\nabla v_{t}$. By (3.21) and the positivity of $\nabla^{2}v-v_{tt}^{-1}\nabla v_{t}\otimes\nabla v_{t}$ (this is because $D^{2}v\geq 0$), it follows that ${\mathcal{L}}_{F_{k}}(v)\geq 0$. ∎

3.2. $C^{1}$ estimates

First we have the following,

Proposition 3.8**.**

Let $a$ be the constant in Proposition 3.7. Then we have,

[TABLE]

Proof.

Since $u_{tt}>0$, it follows that $u_{t}(t,x)$ is increasing in $t$. Hence we only need to argue $u_{t}(0,x)\leq u_{t}(1,x)$ are both bounded. We compute, using Proposition 3.7,

[TABLE]

It is evident that $u_{t}(0,\cdot)\leq u_{1}-u_{0}$ by convexity. Similarly we have $u_{1}-u_{0}\leq u_{t}(1,\cdot)\leq a+u_{1}-u_{0}$.

∎

To derive estimates of $|\nabla u|^{2}$ and second order derivatives, we need some preparation due to the complicated computations. First we need to choose a normalization condition. Note that if $u$ is admissible, then $\tilde{u}=u-c_{1}t-c_{2}$ is also admissible since $A_{u},E_{u}$ do not change at all. Hence if $u$ is a solution, $\tilde{u}=u-c_{1}t-c_{2}$ is also a solution since $\nabla\tilde{u}=\nabla u,D^{2}\tilde{u}=D^{2}u$. The corresponding boundary condition is changed by a constant with $\tilde{u}_{0}=u_{0}-c_{2},\tilde{u}_{1}=u_{1}-c_{1}-c_{2}$ and $\tilde{u}_{t}=u_{t}-c_{1}$. Hence we can choose two sufficiently large constants $c_{1}$ and $c_{2}$ such that $\tilde{u}\leq-1$, and $\tilde{u}_{t}\leq-1$. We can choose such a normalization condition on $u_{0},u_{1}$ such that,

[TABLE]

where $c_{0}$ is the uniform bound we have obtained above for $|u|$ and $|u_{t}|$.

Next we compute ${\mathcal{L}}_{F_{k}}(v)$ for various barrier functions $v$. The philosophy is well-known in nonlinear elliptic theory, to construct various barrier functions $v$ such that

[TABLE]

Such barrier functions serve as the purpose of subsolutions with respect to ${\mathcal{L}}_{F_{k}}$ and play an essential role in the maximum principle argument. The first such function is the $t$-functions,

Proposition 3.9**.**

Suppose $v=v(t)$ is a $t$-function, then

[TABLE]

where we apply Proposition 2.2 in the last step above. In particular,

[TABLE]

The second choice is the function $-u$ itself. We compute ${\mathcal{L}}_{F_{k}}(u)$.

Proposition 3.10**.**

We have,

[TABLE]

Proof.

By (3.20), we compute

[TABLE]

where we have used (3.21). ∎

Remark 3.11**.**

Both the propositions above are derived in [18] for general $k$. We include the computations here for completeness.

We use the operator $D=(\partial_{t},\nabla)$ to denote the gradient on $\mathbb{R}\times M$, where the space derivative $\phi_{k}$ denotes the covariant derivative $\nabla_{k}\phi$. We rewrite (3.20) as,

[TABLE]

where

[TABLE]

Proposition 3.12**.**

We have the following,

[TABLE]

where $Q_{u}$ is a quadratic form on $D\phi,D\psi$ given by

[TABLE]

Moreover, we compute

[TABLE]

An important feature is that $Q_{u}$ is positive definite in the sense that

[TABLE]

Proof.

This is a straightforward computation. The main point is that ${\mathcal{L}}_{F_{k}}$ and $P_{u}$ are second order linear differential operator and the product rule would introduce mixed terms on first derivatives, which lead to the terms $Q_{u}(D\phi,D\psi)+2u_{tt}^{-1}f\phi_{t}\psi_{t}$. Similarly this applies to $e^{\phi}$. Since $E_{u}\in\Gamma^{+}_{k}$, $T_{k-1}(E_{u})$ is positive definite, it follows that

[TABLE]

where $Y=(\sqrt{u_{tt}})\nabla\phi-\phi_{t}\nabla u_{t}(\sqrt{u_{tt}})^{-1}.$ Clearly the positivity of $Q_{u}$ is simply the consequence of the ellipticity of $F_{k}$. ∎

Proposition 3.13**.**

We compute, using (3.30) in Proposition 3.12,

[TABLE]

Proposition 3.14**.**

We compute,

[TABLE]

Proof.

By (3.29), we have

[TABLE]

Since taking time derivative has the same effect of taking variation, this gives

[TABLE]

It is clear that $Q_{u}(Du_{t},Du_{t})=0$. This completes the computation. ∎

Proposition 3.15**.**

We compute

[TABLE]

where we denote,

[TABLE]

Proof.

First we compute, applying (3.29) to $\phi=\psi=u_{i}$,

[TABLE]

Now we compute

[TABLE]

Differentiate equation $\sigma_{k}(E_{u})=fu_{tt}^{k-1}$ we get

[TABLE]

We compute

[TABLE]

Note that

[TABLE]

So

[TABLE]

This completes the computation by combining (3.34) and (3.38). ∎

Remark 3.16**.**

The computations above are essentially derived in [18] for general $k$. We use the quadratic form $Q_{u}$ to simplify the notations and computations. Of course the positivity of $Q_{u}$ is equivalent to the fact that $F_{k}$ is an elliptic operator.

Now we prove the estimate for $|\nabla u|^{2}$. Combining all the computations above, we have the following estimates,

Lemma 3.17**.**

For $\lambda,b\geq 1$ sufficiently large, we have

[TABLE]

where $C_{4}=C_{4}(\lambda,|u|_{C^{0}})$.

Proof.

By Proposition 3.10, we get

[TABLE]

We claim that for a constant $C_{2}\geq 2u_{t}^{2}$,

[TABLE]

We estimate,

[TABLE]

where $Y=\sqrt{u_{tt}}\nabla u-2(\sqrt{u_{tt}})^{-1}u_{t}\nabla u_{t}$. The claim follows since

[TABLE]

Choose $b\geq(\frac{C_{2}}{4}+1)\lambda^{2}e^{-\lambda u}$, then we estimate by Proposition 3.13

[TABLE]

This completes the proof if $\lambda$ is sufficiently large. ∎

Lemma 3.18**.**

There exists a uniform constant $C=C(n,k\sup f,\sup|\nabla f^{\frac{1}{k+1}}|,g,|u_{0}|_{C^{1}},|u_{1}|_{C^{1}})$ such that

[TABLE]

Proof.

We take the barrier function

[TABLE]

where $\lambda,b$ are the constants in Lemma 3.17. We compute

[TABLE]

We have, by Proposition 3.15, that

[TABLE]

Suppose $|\nabla u|$ is large enough, otherwise we have done. Hence by Lemma 3.17 and (3.42), we have

[TABLE]

If $w$ achieves its maximum on the boundary, then we are already done. Otherwise, suppose $w$ achieves its maximum at $p=(t,x)\in(0,1)\times M$. Then ${\mathcal{L}}_{F_{k}}(w)\leq 0$ at $p$. Hence it follows that (at $p$)

[TABLE]

We compute by inequality of arithmetic and geometric mean

[TABLE]

By Proposition 3.1 and Proposition 3.3

[TABLE]

it follows that (at $p$)

[TABLE]

This gives the upper bound of $|\nabla u|$ at $p$, and hence the upper bound of $w$. It is not hard to check the dependence of the constants. ∎

3.3. $C^{2}$ estimate

Now we derive the estimates of second order. The estimates of second order contain the boundary estimates and the interior estimates.

3.3.1. Boundary estimates

The boundary is given by two time slices $\{t=0\}\times M$ and $\{t=1\}\times M$. The tangential-tangential direction, namely $|\nabla^{2}u|$ is immediate by the boundary data $|\nabla^{2}u_{0}|,|\nabla^{2}u_{1}|$. While the usual “harder” part of the normal-normal direction ($u_{tt}$) follows directly from the equation once the tangential-normal direction ($|\nabla u_{t}|$) is bounded,

[TABLE]

Note that $\sigma_{k}(A_{u})\geq\delta>0$ at $t=0$ and $t=1$, for some uniform constant $\delta$ depending only on $u_{0},u_{1}$. Hence one only needs to bound $|\nabla u_{t}|$ on the boundary. Such a uniform estimate has been obtained by Gursky-Streets in [18] for the equation for all $1\leq k\leq n$,

[TABLE]

They stated their results for $E_{u}^{\epsilon}=(1+\epsilon)u_{tt}A_{u}-\nabla u_{t}\otimes\nabla u_{t}$ but $\epsilon$ does not play any role in their argument. We summarize their results as follows,

Theorem 3.1** (Gursky-Streets [18]).**

If $E_{u}\in\Gamma^{+}_{k}$ and $u$ solves (3.43). There exists a uniform constant $C_{3}$, such that

[TABLE]

3.3.2. Interior estimates

Note that $A_{u}\in\Gamma^{+}_{k}$ implies that $\sigma_{1}(A_{u})>0$. Given the uniform bound on $|\nabla u|$,

[TABLE]

This leads to a lower bound of $\Delta u$: there exists a constant $C_{2}$ such that $\Delta u+C_{2}\geq 1.$ Moreover, this gives the equivalence of $\sigma_{1}(A_{u})$ and $\Delta u$ in the sense

[TABLE]

We want to derive upper bound on $u_{tt}$ and $\Delta u+C_{2}$ (equivalently, the upper bound of $\sigma_{1}(A_{u})$), which will imply the full hessian bound of $u$ since $A_{u}\in\Gamma^{+}_{k}$, and

[TABLE]

The bound on $|\nabla u_{t}|$ will follow from (3.14) in Proposition 3.5 for $i=1$, in the sense that

[TABLE]

Gursky-Streets obtained interior $C^{2}$ estimates for (3.43), depending on the parameter $\epsilon^{-1}$. Their computations of ${\mathcal{L}}_{F_{k}}(u_{tt})$ and ${\mathcal{L}}_{F_{k}}(\Delta u)$ in [18] are very involved. Here we offer a variant of such computations and this provides significant simplifications. The complicated nonlinear terms of first order in $A_{u}$ and the curvature of the background metric will bring extra challenge, not only making the computations much more complicated, but also introducing several nonlinear terms which need extra care.

We need some preparations. Given a symmetric matrix $R=(r_{ij})$ of $(n+1)\times(n+1)$, we use $r=(r_{ij})$ for the $n\times n$ portion with $ij\neq 0$ and $Y=(r_{01},\cdots,r_{0n})$. Note that

[TABLE]

We use the standard notation

[TABLE]

Take the matrix $R$ of the form

[TABLE]

Then we write the equation $F_{k}(R)=f$ and its equivalent form $G_{k}(R)=\log f.$ With this notation, we also record the linearization of $F_{k}(R)$. Given a smooth function $\phi$, we have

[TABLE]

We record the derivatives of $F_{k}$.

Proposition 3.19**.**

We have

[TABLE]

We compute, for $i,j\neq 0$,

[TABLE]

Proof.

This is a straightforward computation. ∎

Now we are ready to compute ${\mathcal{L}}_{F_{k}}(u_{tt})$ and ${\mathcal{L}}_{F_{k}}(\Delta u)$.

Proposition 3.20**.**

We have the following,

[TABLE]

Proof.

We compute

[TABLE]

That is

[TABLE]

Now we consider

[TABLE]

The main point is that $A_{u}$, hence $R$ is not linear on $u$. We compute

[TABLE]

Denote ${\mathcal{R}}=2\nabla u_{t}\otimes\nabla u_{t}-|\nabla u_{t}|^{2}g$ and this is the term coming from the nonlinearity of $A_{u}$. Hence we can write, with $\phi=u_{tt}$,

[TABLE]

By (3.45) and (3.48), we get that

[TABLE]

where we use the notation ${\mathcal{R}}_{i0}=0$, for $i=0,1,\cdots,n$. We claim that

[TABLE]

But this is straightforward since $F_{k}=u_{tt}^{1-k}\sigma_{k}(E_{u})$,

[TABLE]

This completes the proof. ∎

Next we compute ${\mathcal{L}}_{F_{k}}(\Delta u)$. Note that

[TABLE]

Proposition 3.21**.**

[TABLE]

where ${\mathcal{R}}_{1}$ is given in (3.54) and (3.53). We have the following,

[TABLE]

with

[TABLE]

where ${\mathcal{S}}_{0}$ is a uniformly bounded term (matrix) and $Rm*\nabla^{2}u$ denotes two terms of contraction of curvature with $\nabla^{2}u$ (which we do not need precise expression).

Proof.

We compute

[TABLE]

Now we compute

[TABLE]

Recall $A_{u}=A+\nabla^{2}u+\nabla u\otimes\nabla u-|\nabla u|^{2}g/2$ and now we compute $\Delta A_{u}$. We need several Bochner-Weitzenbock formula as follows,

[TABLE]

We use $Rm*\nabla^{2}u+\nabla Rm*\nabla u$ to denote contraction of terms which we do not need precise expression. We can then compute

[TABLE]

where ${\mathcal{S}}$ is the remaining matrix of the form

[TABLE]

Denote

[TABLE]

Then we can write

[TABLE]

It then follows that

[TABLE]

Together with (3.52) this completes the proof of (3.49). The computation of $F_{k}^{ij}{\mathcal{R}}_{1,ij}$ in (3.50) is straightforward, noting that

[TABLE]

∎

To estimate $u_{tt}$ and $\Delta u$, we need the following Lemma.

Lemma 3.22**.**

Let $\phi$ be any smooth function. For $n\geq 2k$,

[TABLE]

Proof.

Diagonalize $E_{u}$ with eigenvalues $\lambda=(\lambda_{1},...,\lambda_{n})$, so

[TABLE]

then (3.55) is equivalent to prove

[TABLE]

The last inequality holds from Newton’s inequality (3.7) noting that $\lambda|i\in\mathbb{R}^{n-1}$. ∎

The interior estimate of $u_{tt}$ now becomes immediate ($n\geq 2k)$.

Lemma 3.23**.**

For $n\geq 2k$, there exists a constant $C_{3}$ such that

[TABLE]

Proof.

By the concavity of $G$, Lemma 3.22 and Proposition 3.20, we have

[TABLE]

It then follows that, using (3.32),

[TABLE]

If $u_{tt}+u_{t}^{2}$ achieves its maximum on the boundary, then by Theorem 3.1 and Proposition 3.8 we are done. Otherwise at the maximum point of $u_{tt}+u_{t}^{2}$, we have

[TABLE]

This is sufficient to bound $u_{tt}$ by a uniform constant $C_{3}$, where $C_{3}$ depends on the boundary estimate of $u_{tt}$ and $-f_{tt}f^{-1},|f_{t}|f^{-1}$ in addition. ∎

Proposition 3.24**.**

For $n\geq 2k+1$, there exists a uniform constant $C$ such that

[TABLE]

Proof.

By Lemma 3.5, $f_{1}>0$ then (3.14) yields

[TABLE]

So in Proposition 3.21,

[TABLE]

and

[TABLE]

In Proposition 3.21, combining (LABEL:eqr), (3.60) and Lemma 3.22, together with the concavity of $G_{k}$,

[TABLE]

The last inequality holds since $|T_{k-1}(E_{u})|\leq\sigma_{1}(T_{k-1}(E_{u}))=(n-k+1)\sigma_{k-1}(E_{u})$. Suppose $\triangle u$ obtains its maximum at an interior point $p$ (otherwise done), assume $|\nabla^{2}u(p)|$ is sufficiently large(otherwise done), then at $p$ we get

[TABLE]

By Lemma 3.5 for $i=k$, it means that

[TABLE]

This is sufficient to get a uniform upper bound of $\triangle u$. ∎

The estimates of $\Delta u$ (for $n\geq 2k+1$) is rather straightforward given the strictly lower bound of the quadratic form in Lemma 3.22. When $n=2k$, such a positivity is too weak and the interior estimate of $\Delta u$ is subtle. A key technical ingredient is the following,

Lemma 3.25**.**

[TABLE]

Proof.

Recall

[TABLE]

And we get

[TABLE]

for some uniformly positive constant $\epsilon_{0}\leq 1$ such that $u_{tt}^{-1}\geq 2\epsilon_{0}$. It follows that

[TABLE]

∎

Proposition 3.26**.**

$n=2k$, there exists a uniform constant $C$ such that

[TABLE]

Proof.

Consider test function

[TABLE]

with

[TABLE]

In Proposition 3.21, combining (LABEL:eqr), (3.60), together with the concavity of $G_{k}$, one get

[TABLE]

Furthermore (3.25) and Proposition 3.15 give that

[TABLE]

We note by Lemma 3.25

[TABLE]

Combining above inequalities (3.65)-(3.67) we obtain

[TABLE]

If $v$ achieves its maximum on the boundary, then we are done. Otherwise $v$ obtains its maximum at an interior point $p$, it follows that

[TABLE]

We claim that this is sufficient to bound $|\nabla^{2}u|$ at $p$,

[TABLE]

We can assume $|\nabla^{2}u|(p)\geq 2C_{1}\epsilon^{-1}$ (otherwise done), then

[TABLE]

And by (3.69) we have

[TABLE]

By Proposition 3.5 for $i=k$, we get at $p$

[TABLE]

And

[TABLE]

This establishes claim (3.70). Obviously we have $\Delta u(p)\leq n|\nabla^{2}u|(p)$. Since $v\leq v(p)$, we have obtained

[TABLE]

In other words, we have

[TABLE]

Note that $\Delta u-C_{2}\leq\sigma_{1}(A_{u})\leq\Delta u+C_{2}$, we get

[TABLE]

We observe that

[TABLE]

Hence $K<\sup\sigma_{1}(A_{u})$, and we have proved that

[TABLE]

This completes the proof of the uniformly upper bound of $\Delta u$ i.e. (3.64). ∎

3.4. The existence and uniqueness

In this section we prove Theorem 1 and Theorem 2. Given the estimates in the last section, the proof of Theorem 1 is standard and rather straightforward. We shall keep it brief.

Proof of Theorem 1.

Let $w=(1-t)u_{0}+tu_{1}+at(t-1)$ for $a$ sufficiently large. Then

[TABLE]

For any smooth function $f$, we consider the continuity path for $s\in[0,1]$

[TABLE]

Then for $s=0$, $v^{s}=w$ solves the equation. The openness follows from the ellipticity and the closedness follows from the a priori estimates. Hence the equation has a solution for $s=1$. For any $f>0$, the solution is unique by a straightforward maximum principle argument. ∎

Next we discuss the degenerate equation (1) and prove Theorem 2.

Proof.

We can take $f=s$ for $s\in(0,1]$. Hence we get a unique smooth solution $u^{s}$ for

[TABLE]

The comparison principle implies that $u^{s}\geq u^{\tilde{s}}$ for $s\leq\tilde{s}$. Hence $u^{s}$ converges uniformly to $u$ for $s\rightarrow 0$ with $u\in C^{1,1}$. In particular $u^{s}$ converges to $u$ in $C^{1,\alpha}$ for any $\alpha\in(0,1)$.

Following a standard notion of viscosity solution, $u$ solves the degenerate equation

[TABLE]

in the sense of viscosity solution, see [11, Lemma 6.1, Remark 6.3]. Since $u\in C^{1,1}$, this implies that

[TABLE]

holds in $L^{\infty}$ sense; and in particular $u$ is a strong solution. We complete the proof given the uniqueness proved below. ∎

The uniqueness problem for degenerate elliptic equations can be subtle. Our argument is an adaption of the argument used in Guan-Zhang [16]. First we define precisely the notion of admissible solution for degenerate equations.

Definition 3.27**.**

A $C^{1,1}$ function $u$ is called an admissible solution of

[TABLE]

if $A_{u}\in\overline{\Gamma_{k}^{+}}$ and $u_{tt}\geq 0.$

The key point is the following approximation result.

Lemma 3.28**.**

Suppose $u$ is a $C^{1,1}$ admissible function defined on $M\times[0,1]$ with

[TABLE]

defined in (3.1). For any $\delta>0$, there is an admissible function $u_{\delta}\in C^{\infty}(M\times[0,1])$ such that $0<F_{k}(u_{\delta})\leq\delta$ and $||u-u_{\delta}||_{C^{0}(M\times[0,1])}\leq\delta$.

Proof.

For $\epsilon\in{(0,1)}$, let $v=(1-\epsilon)u$. First we show that $A_{v}\in\Gamma_{k}^{+}$ (in $L^{\infty}$ sense since $u$ might not be a $C^{2}$ function). We have

[TABLE]

Hence we compute

[TABLE]

It is important to notice that for $2k\leq n$, the $n\times n$ matrix $\text{diag}(-1,1,\cdots,1)$ is in $\Gamma_{k}^{+}$ (this is not the case if we drop the assumption $2k\leq n$. For example, $\text{diag}(-1,1,1)$ is not in $\Gamma_{2}^{+}$ with $k=2,n=3$). This implies that

[TABLE]

Hence this follows that

[TABLE]

So $A_{v}\in\Gamma_{k}^{+}$. Moreover By Theorem 2.1, we have

[TABLE]

Denote $w=v+\epsilon t^{2}$. Then

[TABLE]

We can then approximate $w$ by a smooth function $u_{\delta}$ such that

[TABLE]

and $|u-u_{\delta}|\leq\delta$ by choosing $\epsilon$ sufficiently small (depending on $\delta$).

∎

We are ready to establish the uniqueness of the degenerate equation.

Proposition 3.29**.**

A $C^{1,1}$ admissible solution to the degenerate equqation (1) with given boundary data is unique.

Proof.

Suppose there are two such solutions $u_{1},u_{2}$ with the same boundary data. For any $0<\delta<1$, pick any $0<\delta_{1},\delta_{2}<\delta$, by Lemma 3.28, there exist two smooth functions $v_{1},v_{2}$ such that

[TABLE]

in $M\times[0,1]$, $||v_{i}-u_{i}||_{C^{2}(M\times[0,1])}\leq\delta_{i}$ and $0<f_{i}\leq\delta_{i}$ for $i=1,2$. Set $w_{1}=\delta v_{1}+(1-\delta)t^{2}$. A straightforward computation gives that

[TABLE]

We may choose $\delta_{2}$ sufficiently small such that $0<\delta_{2}<F_{k}(w_{1})$. The maximum principle implies $max_{M\times[0,1]}(w_{1}-v_{2})\leq max_{M\times\{0,1\}}(w_{1}-v_{2})$. Thus

[TABLE]

where constant $C$ depends only on $C^{0}$ norm of $u_{1},u_{2}$. Interchange the role of $u_{1}$ and $u_{2}$, we have

[TABLE]

Since $0<\delta<1$ is arbitrary, we conclude that $u_{1}=u_{2}$. ∎

4. Appendix

In this section we introduce the geometric structure briefly related to the Gursky-Streets equation. It was noted [19] that when $n=2k\geq 6$ and $M$ is locally conformally flat, the theory is parallel to the case $n=4=2k$, studied extensively in [18] and [22]. We include the discussions for completeness.

Let $(M^{n},g)$ be a compact Riemannian manifold of dimension $n$, $n\geq 3$. Given a conformal metric $g_{u}=e^{-2u}g$. Recall that the Schouten tensor is given by

[TABLE]

Let $g_{u}=e^{-2u(t)}g$ be a one-parameter family of conformal metrics. Then a simple computation gives that

[TABLE]

where the Hessian is with respect to $g_{u}$. A direct calculation [31] yields

[TABLE]

where $T_{k-1}$ is the Newton transform. Since the Newton transform is a (1,1)-tensor, for the pairing in (4.1) we lower an index of $T_{k-1}(g_{u}^{-1}A_{u})$ and view it as $(0,2)$-tensor, and use the inner product induced by $g_{u}$. For example, if $n=4$ and $k=2$,

[TABLE]

(4.1) yields

[TABLE]

Definition 4.1**.**

For $\phi,\psi\in C^{\infty}(M)$ we define the $\sigma_{k}$ metric as following,

[TABLE]

We will need a key property as following.

Lemma 4.2**.**

If $k=2$ or if the manifold is locally conformally flat , then $T_{k-1}(g^{-1}A)$ is divergence-free.

A straightforward computation gives the following,

Lemma 4.3**.**

If $n=4=2k$ or if $n=2k$ and $M^{n}$ is conformally flat, the geodesic equation is given by

[TABLE]

Here we introduce Andrew’s Poincar$\acute{e}$ inequality, see [10].

Proposition 4.4**.**

Let $(M^{n},g)$ be a closed Riemannian manifold with positive Ricci curvature. Given $\phi\in C^{\infty}(M)$ such that $\int_{M}\phi\;dV=0$, then

[TABLE]

with equality if and only if $\phi\equiv 0$ or $(M^{n},g)$ is isometric to the round sphere.

Corollary 4.5**.**

Let $(M^{n},g)$ be a closed Riemannian manifold such that $A_{g}\in\Gamma^{+}_{k}$ with $n=2k$. Given $\phi\in C^{\infty}(M)$ and let $\underline{\phi}=V_{g}^{-1}\int_{M}\phi\;dV_{g}$, one has

[TABLE]

with equality if and only if $\phi$ is a constant or $(M^{n},g)$ is isometric to the round sphere.

Proof.

We assume that $\int_{M}\phi\;dV_{g}=0$. By Andrew’s Poincar$\acute{e}$ inequality, to show the claim (4.6) it suffices to show

[TABLE]

That is to show the matrix inequality

[TABLE]

Since $Ric$ and $T_{k-1}(A)$ commute, we only need to show that

[TABLE]

Note that $Ric=(n-2)A+\sigma_{1}(g^{-1}A)g$, we choose coordinate at a point $p\in M$ such that $A$ is diagonal, then $Ric$ and $T_{k-1}$ is also diagonal. Assume the eigenvalues of $A$ are $\lambda_{i},i=1,...,n$. Using Proposition 3.1 and Proposition 3.4, we compute

[TABLE]

∎

Remark 4.6**.**

We should mention that Guan, Viaclovsky and Wang [15] proved that when $2k\geq n$, $Ric$ is positive definite if $g\in\Gamma_{k}^{+}$. In particular, they proved that if $M^{n}$ is locally conformally flat and $g\in\Gamma_{k}^{+}$ for $2k\geq n$, then $M^{n}$ is conformal to the round sphere $S^{n}$.

Set $\sigma=\int_{M}\sigma_{k}(g_{u}^{-1}A_{u})\;dV_{u}$, and $\overline{\sigma}=V_{u}^{-1}\sigma$. Let $M^{2k}$ be locally conformally flat. Brendle and Viaclovsky [4] introduced a functional which integrates the 1-form

[TABLE]

Define the $F$-functional as $dF=\alpha$. When $M^{n}$ is locally conformally flat and $2k=n$, we can get

Proposition 4.7**.**

The functional $F$ is formally geodesically convex.

Proof.

It follows from [4] that, for a path of conformal metrics $u=u(t)$,

[TABLE]

Differentiating again along the geodesic path we have

[TABLE]

where the last line follows from Corollary 4.5. ∎

We have proved that the Dirichlet problem

[TABLE]

with the boundary condition $u(0)=u_{0},u(1)=u_{1}$ has a unique smooth solution $u^{s}$ with uniform $C^{1,1}$ estimates for $0<s\leq 1$. When $s\rightarrow 0$, $u^{s}$ converges uniformly in $C^{1,\alpha}$ to the unique $C^{1,1}$ solution of the geodesic equation

[TABLE]

With this approximation, we have the following,

Lemma 4.8**.**

The functional $F$ is convex along the $C^{1,1}$ geodesic and $F$ takes its minimum in ${\mathcal{C}}_{k}^{+}$ at any smooth critical point with $\sigma_{k}(A_{u})=const$.

Now suppose $u_{0}$ and $u_{1}$ are both smooth critical points of $F$ in ${\mathcal{C}}_{k}^{+}$. Connecting $u_{0}$ and $u_{1}$ by the unique $C^{1,1}$ geodesic, then we have the following

Lemma 4.9**.**

Then either $u(t)=u_{0}+at$ or $(M^{n},g_{u(t)})$ is conformally diffeomorphic to the round sphere $S^{n}$.

As a corollary, this proved the following

Theorem 4.1**.**

Suppose $(M^{2k},g)$ is locally conformally flat such that $g\in\Gamma_{k}^{+}$. Then any solution of $\sigma_{k}(A_{u})=const$ in ${\mathcal{C}}_{k}^{+}$ is conformally diffeomorphic to the round sphere $S^{2k}$.

The proof of Theorem 4.1 relies on the convexity of $F$ along the $C^{1,1}$ geodesic. The proof is rather standard by an approximation argument and the details are almost identical to the case $n=4=2k$, as in [22, Appendix]. We skip the details.

We shall mention that A.B. Li and Y. Y. Li [24, 25] proved the conformal invariance of a large class of fully nonlinear equations on $S^{n}$ by studying the Liouville type theorem. Their classical results include Theorem 4.1 as a special case.