Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p<1

TL;DR

This paper proves a uniqueness theorem for solutions to the $L_p$-Christoffel-Minkowski problem when p<1, addressing a fundamental issue in convex geometric analysis by introducing a new auxiliary function.

Contribution

It establishes the first uniqueness result for the $L_p$-Christoffel-Minkowski problem for p<1, using novel auxiliary functions and techniques inspired by curvature flow analysis.

Findings

Proved uniqueness of solutions for p<1

Introduced a new auxiliary function Z

Extended understanding of convex geometric analysis

Abstract

$L_p$-Christoffel-Minkowski problem arises naturally in the $L_p$-Brunn-Minkowski theory. It connects both curvature measures and area measures of convex bodies and is a fundamental problem in convex geometric analysis. Since the lack of Firey's extension of Brunn-Minkowski inequality and constant rank theorem for $p<1$, the existence and uniqueness of $L_p$-Brunn-Minkowski problem are difficult problems. In this paper, we prove a uniqueness theorem for solutions to $L_p$-Christoffel-Minkowski problem with $p<1$ and constant prescribed data. Our proof is motivated by the idea of Brendle-Choi-Daskaspoulos's work on asymptotic behavior of flows by powers of the Gaussian curvature. One of the highlights of our arguments is that we introduce a new auxiliary function $Z$ which is the key to our proof.

Equations

$$h_{K{+}_{p}t\cdot L}^p=h_K^p+t{h}_L^p.$$

$$\frac{d}{dt}{W}_k(K{+}_{p}t\cdot L){|}_{t=0}=\frac{1}{n+1}{\int }_{{\mathbb{S}}^n}{h}_L^p(x)d{S}_{p,k}(K,x),$$

$$d{S}_{p,k}(K,\cdot )=h_K^{1-p}(x)d{S}_k(K,x),$$

$${\sigma }_k(u_{ij}+u{\delta }_{ij})=\psi (x)u^{p-1},$$

$${\sigma }_k(u_{ij}+u{\delta }_{ij})={\sigma }_k(\lambda (x))=\sum _{1\le i_1<i_2<\cdots <i_k\le n}{\lambda }_{i_1}(x){\lambda }_{i_2}(x)\cdots {\lambda }_{i_k}(x),$$

$${\sigma }_k(u_{ij}+u{\delta }_{ij})=u^{p-1}\text{mbox}on{\mathbb{S}}^n,$$

$$(u_{ij}+u{\delta }_{ij})$$

$$K^{\alpha }=⟨X,\nu ⟩$$

$$W(x)=K^{\alpha }{\lambda }_1^{-1}(h_{ij})-\frac{n\alpha -1}{2n\alpha }|X{|}^2=u\cdot {\lambda }_1(b_{ij})-\frac{n\alpha -1}{2n\alpha }(u^2+|Du{|}^2)$$

$$Z(x)=K^{\alpha }\mathrm{tr}(b_{ij})-\frac{n\alpha -1}{2\alpha }|X{|}^2=u\cdot \mathrm{tr}(b_{ij})-\frac{n\alpha -1}{2\alpha }(u^2+|Du{|}^2),$$

$$S^{\alpha }({\kappa }_1,...,{\kappa }_n)=⟨X,\nu ⟩,$$

$$\displaystyle 2\beta\bigg{(}k\sigma_{k}f-n\sum_{i=1}^{n}\sigma_{k-1}(\lambda|i)\lambda^{2}_{i}\bigg{)},$$

$$k{\sigma }_{k}f-n\sum _{i=1}^n{\sigma }_{k-1}(\lambda |i){\lambda }_i^2=0.$$

$$W(x)=u\cdot {\lambda }_1(b_{ij})-\beta (u^2+|Du{|}^2),$$

$$Z(x)=uF(b_{ij})-n\beta (u^2+|Du{|}^2),$$

$$F(b_{ij})=f({\lambda }_1,{\lambda }_2,...,{\lambda }_n)=\sum _{i=1}^n\frac{n{\sigma }_{k-1}(\lambda |i){\lambda }_i^2}{k{\sigma }_k}=\frac{n}{k}[{\sigma }_1-(k+1)\frac{{\sigma }_{k+1}}{{\sigma }_k}],$$

$${\Gamma }_{+}=\{({\lambda }_1,...,{\lambda }_n):{\lambda }_i>0\text{mbox}for1\le i\le n\},$$

$${\sigma }_k(\lambda )=\sum _{1\le i_1<i_2<\cdots <i_k\le n}{\lambda }_{i_1}{\lambda }_{i_2}\cdots {\lambda }_{i_k}.$$

$${\Gamma }_k=\{\lambda \in {\mathbb{R}}^n:{\sigma }_i(\lambda )>0,\forall 1\le i\le k\}.$$

$${\lambda }_1\le \phi \text{mbox}everywhereand{\lambda }_1(x_0)=\phi (x_0)$$

$${\lambda }_n(x_0)\le ...\le {\lambda }_{m+1}(x_0)<{\lambda }_m(x_0)=...={\lambda }_1(x_0).$$

$$b_{ij;l}={\phi }_l{\delta }_{ij}\text{mbox}at{x}_0\text{mbox}for1\le i,j\le m.$$

$${\phi }_{ii}\ge b_{11;ii}+2\sum _{l>m}\frac{(b_{1l;i}{)}^2}{{\lambda }_1-{\lambda }_l},\text{mbox}at{x}_0.$$

$$\frac{{\partial }^{2}F}{\partial w_{ij}\partial w_{st}}{a}_{ij}{a}_{st}=\frac{{\partial }^{2}f}{\partial {\lambda }_p\partial {\lambda }_q}{a}_{pp}{a}_{qq}+2\sum _{p<q}\frac{\frac{\partial f}{\partial {\lambda }_p}-\frac{\partial f}{\partial {\lambda }_q}}{{\lambda }_p-{\lambda }_q}{a}_{pq}^2.$$

$${\sigma }_k^{ij,pq}{b}_{ij;1}{b}_{pq;1}-2{\sigma }_k^{ii}\sum _{l>m}\frac{(b_{1l;i}{)}^2}{{\lambda }_1-{\lambda }_l}$$

$$\displaystyle\sigma_{k}^{ij,pq}b_{ij;1}b_{pq;1}=\sum_{i\neq j}\sigma_{k-2}(\lambda|ij)b_{ii;1}b_{jj;1}+2\sum_{i>j}\bigg{(}\sigma_{k-1}(\lambda|i)-\sigma_{k-1}(\lambda|j)\bigg{)}\frac{(b_{ij;1})^{2}}{\lambda_{i}-\lambda_{j}}.$$

$$b_{ij;1}=b_{1i;j}=0$$

$${\sigma }_k^{ij,pq}{b}_{ij;1}{b}_{pq;1}$$

$$2{\sigma }_k^{ii}\sum _{l>m}\frac{(b_{1l;i}{)}^2}{{\lambda }_1-{\lambda }_l}$$

$$W(x)=u\cdot {\lambda }_1(x)-\beta (u^2+|Du{|}^2),$$

Full text

Keywords: $L_{p}$-Christoffel-Minkowski problem, uniqueness, convex solutions.**

*MSC: Primary 35J15, Secondary 35J60.

---

1. Introduction

Convex geometry plays an important role in the development of fully nonlinear partial differential equations. The classical Minkowski problem and the Christoffel-Minkowski problem in general, are beautiful examples of such interactions. The classical core of convex geometry is the Brunn-Minkowski theory, the Minkowski sum, the mixed volumes. The $L_{p}$-Brunn-Minkowski theory is an extension of the classical Brunn-Minkowski theory. The roots of the $L_{p}$-Brunn-Minkowski theory date back to the middle of the twentieth century, but its active development had to await the emergence of the concept of $L_{p}$-surface area measure in [22] in the early 1990’s.

Let $\mathcal{K}_{0}^{n+1}$ denote the class of convex bodies (compact convex subsets) in $n+1$-dimensional Euclidean space $\mathbb{R}^{n+1}$ that contain the origin in their interiors and $h_{K}:\mathbb{S}^{n}\rightarrow\mathbb{R}$ be the support function of a compact convex subset $K\in\mathbb{R}^{n+1}$, which determines $K$ uniquely and is defined by $h_{K}(x)=\max\{x\cdot X:X\in K\}$ for $x\in\mathbb{S}^{n}$, where $x\cdot X$ is the standard inner product of $x$ and $X$ in $\mathbb{R}^{n+1}$. For each real $p\geq 1$, Firey [9] defined what has become known as the Minkowski-Firey $L_{p}$-combination $K+_{p}t\cdot L$ for $K,L\in\mathcal{K}_{0}^{n+1}$ and $t\geq 0$ by letting

[TABLE]

This led to the notion of $k$-th $L_{p}$-surface area measure, $S_{p,k}(K,\cdot)$, for each body $K\in\mathcal{K}_{0}^{n+1}$, via the variational formula:

[TABLE]

which holds for each $L\in\mathcal{K}_{0}^{n+1}$, $p\geq 1$ and $1\leq k\leq n$. Here $W_{k}(K)$ is the usual quermassintegral for $K$. It was also shown in [22] that for each $K\in\mathcal{K}_{0}^{n+1}$,

[TABLE]

which shows that $k$-th $L_{p}$-surface area measure may be extended to all $p\in\mathbb{R}$ in a completely obvious manner. Here $dS_{k}(K,\cdot)$ is the $k$-th surface area measure of $K$.

The associated $L_{p}$-Christoffel-Minkowski problem in the $L_{p}$-Brunn-Minkowski theory (first studied in [22]) asks: For fixed $p\in\mathbb{R}$, given a Borel measure $\mu$ on $\mathbb{S}^{n}$ (the data) what are necessary and sufficient conditions on the measure $\mu$ to guarantee the existence of a body $K\in\mathcal{K}_{0}^{n+1}$ such that $\mu=S_{p,k}(K,\cdot)$, and if such a body $K$ exists to what extent is $K$ unique?

If the measure $\mu$ has a density function $\psi:\mathbb{S}^{n}\rightarrow[0,+\infty)$, then the partial differential equation that is associated with the $L_{p}$-Christoffel-Minkowski problem (with data $\psi$) is the $k$-Hessian type equation on $\mathbb{S}^{n}$

[TABLE]

where $u_{ij}$ are the second order covariant derivatives with respect to any orthonormal frame $\{e_{1},e_{2},...,e_{n}\}$ on $\mathbb{S}^{n}$, $\delta_{ij}$ is the standard Kronecker symbol and

[TABLE]

with $\lambda(x)=(\lambda_{1}(x),\lambda_{2}(x),\cdot\cdot\cdot,\lambda_{n}(x))$ being the eigenvalues of the matrix $u_{ij}+u\delta_{ij}$.

The $L_{p}$-Minkowski problem ($k=n$) has been extensively studied during the last twenty years, see [2, 7, 22, 24, 23] and see also [26] for the most comprehensive list of results. When $p\geq 1$, the existence and uniqueness of solutions are well understood. However, when $p<1$ the uniqueness of solutions to the $L_{p}$-Minkowski problem is very subtle, and indeed it was shown in [19] that the uniqueness fails when $p<0$ even restricted to smooth origin-symmetric convex bodies. Recently, Brendle-Choi-Daskaspoulos’ work [3] implies the uniqueness holds true for $1>p>-1-n$ and $\psi\equiv 1$, and Chen-Huang-Li-Liu [5] prove the uniqueness for $p$ close to $1$ and even positive function $\psi$.

For $k<n$, $L_{p}$-Christoffel-Minkowski problem is difficult to deal with, since the admissible solution to equation (1.1) is not necessary a geometric solution to $L_{p}$-Christoffel- Minkowski problem if $k<n$. So, one needs to deal with the convexity of the solutions of (1.1). Under a sufficient condition on the prescribed function, Guan-Ma [13] proved the existence of a unique convex solution. The key tool to handle the convexity is the constant rank theorem for fully nonlinear partial differential equations. Later, the equation (1.1) has been studied by Hu-Ma-Shen [20] for $p\geq k+1$ and Guan-Xia [14] for $1<p<k+1$ and even prescribed data, by using the constant rank theorem. See also [15] for the proof of uniqueness and [21] for a simple proof. But for $p<1$, since the lack of Firey’s extension of Brunn-Minkowski inequality (See Corollary 1.3 in [22]) and constant rank theorem, the existence and uniqueness are difficult and challenging problems. As far as I know, the existence and uniqueness for $p<1$ are unknown until now. In this paper, we make some progresses on the uniqueness for $p<1$ and $\psi\equiv 1$.

We consider the uniqueness of strictly convex solutions to the following $L_{p}$-Christoffel-Minkowski problem:

[TABLE]

here the strict convexity of a solution, $u$, means that the matrix

[TABLE]

is positive definite on $\mathbb{S}^{n}$. We mainly get the following result.

Theorem 1.1**.**

Assume $u\in C^{4}(\mathbb{S}^{n})$ is a strictly convex solution to (1.2), then $u\equiv constant$ for $1>p>1-k$.

Remark 1.1**.**

We know from McCoy’s work [25], Theorem 1.1 holds true for $p=1-k$. Thus, Theorem 1.1 in fact holds true for $1>p\geq 1-k$.

Our proof is motivated by the idea of Choi-Daskaspoulos [6] and Brendle-Choi-Daskaspoulos [3] in which they show the self-similar solution of $\alpha$-Gauss curvature flow, .i.e, an embedded, strictly convex hypersurface in $\mathbb{R}^{n+1}$ given by $X:\mathbb{S}^{n}\rightarrow\mathbb{R}^{n+1}$ satisfying the equation

[TABLE]

is a sphere when $\alpha>\frac{1}{n+2}$, where $K$ and $\nu$ are the Gauss curvature and out unit normal of $\Sigma$ respectively. Their result is also equivalent to say that the $L_{p}$-Minkowski problem (1.2)($k=n$) has the unique solution $u\equiv 1$ for $1>p>-n-1$. The idea of their proof is to apply Maximum Principles for the following two important auxiliary functions which are introduced in [6, 3]:

[TABLE]

and

[TABLE]

where $b_{ij}$ is the inverse matrix of the second fundamental form $h_{ij}$ of $\Sigma$, $\lambda_{1}(b_{ij})$ is the biggest eigenvalues of $b_{ij}$, $\lambda_{1}^{-1}(h_{ij})=\lambda_{1}(b_{ij})$, and $u:\mathbb{S}^{n}\rightarrow\mathbb{R}$ is the support function of $\Sigma$. Later, Gao-Li-Ma [11] and Gao-Ma[10] use this two functions above to study the uniqueness of closed self-similar solutions to $\sigma^{\alpha}_{k}$-curvature flow following the idea of [6, 3]. In details, in [11] the authors consider the following general equation

[TABLE]

where $S$ is a 1-homogeneous smooth symmetric function of the principal curvatures $\kappa_{i}$ of the hypersurface $\Sigma$ given by $X:\mathbb{S}^{n}\rightarrow\mathbb{R}^{n+1}$. Under some assumptions on $S$, they show $\Sigma=X(\mathbb{S}^{n})$ is a round sphere for $\alpha\geq 1$. Examples of $S$ include $S=\sigma_{k}^{\frac{1}{k}}(\kappa_{1},...,\kappa_{n})$, but not include $S=(\frac{\sigma_{n}}{\sigma_{n-k}})^{\frac{1}{k}}(\kappa_{1},...,\kappa_{n})$, for which the equation (1.4) is equivalent to $L_{p}$ -Christoffel-Minkowski problem (1.2) with $p=1-\frac{1}{\alpha}$. The main difficulty lies in the non-positivity of the term (2)

[TABLE]

if $f=\sigma_{1}$. (In this case, the revised function $Z$ (1.5) is just the original function $Z$ (1.3).) To overcome this difficulty, the easiest way is to choose $f$ such that

[TABLE]

So, we need to modify the function $Z$. We introduce the following two auxiliary functions: one is the original function $W$

[TABLE]

the other is a new function $Z$

[TABLE]

where $\lambda_{n}\leq...\leq\lambda_{2}\leq\lambda_{1}$ are the eigenvalues of the matrix $b_{ij}=u_{ij}+u\delta_{ij}$, $\beta=\frac{p-1+k}{2k}$ and

[TABLE]

here the third equality can be easily derived by Proposition 2.1(7) and we denote by $\sigma_{k-1}(\lambda|i)=\frac{\partial\sigma_{k}}{\partial\lambda_{i}}$. To our surprise, we find that $f(\lambda_{1},\lambda_{2},...,\lambda_{n})$ is a convex function in the positive cone

[TABLE]

since $\frac{\sigma_{k+1}}{\sigma_{k}}$ is concave in $\Gamma_{+}$ by (1) and (5) in Proposition 2.1. This fact is the key to our proof.

Remark 1.2**.**

We can propose the following questions:

(i) When $k=n$ our result does not cover the previous result in [6, 3], then it is natural to ask if one can improve it?

(ii) Can we construct some non-uniqueness examples of solutions to (1.2) for $p<1-k$?

2. The proof of Theorem 1.2

Let $\lambda=(\lambda_{1},\dots,\lambda_{n})\in\mathbb{R}^{n}$, we recall the definition of elementary symmetric function for $1\leq k\leq n$

[TABLE]

We also set $\sigma_{0}=1$ and $\sigma_{k}=0$ for $k>n$ or $k<0$. Recall that the Gårding’s cone is defined as

[TABLE]

We denote by $\sigma_{k-1}(\lambda|i)=\frac{\partial\sigma_{k}}{\partial\lambda_{i}}$ and $\sigma_{k-2}(\lambda|ij)=\frac{\partial^{2}\sigma_{k}}{\partial\lambda_{i}\partial\lambda_{j}}$. Then, we list some properties of $\sigma_{k}$ which will be used later.

Proposition 2.1**.**

Let $\lambda=(\lambda_{1},\dots,\lambda_{n})\in\mathbb{R}^{n}$ and $1\leq k\leq n$, then we have

(1) $\Gamma_{1}\supset\Gamma_{2}\supset\cdot\cdot\cdot\supset\Gamma_{n}=\Gamma_{+}$;

(2) $\sigma_{k-1}(\lambda|i)>0$ for $\lambda\in\Gamma_{k}$ and $1\leq i\leq n$;

(3) $\sigma_{k}(\lambda)=\sigma_{k}(\lambda|i)+\lambda_{i}\sigma_{k-1}(\lambda|i)$ for $1\leq i\leq n$;

(4) $\sum_{i=1}^{n}\frac{\partial(\frac{\sigma_{k+1}}{\sigma_{k}})}{\partial\lambda_{i}}\geq\frac{n-k}{k+1}$ for $\lambda\in\Gamma_{k+1}$;

(5) $\sigma^{\frac{1}{k}}_{k}$ and \Big{[}\frac{\sigma_{k}}{\sigma_{l}}\Big{]}^{\frac{1}{k-l}} are concave in $\Gamma_{k}$ for $0\leq l<k$;

(6) If $\lambda_{1}\geq\lambda_{2}\geq\cdot\cdot\cdot\geq\lambda_{n}$, then $\sigma_{k-1}(\lambda|1)\leq\sigma_{k-1}(\lambda|2)\leq\cdot\cdot\cdot\leq\sigma_{k-1}(\lambda|n)$ for $\lambda\in\Gamma_{k}$;

(7) $\sum_{i=1}^{n}\sigma_{k-1}(\lambda|i)\lambda^{2}_{i}=\sigma_{1}\sigma_{k}-(k+1)\sigma_{k+1}$.

Proof.

All the properties are well known. For example, see Chapter XV in [18] or [17] for proofs of (1), (2), (3), (6) and (7); see Lemma 2.2.19 in [12] for a proof of (4); see [4] and [18] for a proof of (5). $\hfill\Box$

We choose an orthonormal frame $\{e_{1},e_{2},...,e_{n}\}$ on $\mathbb{S}^{n}$. We use the notations $u_{i}=D_{e_{i}}u$, $u_{ij}=D_{e_{j}}D_{e_{i}}u$, $D_{e_{l}}b_{ij}=b_{ij;l}$ …, and so on, where $D$ is the standard Levi-Civita connection on $\mathbb{S}^{n}$. Set $b_{ij}=u_{ij}+u\delta_{ij}$, we denote by $\lambda_{n}(x)\leq...\leq\lambda_{2}(x)\leq\lambda_{1}(x)$ are the eigenvalues of $\{b_{ij}(x)\}$, arranged in decreasing order. Each eigenvalue defines a Lipschitz continuous function on $\mathbb{S}^{n}$.

We recall the following Lemma which is similar to Lemma 5 in [3] and their proofs are almost the same.

Lemma 2.2**.**

Suppose that $\varphi$ is a smooth function on $\mathbb{S}^{n}$ such that

[TABLE]

for some $x_{0}\in\mathbb{S}^{n}.$ Let $m$ denote the multiplicity of the biggest eigenvalue of $b_{ij}$ at $x_{0}$, so that

[TABLE]

Then, we have

[TABLE]

Moreover,

[TABLE]

Now we list the following well-known result (See Lemma 3.2 in [11] or [1]).

Lemma 2.3**.**

If $W=(w_{ij})$ is a symmetric real matrix, $\lambda_{i}=\lambda_{i}(W)$ is one of its eigenvalues ($i=1,...,n$) and $F=F(W)=f(\lambda(W))$ is a symmetric function of $\lambda_{1},...,\lambda_{n}$, then for any real symmetric matrix $A=(a_{ij})$, we have the following formulas:

[TABLE]

We also need Lemma 4.4 in [11] which statement as follows.

Lemma 2.4**.**

Under the assumptions of Lemma 2.2, we have at $x_{0}$

[TABLE]

Proof.

For convenience, we give a proof here. We know from Lemma 2.3

[TABLE]

Since we have by (2.1)

[TABLE]

for $2\leq i\leq m$, we can obtain

[TABLE]

and

[TABLE]

the lemma follows by adding the above two equations together. $\hfill\Box$

Now, we begin to prove Theorem 1.1. Set

[TABLE]

where $\beta=\frac{p-1+k}{2k}$ and $\lambda_{n}(x)\leq...\leq\lambda_{1}(x)$ are the eigenvalues of $\{b_{ij}(x)\}$. Since $1>p>1-k$, so $0<\beta<\frac{1}{2}$. Our proof is divided into two steps.

Step 1: we will prove

[TABLE]

for any $x_{0}\in\{x\in\mathbb{S}^{n}:W(x)=\max_{\mathbb{S}^{n}}W\}$.

Assume $W(x)$ attains its maximum at $x_{0}$. As above, we denote by $m$ the multiplicity of the biggest eigenvalue of $b_{ij}$ at $x_{0}$. Let us define a smooth function $\varphi$ such that

[TABLE]

Since $W$ attains its maximum at $x_{0}$, we have $\varphi(x)\geq\lambda_{1}(x)$ everywhere and $\lambda_{1}=\varphi$ at $x_{0}$. Choose an orthonormal frame at $x_{0}$ such that

[TABLE]

with

[TABLE]

Since $u\cdot\varphi-\beta(u^{2}+|Du|^{2})=constant$, then

[TABLE]

and

[TABLE]

Taking (2.2)’s value at $x_{0}$, we have by (2.1)

[TABLE]

Thus,

[TABLE]

which implies together with (2.1) and $0<2\beta<1$, $u_{i}(x_{0})=0$ for $2\leq i\leq m$. Taking (2.3)’s value at $x_{0}$ results in

[TABLE]

Thus, we obtain at $x_{0}$ using Lemma 2.2

[TABLE]

here we use the following Ricci identity (see [16] or (1.30) in [8])

[TABLE]

to get the last inequality. Differentiating the equation (1.2) shows

[TABLE]

Differentiating it again

[TABLE]

Due to the concavity of $\sigma^{\frac{1}{k}}_{k}(\lambda)$ (see (5) in Proposition 2.1),

[TABLE]

which results in together with Lemma 2.4 by noticing that the forth and fifth terms in the right hand of the equation in Lemma 2.4 are negative (this fact can be easily seen by Proposition 2.1 (6))

[TABLE]

Then, we arrive at $x_{0}$ by noticing that $u_{i}(x_{0})=0$ for $2\leq i\leq m$

[TABLE]

where we use the following inequality to get the last but one inequality

[TABLE]

which can be easily proved in view of the assumption on the positive definite of $u_{ij}+u\delta_{ij}$ and Proposition 2.1 (1)(2)(3). Thus, $\lambda_{1}(x_{0})=\lambda_{2}(x_{0})=...=\lambda_{n}(x_{0})$ and $|Du|(x_{0})=0$.

Step 2: we want to show that $\{x\in\mathbb{S}^{n}:W(x)=\max_{\mathbb{S}^{n}}W\}$ is an open set.

We define

[TABLE]

where

[TABLE]

here the third equality is derived by Proposition 2.1(7). Clearly, $f(\lambda_{1},\lambda_{2},...,\lambda_{n})$ is a 1-homogeneous convex function by Proposition 2.1 (1)(5) and satisfies $f(1,1,...,1)=n$ , since $u$ is strictly convex. We will prove for any $x_{0}\in\{x\in\mathbb{S}^{n}:W(x)=\max_{\mathbb{S}^{n}}W\}$, there exists a small neighborhood $U(x_{0})$ of $x_{0}$ such that

[TABLE]

and

[TABLE]

Denoting by $f_{i}=\frac{\partial f}{\partial\lambda_{i}}$ and $f_{ij}=\frac{\partial^{2}f}{\partial\lambda_{i}\partial\lambda_{j}}$. Since $f_{i}(\lambda(x_{0}))=1$, we can choose $U(x_{0})$ such that $f_{i}$ is positive in $U(x_{0})$. For any $x\in U(x_{0})$, we choose a coordinate at $x$ such that

[TABLE]

Then, we have at $x$

[TABLE]

and

[TABLE]

in view of the Ricci identity (see [16] or (1.30) in [8])

[TABLE]

and we use the convexity of $f$ to get the last inequality. Differentiating the equation (1.2) shows

[TABLE]

Differentiating it again

[TABLE]

Due to the concavity of $\sigma^{\frac{1}{k}}_{k}$ (see (5) in Proposition 2.1),

[TABLE]

and the positivity of $f_{i}$ in $U(x_{0})$, we arrive

[TABLE]

At $x_{0}$, we have $\lambda_{1}=\lambda_{2}=...=\lambda_{n}$. Thus, at $x_{0}$

[TABLE]

Thus,

[TABLE]

So,

[TABLE]

for $1>p>1-k$. Thus, there exists a small neighborhood $U(x_{0})$ such that

[TABLE]

Moreover, we obtain by (4) in Proposition 2.1

[TABLE]

which implies

[TABLE]

Thus, we have by $p>1-k$

[TABLE]

Thus, combining (2) and (2.6), we can find $U(x_{0})$ such that

[TABLE]

Since $f_{i}(\lambda_{0})=1$, we can choose $U(\lambda_{0})$ such that $f$ is increasing with each $\lambda_{i}$ in $U(\lambda_{0})$, where $\lambda_{0}=(\lambda_{1}(x_{0}),...,\lambda_{n}(x_{0}))$. Then, we can choose $U(x_{0})$ such that $\{\lambda(x):x\in U(x_{0})\}\subset U(\lambda_{0})$. So, we have

[TABLE]

which implies

[TABLE]

Thus, we have by the strong maximum principle

[TABLE]

for any $x$ in $U(x_{0})$, which implies

[TABLE]

for any $x$ in $U(x_{0})$. Thus, $W(x)\equiv\max_{\mathbb{S}^{n}}W$ for any $x\in\mathbb{S}^{n}$. So, $Du=0$ which implies $u=constant$. Thus, we complete our proof.

Acknowledgement: Parts of this work were done, while the author was visiting the mathematical institute of Albert-Ludwigs-Universität Freiburg in Germany. He would like to express his deep gratitude to Prof. Guofang Wang for invitation, continuous support, encouragement, and some important suggestions on this paper. He also thanks the mathematical institute of Albert-Ludwigs-Universität Freiburg for its hospitality. Moreover, he also thanks Prof. Chuanqiang Chen for some suggestions on this paper. Lastly, he is grateful to the reviewer’s valuable comments, which is helpful for improving the manuscript.