A stronger constant rank theorem

TL;DR

This paper establishes stronger constant rank theorems for convex solutions of a class of semilinear equations related to Liouville equations, showing conditions under which the Hessian's rank or certain symmetric polynomials are constant or zero.

Contribution

It extends constant rank theorems for convex solutions of semilinear equations, providing new conditions on the parameters for the Hessian's rank or symmetric polynomials to be constant or vanish.

Findings

If A ≤ 2 and σ₂(D²u) has a local minimum, then D²u has constant rank 1.

If A ≤ n/(n-1) and σₙ(D²u) has a local minimum, then σₙ(D²u) is identically zero.

The results generalize previous rigidity theorems for solutions to Liouville-type equations.

Abstract

Motivated from one-dimensional rigidity results of entire solutions to Liouville equation, we consider the semilinear equation \begin{align} \label{liouvilleequationab} \Delta u=G(u) \quad \mbox{in $\mathbb{R}^n$}, \end{align}where $G>0, G'<0$ and $GG^{''}\le A(G')^2$, with $A>0$. Let $u$ be a smooth convex solution and $\sigma_k(D^2 u)$ be the $k$-th elementary symmetric polynomial with respect to $D^2u$. We prove stronger constant rank theorems in the following sense. (1) When $A\le 2$, if $\sigma_2(D^2u)$ takes a local minimum, then $D^2 u$ has constant rank $1$. (2) When $A\le \frac{n}{n-1}$, if $\sigma_n(D^2 u)$ takes a local minimum, then $\sigma_n(D^2 u)$ is always zero in the domain.