A stronger constant rank theorem

Abstract

Motivated from one-dimensional rigidity results of entire solutions to Liouville equation, we consider the semilinear equation \begin{align} \label{liouvilleequationab} Δu=G(u) \quad \mbox{in ${\mathbb{R}}^n$}, \end{align}where $G>0,G^{\prime }<0$ and $G{G}^{{}^{\prime \prime }}\le A(G^{\prime }{)}^2$, 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 $D^{2}u$. We prove stronger constant rank theorems in the following sense. (1) When $A\le 2$, if ${\sigma }_2(D^{2}u)$ 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.