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} \Delta u=G(u) \quad \mbox{in $\mathbb{R}<sup>n$}, \end{align}where $G&gt;0, G&#39;&lt;0$ and $GG^{{&#39;&#39;}\le} A(G&#39;)<sup>2$, with $A&gt;0$. Let $u$ be a smooth convex solution and $\sigma_k(D² u)$ be the $k$-th elementary symmetric polynomial with respect to $D<sup>2u$. We prove stronger constant rank theorems in the following sense. (1) When $A\le 2$, if $\sigma_2(D<sup>2u)$ takes a local minimum, then $D² u$ has constant rank $1$. (2) When $A\le \frac{n}{n-1}$, if $\sigma_n(D² u)$ takes a local minimum, then $\sigma_n(D² u)$ is always zero in the domain.