Stable solutions of symmetric systems on Riemannian manifolds

TL;DR

This paper investigates stable solutions of symmetric elliptic systems on Riemannian manifolds, establishing inequalities and Liouville theorems that lead to flatness results for solutions' level sets.

Contribution

It introduces new stability and Poincaré inequalities for symmetric systems on manifolds and applies them to prove Liouville theorems and geometric flatness results.

Findings

Established stability inequality for solutions.

Proved Liouville theorems under certain conditions.

Showed flatness of level sets for stable solutions.

Abstract

We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold $\mathbb{M}$ without boundary, \begin{equation*} -\Delta_g u_i = H_i(u_1,\cdots,u_m) \ \ \text{on} \ \ \mathbb{M}, \end{equation*} when $\Delta_g$ stands for the Laplace-Beltrami operator, $u_i:\mathbb{M}\to \mathbb R$ and $H_i\in C^1(\mathbb R^m) $ for $1\le i\le m$. This system is called symmetric if the matrix of partial derivatives of all components of $H$, that is $\mathbb H(u)=(\partial_j H_i(u))_{i,j=1}^m$, is symmetric. We prove a stability inequality and a Poincar\'{e} type inequality for stable solutions using the Bochner-Weitzenb\"{o}ck formula. Then, we apply these inequalities to establish Liouville theorems and flatness of level sets for stable solutions of the above symmetric system, under certain assumptions on the manifold and on solutions.