Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stable solutions of symmetric systems on Riemannian manifolds   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.

7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
16
The categories of this paper are: stat
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol