Liouville theorems for 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) \end{equation*} where \(\Delta_g\) stands for the Laplace-Beltrami operator and \(H_i\in C^1(\mathbb R^m) \to \mathbb R\). This system is called symmetric if the matrix of partial derivatives of all components of \(H\), that is \(\mathcal H(u)=(\partial_j H_i(u))_{i,j=1}^m\), is symmetric. We prove a stability inequality and a Poincar\'{e} type inequality for stable solutions. 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.