Serrin's over-determined Problem on Riemannian Manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold of dimension $N$, $N\geq 2$. In this paper, we prove that there exists a family of domains $(\Omega_\varepsilon)_{\varepsilon\in(0,\varepsilon_0)}$ and functions $u_\varepsilon$ such that $ -\Delta_{g} u_\varepsilon=1 \quad \textrm{ in } \Omega_\varepsilon, \quad u_\varepsilon=0 \quad\textrm{ on }\partial\Omega_\varepsilon, \quad {g}(\nabla_{ {g}} {u_\varepsilon}, {\nu}_\varepsilon)=-\frac{\varepsilon}{N} \quad \textrm{ on }\partial\Omega_\varepsilon, $ where $\nu_\varepsilon$ is the unit outer normal of $\partial\Omega_\varepsilon$. The domains $\Omega_\varepsilon$ are smooth perturbations of geodesic balls of radius $\varepsilon$ centered at some point $p_0$. If, in addition, $p_0$ is a non-degenerate critical point of the scalar curvature of $g$ then, the family $(\partial\Omega_\varepsilon)_{\varepsilon\in(0,\varepsilon_0)}$ constitutes a smooth foliation of a neighborhood of $p_0$. By considering a family of domains $\Omega_\varepsilon$ in which the above overdetermined system is satisfied, we also prove that if this family converges to some point $p_0$ in a suitable sense as $\varepsilon\to 0$, then $p_0$ is a critical point of the scalar curvature. A Taylor expansion of he energy rigidity for the torsion problem is also given.