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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 $(Ω_\varepsilon)_{\varepsilon\in(0,\varepsilon_0)}$ and functions $u_\varepsilon$ such that $ -Δ_{g} u_\varepsilon=1 \quad \textrm{ in } Ω_\varepsilon, \quad u_\varepsilon=0 \quad\textrm{ on }\partialΩ_\varepsilon, \quad {g}(\nabla_{ {g}} {u_\varepsilon}, ν_\varepsilon)=-\frac{\varepsilon}{N} \quad \textrm{ on }\partialΩ_\varepsilon, $ where $ν_\varepsilon$ is the unit outer normal of $\partialΩ_\varepsilon$. The domains $Ω_\varepsilon$ 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 $(\partialΩ_\varepsilon)_{\varepsilon\in(0,\varepsilon_0)}$ constitutes a smooth foliation of a neighborhood of $p_0$. By considering a family of domains $Ω_\varepsilon$ 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.