Serrin's over-determined Problem on Riemannian Manifolds

TL;DR

This paper extends Serrin's over-determined boundary value problem to Riemannian manifolds, constructing solutions on perturbed geodesic balls and linking critical points of scalar curvature to solution existence.

Contribution

It proves existence of solutions on perturbed geodesic balls in Riemannian manifolds and relates solution concentration points to scalar curvature critical points.

Findings

Existence of solutions on perturbed geodesic balls.

Family of solutions forms a foliation near non-degenerate scalar curvature critical points.

Convergence of solution domains implies the point is a scalar curvature critical point.

Abstract

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.