Rigidity of Serrin-type problems via integral identities

TL;DR

This work extends the classical Serrin overdetermined problem to Riemannian manifolds, deriving rigidity results through integral identities and a P-function framework. It generalizes Euclidean results to settings with $\mathrm{Ric} \ge (n-1)kg$ and addresses annular domains in Einstein manifolds with a closed conformal vector field, producing a Heintze–Karcher inequality and a Soap Bubble-type rigidity. A key contribution is a P-function-based rigidity mechanism: under $f'\le nk$ and suitable curvature bounds, subharmonicity of $P$ yields that equality cases force radial symmetry and that $\Omega$ must be a metric ball; a sharp gradient bound provides a practical rigidity criterion. The annular analysis delivers Minkowski-type identities and umbilicity results, showing that, in space forms, the domain must be a standard annulus with explicit radial solutions, thereby linking geometric shape to overdetermined boundary data in curved spaces.

Abstract

In this short note, we deal with Serrin-type problems in Riemannian manifolds. Firstly, we provide a Soap Bubble type theorem and rigidity results. In another direction, we obtain a rigidity result addressed to annular regions in Einstein manifolds endowed with a conformal vector field.

Theorems & Definitions (12)

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• proof : Proof of Theorem \ref{['1']}
• proof : Proof of Theorem \ref{['2']}
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• proof : Proof of Theorem \ref{['3']}
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• proof : Proof of Theorem \ref{['4']}