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arXiv · 2608.23184

Semilinear overdetermined problems, a divergence formula, and geometric inequalities

Benedito Leandro·Ilton Menezes·Rafael Novais
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Abstract

Considering an $n$-dimensional compact Riemannian manifold with a boundary that satisfies a Serrin-type problem, we prove sharp upper and lower bounds for the area of such a boundary. Then, we present the main result of this paper: a divergence formula for a special vector field on a given Riemannian manifold. This divergence formula is closely related to sub-static manifolds and related metrics, e.g., $V$-static, static, and electrostatic manifolds. We show Minkowski-type inequalities for $V$-sub-static manifolds under different Neumann boundary conditions. Moreover, we prove an area-charge inequality for compact (and noncompact) electrostatic manifolds. The main application of the divergence formula presented in this work generalizes the classical result of Boucher--Gibbons--Horowitz: we prove that an asymptotically hyperbolic space of a static spacetime satisfying the null convergence condition must be the hyperbolic space.

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Benedito Leandro, Ilton Menezes, Rafael Novais. 2026-09-02. Semilinear overdetermined problems, a divergence formula, and geometric inequalities. https://arxiv.org/abs/2608.23184

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