Semilinear overdetermined problems, a divergence formula, and geometric inequalities

@misc{leandro2026semilinear,
  title = {{Semilinear overdetermined problems, a divergence formula, and geometric inequalities}},
  author = {Benedito Leandro and Ilton Menezes and Rafael Novais},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.23184},
  eprint = {2608.23184},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.23184},
}

TY  - UNPB
TI  - Semilinear overdetermined problems, a divergence formula, and geometric inequalities
AU  - Benedito Leandro
AU  - Ilton Menezes
AU  - Rafael Novais
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2608.23184
UR  - https://arxiv.org/abs/2608.23184
AB  - 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.
ER  -