A Minkowski type inequality with free boundary in space forms
Jinyu Guo
TL;DR
This work addresses a Minkowski-type inequality for domains in space forms with a free boundary on an umbilical hypersurface, extending classical results to the free-boundary setting. The authors formulate and solve a mixed boundary value problem for a corrective function using a weight function $V$ that satisfies $\bar{\nabla}^2 V = -K V \bar{g}$ and implement a weighted Reilly formula to derive the inequality $$(\int_\Sigma V\, dA)^2 \geq \frac{n}{n-1} \int_\Omega V\, d\Omega \int_\Sigma H V\, dA.$$ Equality yields rigidity, forcing $\Sigma$ to be part of an umbilical hypersurface, and the approach yields further results including weighted Alexandrov-Fenchel-type inequalities and almost-Schur-type estimates in space forms. The paper thus provides a unified PDE-geometry method to extend Minkowski-type inequalities to free-boundary domains, with detailed case analyses depending on the ambient curvature and the chosen umbilical boundary. These results enhance understanding of free boundary geometry in constant curvature spaces and suggest stability-type consequences for associated curvature functionals.
Abstract
In this paper, we consider a Minkowski inequality for a domain supported on any umbilical hypersurface with free boundary in space forms. We generalize the main result in \cite{Xia} into free boundary case and obtain a free boundary version of optimal weighted Minkowski inequality in space forms.