Stability of Minkowski-type inequalities in certain warped product spaces

TL;DR

This paper investigates the stability of Minkowski-type inequalities in warped product spaces, providing bounds on geometric quantities and demonstrating stability in specific spacetime models, with implications for geometric analysis.

Contribution

It introduces new stability estimates for Minkowski inequalities in warped product spaces and proves their stability in RN-AdS and AdS-Schwarzschild manifolds.

Findings

Established a stability estimate relating the traceless second fundamental form to Minkowski inequality deficits.

Proved the stability of Minkowski inequalities in RN-AdS and AdS-Schwarzschild manifolds.

Derived a new rigidity result for locally conformally flat manifolds.

Abstract

This paper explores the stability of Minkowski-type inequalities for hypersurfaces in warped product spaces. We establish a stability estimate that bounds the norm of the traceless second fundamental form of the hypersurface in terms of the deficit in the Minkowski inequalities satisfied by the hypersurface. Additionally, we prove the stability of Minkowski inequalities in specific cases of the Reissner-Nordstr\"om Anti-de Sitter (RN-AdS) and Anti-de Sitter Schwarzschild (AdS-Schwarzschild) manifolds, which serve as examples of warped products. We also establish a new rigidity result for locally conformally flat manifolds to understand the stability of these inequalities.