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Stability of Minkowski-type inequalities in certain warped product spaces

Prachi Sahjwani

TL;DR

The paper develops stability results for Minkowski-type inequalities for hypersurfaces in warped product spaces and in RN-AdS/AdS-Schwarzschild manifolds. It combines a curvature-flow framework with a rigidity result for locally conformally flat ambient spaces to bound the traceless second fundamental form by the deficit in the inequality, and then deduces Hausdorff closeness to radial slices or coordinate spheres via a conformal-Euclidean rigidity argument. The approach yields quantitative estimates under varied curvature/convexity hypotheses and extends to physically meaningful spacetimes, highlighting the stability of geometric inequalities in curved ambient settings relevant to general relativity. Overall, the work provides a unified mechanism to pass from almost-equality in Minkowski-type inequalities to near-radial symmetry of the hypersurfaces, with explicit dependence on ambient geometry and flow-induced quantities.

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-Nordström 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.

Stability of Minkowski-type inequalities in certain warped product spaces

TL;DR

The paper develops stability results for Minkowski-type inequalities for hypersurfaces in warped product spaces and in RN-AdS/AdS-Schwarzschild manifolds. It combines a curvature-flow framework with a rigidity result for locally conformally flat ambient spaces to bound the traceless second fundamental form by the deficit in the inequality, and then deduces Hausdorff closeness to radial slices or coordinate spheres via a conformal-Euclidean rigidity argument. The approach yields quantitative estimates under varied curvature/convexity hypotheses and extends to physically meaningful spacetimes, highlighting the stability of geometric inequalities in curved ambient settings relevant to general relativity. Overall, the work provides a unified mechanism to pass from almost-equality in Minkowski-type inequalities to near-radial symmetry of the hypersurfaces, with explicit dependence on ambient geometry and flow-induced quantities.

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-Nordström 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.
Paper Structure (9 sections, 6 theorems, 150 equations)

This paper contains 9 sections, 6 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Rigidity Result
  3. Preliminaries for Minkowski Inequalities in Warped Product Spaces
  4. Stability of Minkowski's Inequalities in Warped Product Spaces
  5. Proof of \ref{['theorem 1']}
  6. Proof of \ref{['theorem 2']}
  7. Proof of \ref{['theorem 3']}
  8. Stability of Minkowski Inequalities in RN-AdS and AdS-Schwarzschild manifolds
  9. Proof of \ref{['theorem 4']} and \ref{['theorem 5']}

Key Result

Theorem 1.4

Let $n=2$, and in addition to assumption, suppose that $M$ satisfies Let $\Sigma \subset M$ be a strictly convex graph over $\mathbb{S}^{n}$. Then there exists a constant $C= C(n, \|\omega\|_{\infty}, \|\nabla\omega\|_{\infty}, \textrm{Vol}(\Sigma))$ such that where $S_{r}$ is the radial slice and $\omega$ is the conformal factor of the metric as in remark conformal .

Theorems & Definitions (21)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 11 more