Stability of Minkowski-type inequalities in certain warped product spaces.

Abstract

This paper proves quantitative stability estimates for five Minkowski-type inequalities for hypersurfaces in warped product spaces. In each case we show that if a hypersurface nearly achieves equality, it must be geometrically close, in the Hausdorff sense, to a radial slice. The ambient spaces are warped products $(a,b)\times S^n$ with metric $\text{d}{r}^2+{\lambda }^2\left(r\right)g_{S^n}$ , as well as the Reissner-Nordström Anti-de Sitter (RN-AdS) and Anti-de Sitter Schwarzschild (AdS-Schwarzschild) manifolds. The proofs combine two ingredients: a quantitative analysis of locally constrained inverse curvature flows, which yields bounds on the traceless second fundamental form in terms of the deficit in the inequality, and a new rigidity theorem for hypersurfaces in locally conformally flat spaces, which converts such bounds into Hausdorff closeness to a radial slice.