Stability of the quermassintegral inequalities in hyperbolic space

TL;DR

This paper establishes a stability estimate for quermassintegral inequalities in hyperbolic space, quantifying how close a hypersurface is to a geodesic sphere based on the inequality deficit, with explicit exponents.

Contribution

It provides the first explicit stability estimate for these inequalities in hyperbolic space, independent of dimension, using new curvature estimates for inverse flows.

Findings

Stability estimate relating Hausdorff distance to the inequality deficit

Explicit exponent of the deficit independent of dimension

Applicable to domains with bounded inradius and curvature quotient

Abstract

For the quermassintegral inequalities of horospherically convex hypersurfaces in the $(n+1)$-dimensional hyperbolic space, where $n\geq 2$, we prove a stability estimate relating the Hausdorff distance to a geodesic sphere by the deficit in the quermassintegral inequality. The exponent of the deficit is explicitly given and does not depend on the dimension. The estimate is valid in the class of domains with upper and lower bound on the inradius and an upper bound on a curvature quotient. This is achieved by some new initial value independent curvature estimates for locally constrained flows of inverse type.