Stability of quermassintegral and Minkowski-type inequalities in warped product spaces

Sahjwani, Prachi 2025. Stability of quermassintegral and Minkowski-type inequalities in warped product spaces. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

In this thesis, we establish new stability estimates for geometric inequalities in warped product spaces. First, we focus on the quermassintegral inequalities for horospherically convex hypersurfaces in (n+1)-dimensional hyperbolic space (n ≥ 2). We show that the Hausdorff distance to a geodesic sphere can be controlled by the deficit in the quermassintegral inequality, with an exponent independent of the dimension. This result relies on new curvature estimates for locally constrained flows of inverse type, under some bounds on inradius and curvature quotients. We also study the stability of Minkowski-type inequalities in warped product spaces, which are locally conformally flat with a bounded conformal factor. In these spaces, we prove a stability estimate that bounds the traceless second fundamental form of a hypersurface in terms of the deficit in the Minkowski-type inequalities. We also establish similar stability results in RN-AdS Schwarzschild and AdS-Schwarzschild spacetimes. Alongside these results, we obtain a new rigidity theorem for conformally flat spaces, which is crucial for understanding how near-equality in these inequalities forces hypersurfaces to remain close to standard “model” shapes.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Schools > Mathematics
Uncontrolled Keywords:	1. Quermassintegral inequalities 2. Minkowski-type inequalities 3. Hyperbolic space 4. Warped-product space 5. Inverse curvature flows 6. Curvature flow
Date of First Compliant Deposit:	8 December 2025
Last Modified:	15 Dec 2025 10:33
URI:	https://orca.cardiff.ac.uk/id/eprint/182981

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