Research Interests

• Geometric Analysis

• Differential Geometry

• Partial Differential Equations

Thesis

Stability In Physical Systems Governed By Curvature Quantities

My research lies at the intersection of geometric analysis, partial differential equations (PDEs), and differential geometry, focusing on the stability of geometric inequalities in non-Euclidean spaces such as hyperbolic and warped product spaces. These geometric inequalities, including quermassintegral and Minkowski inequalities, are vital in understanding relationships between curvature, volume, and surface area in curved geometries. My work explores how small deviations from these inequalities affect the shape and stability of hypersurfaces, with applications in theoretical mathematics and the physics of general relativity.

In particular, I study the stability of quermassintegral inequalities for horospherically convex hypersurfaces in hyperbolic space, using curvature flows to estimate how close my hypersurface is to a sphere when the inequality is almost an equality. Additionally, I am investigating Minkowski inequalities in RN-AdS-Schwarzschild space, which is relevant to the structure of black holes. This research contributes to understanding how curvature and geometric properties influence stability in physical systems, providing new insights in both mathematical theory and general relativity.

Publications and Preprints

• With J. Scheuer, Stability of quermassintegral inequalities in hyperbolic space, (2023), Journal of Geometric Analysis 34, article number: 13. Link 

funding

Engineering and Physical Sciences Research Council (EPSRC) Grant details