2026 Theses Doctoral

Rigidity and Comparison Problems in Scalar Curvature Geometry

Wang, Yipeng

Lower bounds on scalar curvature exhibit a striking interplay between flexibility and rigidity. This thesis studies several rigidity and comparison problems in scalar curvature geometry, with connections to Dirac boundary value problems, geometric inequalities, and macroscopic geometry.

In Chapter 2, we prove a Llarull-type rigidity theorem for log-concave spherical bands. As a consequence, we establish a special case of a conjecture of Gromov concerning Llarull's theorem on a punctured sphere. The proof uses a new holographic index theorem. In Chapter 3, we study Gromov's dihedral rigidity problem for acute polytopes, combining a delicate smoothing argument with an estimate of Fefferman and Phong. In Chapter 4, we study fill-ins with scalar curvature bounded from below. We give a new proof of an estimate involving the first eigenvalue of the boundary Dirac operator, and prove an asymptotically sharp estimate for the total mean curvature of toroidal fill-ins with scalar curvature bounded below by −𝑛(𝑛 − 1). In Chapter 5, we extend several classical rigidity theorems to stabilized scalar curvature. One ingredient in the argument is a new quantity that is monotone along the Ricci flow coupled to the heat equation. In Chapter 6, we prove an effective volume growth estimate for three-manifolds with nonnegative Ricci curvature and strictly positive scalar curvature.