Member Seminar

Speaker: Puskar Mondal

Title: Positive mass and rigidity theorems in Riemannian geometry

Abstract: Positive mass theorem proved by Schoen-Yau, Witten, Taubes-Parker is one of the most important results in scalar curvature geometry in asymptotically flat settings. Since then several versions have been proven and generalized to other geometries such as asymptotically hyperbolic manifolds. The analogous theorem for strictly positive curvature geometries is absent. There have been counterexamples but a precise quantification does not exist.I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that $n+1~(n\geq 2)$ dimensional spherical caps with constant positive mean curvature totally umbilic boundaries are rigid under smooth perturbations, and such rigidity results fail for the hemisphere. The assertion of this result is based on the notion of a real Killing connection and solution of the boundary value problem associated with its Dirac operator. Additionally, an improved eigenvalue estimate for the Dirac operator on hypersurfaces in positively curved manifolds is obtained.