Geometric analysis of scalar curvature rigidity

초록

In this talk, I will talk about the following scalar curvature rigidity problem: Let (M,g1) be an n-dimensional complete Riemannian manifold. Given a compact smooth domain Ω with ∂Ω, can one find a conformal metric g whose scalar curvature R[g]≥R[g1] on Ω and the mean curvature H[g]≥H[g1] on ∂Ω with g=g1 on ∂Ω? We review the developments of this problem and present a rigidity result showing that g=g1 on some smooth domains in a general Riemannian manifold using a conformal invariant.