Analysis and Geometric Analysis Seminar
In Riemannian manifold (M^n, g), it is well-known that its minimizing hypersurface is smooth when n<=7, and singular when n>=8. This is one of the major difficulties in generalizing many interesting results to higher dimensions, including the Riemannian Penrose inequality. In particular, in dimension 8, the minimizing hypersurface has isolated singularities, and Nathan Smale constructed a local perturbation process to smooth out the singularities. However, Smale’s perturbation will also produce a small region with possibly negative scalar curvature. In order to apply this perturbation in general relativity, we constructed a local deformation prescribing the scalar curvature and the mean curvature simultaneously. In this talk, we will discuss how the weighted function spaces help us localize the deformation in complete manifolds with boundary, assuming certain generic conditions. We will also discuss some applications of this result in general relativity.