Paul Allen, Lewis & Clark College

Title: Boundary Regularity for the Singular Yamabe Problem

Abstract: Given a conformally compact Riemannian manifold,
the singular Yamabe problem seeks to find a conformally related
geometry with constant scalar curvature. Results of Andersson,
Chrusciel, and Friedrich show that even if the original geometry has a
smooth conformal compactification, the conformally related metric
might exhibit singular behavior at conformal infinity. In this talk I
present a framework, developed jointly with Isenberg, Lee, and
Stavrov, in which solving the Yamabe problem is "closed' in the sense
that solutions lie in the same regularity class as the original
metric. This framework has further applications to geometric analysis
problems in general relativity, which I discuss as time permits.