Daren Cheng, Department of Pure Mathematics, University of Waterloo
"Incompressible minimal surfaces and topological consequences of positive scalar curvature (Part 4)"
Suppose M is a closed 3-manifold whose fundamental group contains a copy of the fundamental group of T^2 and fix a metric h on M. Given a metric g on T^2, in the previous talk I described how Schoen and Yau (Annals, 1979) produced an energy minimizer among maps from (T^2, g) into (M, h) that induce the same \pi_1-action as a given smooth map from T^2 to M, which we can choose to be \pi_1-injective thanks to the assumption on \pi_1(M).
In this talk I will explain the second stage of the minimization process, in which one varies the domain metric g. It is to prove convergence in this step that the \pi_1-injectivity of the maps found above is needed. The result is that (M, h) admits a branched immersed stable minimal torus. As mentioned last time, this prevents the ambient metric h from having positive scalar curvature, modulo ruling out the branch points, which we will not address in the talk. Instead, time permitting, I will elaborate on some of the regularity results used in the process for energy-minimizing maps from two-dimensional domains.