Homotopy classes of harmonic maps of the stratified 2-spheres and applications to geometric flows

Abstract

We show that the set of harmonic maps from the 2-dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes. We apply this result to construct infinitely many harmonic map flows and mean curvature flows of 2-sphere in the connected sum of two closed 3-dimensional manifolds M1≠S3 and M2≠S3,RP3, which must develop finite time singularity.