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

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$M_1\ne S^3$ and _M2≠S³,RP³$M_2\ne S^3,\mathbb{R}{\mathbb{P}}^3$, which must develop finite time singularity.