THE FORMATION OF SINGULARITIES IN THE HARMONIC MAP HEAT FLOW WITH BOUNDARY CONDITIONS

Abstract

Let $M$ be a compact manifold with boundary and $N$ be compact manifold without boundary. Let $u(x,t)$ be a smooth solution of the harmonic heat equation from $M$ to $N$ with Dirichlet or Neumann condition. Suppose that $M$ is strictly convex, we will prove a monotonicity formula for $u$. Moreover, if $T$ is the blow-up time for $u$, and $\sup_M |Du|^2(x,t) \leq C/(T-t)$, we prove that a subsequence of the rescaled solutions converges to a homothetically shrinking soliton.