The Radius of Vanishing Bubbles in Equivariant Harmonic Map Flow from $D^2$ to $S^2$

Abstract

We derive an upper bound for the radius $R(t)$ of a vanishing bubble in a family of equivariant maps $F_t:D^2\to S^2$ which evolve by the harmonic map flow. The self-similar “type 1” radius would be $R(t)=C\sqrt{T-t}$. We prove that $R(t)=o(T-t)$.