The Blow-up Locus of Heat Flows for Harmonic Maps

Let M and N be two compact Riemannian manifolds. Let u k (x, t) be a sequence of strong stationary weak heat flows from M.R + to N with bounded energies. Assume that u k.u weakly in H 1, 2(M.R +, N) and that .t is the blow-up set for a fixed t > 0. In this paper we first prove .t is an H m.2-rectifiable set for almost all t.R +. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, .t is a distance solution of the (m. 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set .t<0.t. {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.