Jiayu Li, Gang Tian
Let M and N be two compact Riemannian manifolds. Let uk (x, t) be a sequence of strong stationary weak heat flows from M×R+ to N with bounded energies. Assume that uk→u weakly in H1,2(M×R+, N) and that Σt is the blow-up set for a fixed t > 0. In this paper we first prove Σt is an Hm-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, Σt 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<0Σt× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.
