The Blow-up Locus of Heat Flows for Harmonic Maps

Let M and N be two compact Riemannian manifolds. Let μ_k(x, t) be a sequence of strong stationary weak heat flows from M × R⁺ to N with bounded energies. Assume that μ_k → u weakly in H^1,2(M × R⁺, N) and that Σ^t is the blow-up set for a fixed t > 0. In this paper we first prove Σ^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, Σ^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 U_t<oΣ^t x {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.