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<sup>+</sup> to N with bounded energies.Assume that u<sub>k</sub>→u...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<sup>+</sup> to N with bounded energies.Assume that u<sub>k</sub>→u weakly in H<sup>1.2</sup>（M×R<sup>+</sup>,N）and that ∑<sup>t</sup> is the blow-up set for a fixed t】0.In this paper we first prove ∑<sup>t</sup> is an H<sup>m-2</sup>-rectifiable set for almost all t ∈ R<sup>+</sup>.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,∑<sup>t</sup> 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 ∪<sub>t</sub>【0<sub>∑</sub><sup>t</sup>×{t}.We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.展开更多
文摘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<sup>+</sup> to N with bounded energies.Assume that u<sub>k</sub>→u weakly in H<sup>1.2</sup>（M×R<sup>+</sup>,N）and that ∑<sup>t</sup> is the blow-up set for a fixed t】0.In this paper we first prove ∑<sup>t</sup> is an H<sup>m-2</sup>-rectifiable set for almost all t ∈ R<sup>+</sup>.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,∑<sup>t</sup> 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 ∪<sub>t</sub>【0<sub>∑</sub><sup>t</sup>×{t}.We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.
