The Blow-up Locus of Heat Flows for Harmonic Maps

Jiayu Li, Gang Tian

Research output: Contribution to journalArticle

20 Scopus citations

Abstract

Let M and N be two compact Riemannian manifolds. Let μk(x, t) be a sequence of strong stationary weak heat flows from M × R+ to N with bounded energies. Assume that μk → 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 Ut<oΣt x {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.

Original languageEnglish (US)
Pages (from-to)29-62
Number of pages34
JournalActa Mathematica Sinica, English Series
Volume16
Issue number1
DOIs
StatePublished - Jan 1 2000

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Blow-up locus
  • Brakke motion
  • Heat flow
  • Mean curvature flow

Fingerprint Dive into the research topics of 'The Blow-up Locus of Heat Flows for Harmonic Maps'. Together they form a unique fingerprint.

  • Cite this