### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 29-62 |

Number of pages | 34 |

Journal | Acta Mathematica Sinica, English Series |

Volume | 16 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2000 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

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

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## Cite this

*Acta Mathematica Sinica, English Series*,

*16*(1), 29-62. https://doi.org/10.1007/s101149900024