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Well-posedness of mean curvature flow

Yongheng Han

Abstract

In this paper, using heat kernel estimates and contraction mapping principle, we give a new proof of the existence and uniqueness of mean curvature flow starting from hypersurface with bounded second fundamental form. Moreover, we show the continuous dependence of mean curvature flow on initial data.

Well-posedness of mean curvature flow

Abstract

In this paper, using heat kernel estimates and contraction mapping principle, we give a new proof of the existence and uniqueness of mean curvature flow starting from hypersurface with bounded second fundamental form. Moreover, we show the continuous dependence of mean curvature flow on initial data.
Paper Structure (10 sections, 31 theorems, 144 equations)

This paper contains 10 sections, 31 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Graphic mean curvature flow equations
  3. MCF equation over a manifold
  4. MCF equation over evolving metric
  5. Heat kernel estimates
  6. Harmonic radius
  7. Heat kernel over a manifold
  8. Heat kernel over evolving metric
  9. Existence and uniqueness
  10. Continuous dependence

Key Result

Theorem 1.1

Let $\mathbf{x}_0:M^n\to \mathbb{R}^{n+1}$ be an isometrically immersed Riemannian manifold with bounded second fundamental form $|A|\leq \kappa$. Then there exists a sufficient small constant $T=T(\kappa,n)>0$ and a family of immersions $\{\mathbf{x}(\cdot,t)\}_{t\in [0,T]}$ that solves the mean cu

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 3.1
  • Lemma 3.2
  • ...and 44 more