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Extrinsic bi-Conformal Heat Flow and its smoothness

Woongbae Park

Abstract

In this paper we introduce conformal heat flow of (extrinsic) biharmonic maps on $4$-manifold, simply called bi-conformal heat flow (bi-CHF), and study its properties. Similar to other CHF of harmonic maps and regularized $n$-harmonic maps, (CHF and regularized $n$-CHF respectively), we obtain global smoothness and no finite time singularity.

Extrinsic bi-Conformal Heat Flow and its smoothness

Abstract

In this paper we introduce conformal heat flow of (extrinsic) biharmonic maps on -manifold, simply called bi-conformal heat flow (bi-CHF), and study its properties. Similar to other CHF of harmonic maps and regularized -harmonic maps, (CHF and regularized -CHF respectively), we obtain global smoothness and no finite time singularity.
Paper Structure (7 sections, 35 theorems, 267 equations)

This paper contains 7 sections, 35 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Local energy estimate
  4. $C^0$ estimate for $df$
  5. Higher order estimate
  6. Short time existence
  7. Global regularity

Key Result

Theorem 1.3

Let $f_0 \in W^{6,2}(M,N)$. Then there exists a smooth solution $(f,u)$ of eq1 on $M \times [0,\infty)$ with initial condition $f(0)=f_0, u(0)=0$.

Theorems & Definitions (74)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 64 more