Extrinsic bi-Conformal Heat Flow and its smoothness

TL;DR

This paper introduces the bi-conformal heat flow for extrinsic biharmonic maps on 4-manifolds, demonstrating its global smoothness and absence of finite-time singularities, extending the understanding of conformal heat flows.

Contribution

It defines the bi-conformal heat flow for extrinsic biharmonic maps and proves its global smoothness and singularity-free behavior.

Findings

Global smoothness of bi-conformal heat flow

No finite time singularity occurs

Extension of heat flow theory to biharmonic maps

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.