Regularized $n$-Conformal heat flow and global smoothness

TL;DR

This paper introduces a regularized conformal heat flow for n-harmonic maps on n-dimensional manifolds, demonstrating it avoids finite-time singularities unlike traditional flows, thereby ensuring global smoothness.

Contribution

It extends the understanding of conformal heat flows by proving the absence of finite-time singularities in the regularized n-conformal case, unlike previous harmonic map flows.

Findings

Regularized n-conformal heat flow avoids finite-time singularities.

The flow maintains smoothness globally in time.

Comparison with traditional harmonic map flow highlights improved behavior.

Abstract

In this paper, we introduce the regularized conformal heat flow of $n$-harmonic maps, or simply regularized $n$-conformal heat flow from $n$-dimensional Riemannian manifold. This is a system of evolution equations combined with regularized $n$-harmonic map flow and a metric evolution equation in conformal direction. For $n=2$, the conformal heat flow does not develop finite time singularity unlike usual harmonic map flow \cite{P23} (Park, 2024). In this paper, we show the analogous result, that regularized $n$-conformal heat flow does not develop finite time singularity unlike the (regularized) $n$-harmonic map flow.