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Regularized $n$-Conformal heat flow and global smoothness

Woongbae Park

TL;DR

The work presents a regularized higher-order flow, the $n$-conformal heat flow, for maps between Riemannian manifolds by coupling a regularized $n$-energy with a conformal factor $u$. For $n\le 4$ and $\varepsilon\in(0,1]$, the authors prove global existence of a smooth solution to the regularized system with no finite-time blow-up, and they show that the regularized model inherits energy monotonicity and enjoys a systematic hierarchy of a priori estimates. The analysis develops local energy estimates, higher-order control, $L^q$ and $L^{\infty}$ bounds (via Moser iteration), and a fixed-point argument for short-time existence, culminating in global regularity by excluding energy concentration at finite times. The results generalize the CHF paradigm to the $n$-harmonic setting and indicate that regularization prevents bubbling, suggesting the limit $\varepsilon\to0$ preserves smoothness in the conformal formulation.

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.

Regularized $n$-Conformal heat flow and global smoothness

TL;DR

The work presents a regularized higher-order flow, the -conformal heat flow, for maps between Riemannian manifolds by coupling a regularized -energy with a conformal factor . For and , the authors prove global existence of a smooth solution to the regularized system with no finite-time blow-up, and they show that the regularized model inherits energy monotonicity and enjoys a systematic hierarchy of a priori estimates. The analysis develops local energy estimates, higher-order control, and bounds (via Moser iteration), and a fixed-point argument for short-time existence, culminating in global regularity by excluding energy concentration at finite times. The results generalize the CHF paradigm to the -harmonic setting and indicate that regularization prevents bubbling, suggesting the limit preserves smoothness in the conformal formulation.

Abstract

In this paper, we introduce the regularized conformal heat flow of -harmonic maps, or simply regularized -conformal heat flow from -dimensional Riemannian manifold. This is a system of evolution equations combined with regularized -harmonic map flow and a metric evolution equation in conformal direction. For , 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 -conformal heat flow does not develop finite time singularity unlike the (regularized) -harmonic map flow.

Paper Structure

This paper contains 8 sections, 33 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Local energy estimate
  4. Higher order estimate
  5. $L^q$ estimate
  6. $L^\infty_{loc}$ estimate
  7. Short time existence
  8. Global regularity

Key Result

Theorem 1.1

H97 For any $f_0 \in W^{1,n}(M,N)$, there exists a weak solution $f : M \times [0,\infty) \to N$ of n harmonic flow that satisfies that $\nabla f \in C^{0,\alpha}$ away from at most finitely many singular points $(x_k,T_k) \in M \times [0,\infty)$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 52 more