Finite time blowup of the $n$-harmonic flow on $n$-manifolds

TL;DR

This paper proves that the $n$-harmonic flow on $n$-manifolds develops finite time singularities, extending no-neck results and confirming a conjecture about finite-time blowup in higher dimensions.

Contribution

It generalizes the no-neck result to $n$-harmonic flows and constructs an example demonstrating finite-time blowup on $n$-dimensional manifolds.

Findings

No-neck formation during blowup for $n$-harmonic flow.

Finite-time blowup occurs for $n$-harmonic map flow when $n \\geq 3$.

Confirmed a conjecture by Hungerbühler about blowup behavior.

Abstract

We generalize the no-neck result of Qing-Tian \cite{QT} to show that there is no neck during blowing up for the $n$-harmonic flow as $t\to\infty$. As an application of the no-neck result, we settle a conjecture of Hungerb\"uhler \cite {Hung} by constructing an example to show that the $n$-harmonic map flow on an $n$-dimensional Riemannian manifold blows up in finite time for $n\geq 3$.