Critical $O(d)$-equivariant biharmonic maps

TL;DR

This paper investigates $O(d)$-equivariant biharmonic maps in the critical dimension, demonstrating blowup in the heat flow and classifying equivariant biharmonic maps, revealing new phenomena absent in harmonic maps.

Contribution

It provides the first example of blowup in biharmonic map heat flow and classifies equivariant biharmonic maps, showing they can wind around the target sphere arbitrarily many times.

Findings

Blowup occurs in the biharmonic map heat flow from $B^4(0,1)$ to $S^4$.

Classified all $O(4)$-equivariant biharmonic maps from $R^4$ to $S^4$.

Existence of equivariant biharmonic maps winding arbitrarily many times around $S^4$.

Abstract

We study $O(d)$-equivariant biharmonic maps in the critical dimension. A major consequence of our study concerns the corresponding heat flow. More precisely, we prove that blowup occurs in the biharmonic map heat flow from $B^4(0, 1)$ into $S^4$. To our knowledge, this was the first example of blowup for the biharmonic map heat flow. Such results have been hard to prove, due to the inapplicability of the maximum principle in the biharmonic case. Furthermore, we classify the possible $O(4)$-equivariant biharmonic maps from $\mathbf{R}^4$ into $S^4$, and we show that there exists, in contrast to the harmonic map analogue, equivariant biharmonic maps from $B^4(0,1)$ into $S^4$ that wind around $S^4$ as many times as we wish. We believe that the ideas developed herein could be useful in the study of other higher-order parabolic equations.