Critical O(d)-equivariant biharmonic maps

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 flowfrom B⁴(0, 1) into S⁴. 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 R⁴ into S⁴, and we show that there exists, in contrast to the harmonic map analogue, equivariant biharmonic maps from B⁴(0, 1) into S⁴ that wind around S⁴ 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.