Finite-time blowup of n-harmonic map flow

Initiated by Eells and Sampson in the 1960’s, our problem of interest is how to classify maps between manifolds by harmonic maps. Specifically, can a smooth harmonic map representative be found in a fixed homotopy class of maps between manifolds?

By introducing the heat flow method, Eells-Simpson gave an affirmative answer to this question when given the sectional curvature of the target manifold is non-positive. However, Chang, Ding and Ye constructed a counter-example which showed that the harmonic map flow could blowup in finite-time from a two-dimensional manifold (i.e. $n=2$). For $n>2$, there is no good answer to the Eells-Sampson question. Related to this, Hungerbühler studied the n-harmonic map flow and conjectured that this flow would blowup in finite-time.

This problem has remained open for nearly a quarter of a century. We studied this challenging problem and confirmed Hungerbühler’s conjecture by providing an example.