# 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.

### About Pure mathematics seminars

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 3pm to 3.50pm.

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

#### Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

#### Contact us

To volunteer to talk or to suggest a speaker, email Ole Warnaar or Travis Scrimshaw.