Saturday 20 February 2021

Thom's Gradient Conjecture for Parabolic Systems and the Yang-Mills Flow

Thom's gradient conjecture, proved in this paper, asserts that convergent gradient flows of analytic functions on $\mathbb{R^n}$ cannot spiral forever. More precisely, the projection of the flow onto the unit sphere must converge.

In my paper linked below, I show that this result holds also for gradient flows of analytic functions on infinite dimensional Hilbert spaces, provided that the second derivative is a Fredholm operator. This is similar in spirit to the extension by L. Simon of the Lojasiewicz inequality to the same domain. I also show that the result holds for geometric flows with a Gauge symmetry, such as the Yang-Mills flow.

Thom's Gradient Conjecture for Parabolic Systems and the Yang-Mills Flow