Iterated logarithms and gradient flows

TL;DR

This paper applies advanced mathematical tools like balanced weight filtrations and iterated logarithms to analyze the asymptotic behavior of the Yang--Mills flow on holomorphic bundles, revealing new monotonicity properties and conjectural analogs.

Contribution

It provides a complete asymptotic description of the Yang--Mills flow on Riemann surfaces and introduces a new monotonicity property applicable in arbitrary dimensions.

Findings

Asymptotic description of Yang--Mills flow on Riemann surfaces

Identification of a monotonicity property in arbitrary dimensions

Heuristic support for a conjectural curve shortening flow analog

Abstract

We consider applications of the theory of balanced weight filtrations and iterated logarithms, initiated in arXiv:1706.01073, to PDEs. The main result is a complete description of the asymptotics of the Yang--Mills flow on the space of metrics on a holomorphic bundle over a Riemann surface. A key ingredient in the argument is a monotonicity property of the flow which holds in arbitrary dimension. The A-side analog is a modified curve shortening flow for which we provide a heuristic calculation in support of a detailed conjectural picture.