Gradient flow of the norm squared of a moment map over Kahler manifolds

TL;DR

This paper proves the global existence of smooth solutions for the gradient flow equations of the vortex functional on compact Kähler manifolds, extending Morse theory insights to gauged holomorphic maps.

Contribution

It introduces a heat flow approach to analyze the moduli space of gauged holomorphic maps, building on Atiyah-Bott and Hong's methods, and establishes global existence results.

Findings

Global existence of smooth solutions for the gradient flow equations

Extension of Morse theory techniques to gauged holomorphic maps

Application of Hong's method to vortex functional on Kähler manifolds

Abstract

Inspired by Wilkin's work [23, 24] on Morse theory for the moduli space of Higgs bundles, we study the moduli space of gauged holomorphic maps by a heat flow approach in the spirit of Atiyah and Bott in a series of papers. In this paper, applying the method of Hong [9], we establish the global existence of smooth solutions of the gradient ow equations of the vortex functional over a compact Kahler manifold.