Gradient flow of the norm squared of a moment map

TL;DR

This paper discusses Duistermaat's proof that the gradient flow of the squared norm of a moment map deforms a manifold onto the zero level set, adapting Lojasiewicz's argument for analytic functions.

Contribution

It provides a detailed proof of Duistermaat's result, extending Lojasiewicz's argument to locally analytic functions in the context of moment maps.

Findings

Gradient flow defines a deformation retract onto the zero level set

Extension of Lojasiewicz's argument to locally analytic functions

Clarification of the geometric structure of moment map level sets

Abstract

We present a proof due to Duistermaat that the gradient flow of the norm squared of the moment map defines a deformation retract of the appropriate piece of the manifold onto the zero level set of the moment map. Duistermaat's proof is an adaptation of Lojasiewicz's argument for analytic functions to functions which are locally analytic.