Moment map, convex function and extremal point

TL;DR

This paper develops a theory connecting moment maps with invariant convex functions on Lie algebra duals, analyzing critical points and applying the framework to interpret Kähler-Ricci solitons as generalized extremal metrics.

Contribution

It introduces a new approach combining moment maps with convex functions, extending the understanding of extremal points and their applications in Kähler geometry.

Findings

Critical points of composed functions relate to extremal metrics.

The framework generalizes existing concepts in symplectic and Kähler geometry.

Kähler-Ricci solitons are interpreted as special cases within this theory.

Abstract

The moment map $\mu$ is a central concept in the study of Hamiltonian actions of compact Lie groups $K$ on symplectic manifolds. In this short note, we propose a theory of moment maps coupled with an $\mathrm{Ad}_K$-invariant convex function $f$ on $\mathfrak{k}^{\ast}$, the dual of Lie algebra of $K$, and study the properties of the critical point of $f\circ\mu$. Our motivation comes from Donaldson \cite{Donaldson2017} which is an example of infinite dimensional version of our setting. As an application, we interpret K\"ahler-Ricci solitons as a special case of the generalized extremal metric.