Moment map, convex function and extremal point

2022/08/07 by King Leung Lee, Lee, King Leung, Jacob Sturm +3 · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2208.03724

openalex publication_date 2022/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The moment map μ 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 Ad_K-invariant convex function f on \mathfrakk^∗, the dual of Lie algebra of K, and study the properties of the critical point of f∘μ. Our motivation comes from Donaldson \citeDonaldson2017 which is an example of infinite dimensional version of our setting. As an application, we interpret Kähler-Ricci solitons as a special case of the generalized extremal metric.

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• Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopes

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