# Convexity theorems for the gradient map on probability measures

## TL;DR

This paper investigates the convexity properties of the gradient map associated with a Lie group action on probability measures on a submanifold of a Kähler manifold, extending known results from Abelian to non-Abelian groups.

## Contribution

It establishes convexity theorems for the gradient map in the Abelian case and explores potential extensions to non-Abelian groups.

## Key findings

- Convexity results for Abelian group actions on probability measures.
- Analysis of extension possibilities to non-Abelian groups.
- Foundations for further research on gradient maps in geometric analysis.

## Abstract

Given a K\"ahler manifold $(Z,J,\omega)$ and a compact real submanifold $M\subset Z$, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group ${\rm G}$ on the space of probability measures on $M.$ In particular, we prove convexity results for such map when ${\rm G}$ is Abelian and we investigate how to extend them to the non-Abelian case.