Log-concavity of complexity one Hamiltonian torus actions

TL;DR

This paper proves that for certain symplectic manifolds with a Hamiltonian torus action, the density function of the Duistermaat-Heckman measure is log-concave on the moment map image, revealing a new convexity property.

Contribution

It establishes the log-concavity of the Duistermaat-Heckman measure density for complexity one Hamiltonian torus actions, a novel geometric insight.

Findings

Density function of Duistermaat-Heckman measure is log-concave.

Supports convexity properties of symplectic manifolds with torus actions.

Enhances understanding of measure distributions in symplectic geometry.

Abstract

Let $(M,\omega)$ be a closed $2n$-dimensional symplectic manifold equipped with a Hamiltonian $T^{n-1}$-action. Then Atiyah-Guillemin-Sternberg convexity theorem implies that the image of the moment map is an $(n-1)$-dimensional convex polytope. In this paper, we show that the density function of the Duistermaat-Heckman measure is log-concave on the image of the moment map.