Ricci curvature, the convexity of volume and minimal Lagrangian submanifolds

TL;DR

This paper explores the relationship between Ricci curvature and volume convexity in toric Kähler geometry, extending to quasi-homogeneous manifolds and minimal Lagrangian submanifolds, revealing new geometric insights.

Contribution

It establishes a direct link between Ricci curvature signs and volume convexity, and discusses existence results for minimal Lagrangian submanifolds in broader contexts.

Findings

Ricci curvature sign corresponds to volume convexity in toric Kähler geometry

Convexity properties relate to the existence of minimal Lagrangian submanifolds

Extensions to quasi-homogeneous manifolds show similar relationships

Abstract

We show that, in toric Kaehler geometry, the sign of the Ricci curvature corresponds exactly to convexity properties of the volume functional. We also discuss analogous relationships in the more general context of quasi-homogeneous manifolds, and existence results for minimal Lagrangian submanifolds.