A volume stability theorem on toric manifolds with positive Ricci curvature

TL;DR

This paper proves a volume stability theorem for toric manifolds with positive Ricci curvature, showing that if their volume is close to that of complex projective space, they are biholomorphic to it.

Contribution

It establishes a new volume stability result for toric manifolds with Ricci curvature bounds, connecting geometric closeness to biholomorphic equivalence.

Findings

Toric manifolds with Ricci curvature ≥ 1 and volume close to that of CP^n are biholomorphic to CP^n.

The volume stability theorem links geometric and complex-analytic properties.

The result extends understanding of the structure of manifolds with positive Ricci curvature.

Abstract

In this short note, we will prove a volume stability theorem which says that if an n-dimensional toric manifold $M$ admits a $\mathbb{T}^n$ invariant K\"ahler metric $\omega$ with Ricci curvature no less than 1 and its volume is close to the volume of $\mathbb{CP}^n$, $M$ is bi-holomorphic to $\mathbb{CP}^n$.