Convergence of the calabi flow on toric varieties and related Kaehler manifolds

TL;DR

This paper proves convergence results for the Calabi flow on toric varieties under curvature bounds, establishes exponential convergence to extremal metrics, and confirms conjectures related to K-stability and curvature bounds in Kähler geometry.

Contribution

It demonstrates convergence of the Calabi flow on toric varieties assuming curvature bounds, and proves related conjectures in Kähler geometry.

Findings

Calabi flow converges exponentially to extremal metrics under curvature bounds

Bounded curvature implies uniform bounds on symplectic potentials

Provides proof of Donaldson's conjecture and supports conjectures by Apostolov et al.

Abstract

Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and $ \int_{\partial P} u ~ d \sigma < C_1, $ then there exists a constant $C_2$ depending only on $C_1$ and $P$ such that $\max_P u < C_2$. As an application, we show that if $(X,P)$ is analytic uniform $K$-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Riemannian curvature is uniformly bounded along the Calabi flow. Also we provide a proof of a conjecture of Donaldson. Finally, assuming that the curvature is bounded along the Calabi flow, our method would provide a proof of a conjecture due to Apostolov, Calderbank, Gauduchon and Tonnesen-Friedman.