Toric Surfaces, K-Stability and Calabi Flow

TL;DR

This paper establishes bounds on the geometry of toric surfaces under curvature conditions and demonstrates exponential convergence of the Calabi flow to extremal metrics assuming stability and bounded curvature.

Contribution

It proves diameter bounds and the validity of Donaldson's M-condition for toric surfaces under curvature bounds, and links K-stability to exponential convergence of the Calabi flow.

Findings

Diameter of toric surfaces is bounded under curvature and boundary integral constraints.

Donaldson's M-condition holds for the symplectic potential under these bounds.

Calabi flow converges exponentially to extremal metrics given stability and curvature bounds.

Abstract

Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$. Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $\int_{\partial P} u ~ d \sigma < C_2, $ then there exists a constant $C_3$ depending only on $C_1, C_2$ and $P$ such that the diameter of $X$ is bounded by $C_3$. Moreoever, we can show that there is a constant $M > 0$ depending only on $C_1, C_2$ and $P$ such that Donaldson's $M$-condition holds for $u$. As an application, we show that if $(X,P)$ is (analytic) relative $K$-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.