Calabi flow on toric varieties with bounded Sobolev constant, I

TL;DR

This paper proves that on toric varieties, the Calabi flow remains bounded in the $C^0$-norm under bounded Sobolev constants, and converges exponentially to extremal Kähler metrics under stability and curvature bounds.

Contribution

It establishes uniform bounds for the Calabi flow and convergence results on toric varieties with bounded Sobolev constants, extending to quasi-proper Kähler manifolds.

Findings

The $C^0$-norm of the Calabi flow is uniformly bounded given bounded Sobolev constants.

Under uniform $K$-stability, the modified Calabi flow converges exponentially fast to an extremal Kähler metric.

Results extend to quasi-proper Kähler manifolds under certain conditions.

Abstract

Let $(X, P)$ be a toric variety. In this note, we show that the $C^0$-norm of the Calabi flow $\varphi(t)$ on $X$ is uniformly bounded in $[0, T)$ if the Sobolev constant of $\varphi(t)$ is uniformly bounded in $[0, T)$. We also show that if $(X, P)$ is uniform $K$-stable, then the modified Calabi flow converges exponentially fast to an extremal K\"ahler metric if the Ricci curvature and the Sobolev constant are uniformly bounded. At last, we discuss an extension of our results to a quasi-proper K\"ahler manifold.