Comparison of the Calabi and Mabuchi geometries and applications to geometric flows

TL;DR

This paper studies the metric geometry of the space of K"ahler metrics using Calabi and Mabuchi structures, revealing topology coincidences on finite entropy spaces and applying findings to convergence of geometric flows.

Contribution

It introduces the $L^{p,q}$-Calabi Finsler structure, compares Calabi and Mabuchi topologies, and applies results to K"ahler--Ricci and Calabi flows.

Findings

Calabi and Mabuchi topologies do not dominate each other generally.

On the finite entropy space, these topologies coincide after strengthening.

New convergence results for K"ahler--Ricci and weak Calabi flows.

Abstract

Suppose $(X,\omega)$ is a compact K\"ahler manifold. We introduce and explore the metric geometry of the $L^{p,q}$-Calabi Finsler structure on the space of K\"ahler metrics $\mathcal H$. After noticing that the $L^{p,q}$-Calabi and $L^{p'}$-Mabuchi path length topologies on $\mathcal H$ do not typically dominate each other, we focus on the finite entropy space $\mathcal E^{Ent}$, contained in the intersection of the $L^p$-Calabi and $L^1$-Mabuchi completions of $\mathcal H$ and find that after a natural strengthening, the $L^p$-Calabi and $L^1$-Mabuchi topologies coincide on $\mathcal E^{Ent}$. As applications to our results, we give new convergence results for the K\"ahler--Ricci flow and the weak Calabi flow.