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

Abstract

Suppose $(X,\omega )$ is a compact Kähler manifold. We introduce and explore the metric geometry of the $L^{p,q}$-Calabi Finsler structure on the space of Kähler metrics $\mathcal{H}$. After noticing that the $L^{p,q}$-Calabi and $L^{p^{\prime }}$-Mabuchi path length topologies on $\mathcal{H}$ do not typically dominate each other, we focus on the finite entropy space ${\mathcal{E}}^{\mathrm{Ent}}$, 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 ${\mathcal{E}}^{\mathrm{Ent}}$. As applications to our results, we give new convergence results for the Kähler–Ricci flow and the weak Calabi flow.