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

Suppose (X,ω) is a compact Kähler manifold. We introduce and explore the metric geometry of the Lp,q-Calabi Finsler structure on the space of Kähler metrics H. After noticing that the Lp,q-Calabi and Lp′-Mabuchi path length topologies on H do not typically dominate each other, we focus on the finite entropy space EEnt, contained in the intersection of the Lp-Calabi and L1-Mabuchi completions of H and find that after a natural strengthening, the Lp-Calabi and L1-Mabuchi topologies coincide on EEnt. As applications to our results, we give new convergence results for the Kähler-Ricci flow and the weak Calabi flow.