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

Darvas, Tamás

doi:10.1016/j.anihpc.2016.09.002

Darvas, Tamás

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1131-1140.

Résumé

Suppose $(X, ω)$ 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 $H$ . After noticing that the $L^{p, q}$ -Calabi and $L^{p^{'}}$ -Mabuchi path length topologies on $H$ do not typically dominate each other, we focus on the finite entropy space $E^{Ent}$ , contained in the intersection of the $L^{p}$ -Calabi and $L^{1}$ -Mabuchi completions of $H$ and find that after a natural strengthening, the $L^{p}$ -Calabi and $L^{1}$ -Mabuchi topologies coincide on $E^{Ent}$ . As applications to our results, we give new convergence results for the Kähler–Ricci flow and the weak Calabi flow.

MR Zbl

DOI : 10.1016/j.anihpc.2016.09.002

Mots clés : Ricci flow, Kahler geometry, Calabi metric, Mabuchi metric

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     author = {Darvas, Tam\'as},
     title = {Comparison of the {Calabi} and {Mabuchi} geometries and applications to geometric flows},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1131--1140},
     publisher = {Elsevier},
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Darvas, Tamás. Comparison of the Calabi and Mabuchi geometries and applications to geometric flows. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1131-1140. doi : 10.1016/j.anihpc.2016.09.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2016.09.002/

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