# Convexity of the extended Mabuchi energy and the large time behaviour of the weak Calabi flow

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Chinh Lu

created by daniele on 11 Nov 2015

16 nov 2015
-- 14:00

Aula Bianchi Scienze, SNS, Pisa

Seminario di Matematica, Scuola Normale Superiore di Pisa

**Abstract.**

Let $X$ be a compact Kahler manifold. In this talk we first establish the convexity of the Mabuchi energy along finite energy geodesics in the competion of the space of Kahler metrics (the case of $C^{1,1}$ geodesics was
proved by Berman-Berndtsson). Secondly, following the program of J. Streets we use this to study the asymptotics of the weak Calabi flow which is the gradient flow of the Mabuchi energy in the $\mathrm{CAT}(0)$ space of Kahler metrics. This flow exists for all times and coincide with the smooth one whenever the latter exists. We show that it either diverges or it converges, in the pluripotential sense, to some minimiser of the
functional. This gives the first concrete result about the long time convergence of this flow on general Kahler manifolds, partially confirming a conjecture of Donaldson. This is joint work with Robert Berman and Tamas
Darvas arXiv:1510.01260.