A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics

By Robert J. Berman. 2010

179 citations recorded by the source corpus.

We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kahler manifold X. This functional can be seen as a generalization of Mabuchi's K-energy functional and its twisted versions to more singular situations. Applications to Monge-Ampere equations of mean field type, twisted Kahler-Einstein metrics and Moser-Trudinger type inequalities on Miller manifolds are given. Tian's alpha-invariant is generalized to singular measures, allowing in particular a proof of the existence of Kahler-Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning Miller metrics with conical singularities confirms a recent conjecture of Donaldson). As another application we show that if the Calabi flow in the (anti-)canonical class exists for all times then it converges to a Kahler-Einstein metric, when a unique one exists, which is in line with a well-known conjecture. (C) 2013 Elsevier Inc. All rights reserved.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3bfd83d1ea15d498e5daaf66dccca8b9d3a0c3f8

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In Applied Mathematics

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Citations in the corpus

K‐Stability and Kähler‐Einstein Metrics (2012, background): “They observed that the main existence theorem in [29], coupled with an estimate on log- ̨ invariants in [4], implies the existence of conic Kähler-Einstein metrics with cone angle 2 ˇ so long as D 1 .”
Tian's properness conjectures and Finsler geometry of the space of Kahler metrics (2015, methodology): “Property (P3) is a regularity requirement for minimizes, and is inspired by the work of Berman and Berman–Witt-Nyström [5, 11].”
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, background): “Moreover, φ and φ D are locally integrable, so that ω is admissible and Ric This proposition shows that the diﬀerent deﬁnitions of what should be a singular K¨ahler-Einstein metric, appearing e.g. in [ Ber13, BEGZ10, CGP13, EGZ09 ] etc. coincide.”
Existence and deformations of Kähler–Einstein metrics on smoothable $\mathbb{Q}$-Fano varieties (2014, background): “Both functionals can be defined for more general class of functions, see [4], [6].”
A variational approach to the Yau–Tian–Donaldson conjecture (2015, methodology): “ollary 1.8 shows that Uis a psh geodesic. 2. Twisted Kahler-Einstein currents and coercivity In this section, we review the thermodynamical formalism for twisted Kahler-Einstein currents, following [Berm13], and provide a coercivity criterion for certain functionals on E1. 2.1. Twisted Kahler-Einste”