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A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics

Robert Berman (Institutionen för matematiska vetenskaper, matematik)
Advances in Mathematics (0001-8708). Vol. 248 (2013), p. 1254-1297.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: Monge-Ampere equation, Kahler-Einstein manifolds, Variational methods, SCALAR CURVATURE, HOLDER CONTINUITY, COMPLEX-SURFACES, K-ENERGY, MANIFOLDS, SPACE, CONVERGENCE, EXISTENCE, FLOW



Denna post skapades 2013-11-08. Senast ändrad 2016-07-01.
CPL Pubid: 186267

 

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)

Ämnesområden

Matematik

Chalmers infrastruktur