A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics
Artikel i vetenskaplig tidskrift, 2013

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.

EXISTENCE

Variational methods

COMPLEX-SURFACES

SCALAR CURVATURE

SPACE

Kahler-Einstein manifolds

CONVERGENCE

K-ENERGY

FLOW

Monge-Ampere equation

HOLDER CONTINUITY

MANIFOLDS

Författare

Robert Berman

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

Advances in Mathematics

0001-8708 (ISSN) 1090-2082 (eISSN)

Vol. 248 1254-1297

Ämneskategorier

Matematik

DOI

10.1016/j.aim.2013.08.024