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

Abstract

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-Ampère equations of mean field type, twisted Kahler-Einstein metrics and Moser-Trudinger type inequalities on Kahler manifolds are given. Tian's α- invariant is generalized to singular measures, allowing i particular the construction of Kahler-Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (as proposed in a very recent program of Donaldson). As another application we show that if the Calabi flow in the (anti-) canonical class exits for all times then it converges to a Kahler-Einstein metric, when one exists. Applications to the probabilistic/statistical mechanical approach to the construction of Kahler-Einstein metrics, very recently introduced by the author, will appear elsewhere.

v2: 51 pages. No figures. The main new feature in v.2 (apart from some stylistic changes is that some new developments in connection to Donaldson's program have been taken into account. v3: 52 pages. References added, a few stylistic changes and remark 44 expanded. v4: 49 pages. Exposition improved. v5: the first point in Cor 7.2 in the Appendix has been corrected (submitted version)

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• The Kahler-Ricci flow through singularities
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• Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics
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• Kaehler-Einstein metrics on orbifolds and Einstein metrics on spheres
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Cited by in corpus (11)

• A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem
• On the moduli of Kahler-Einstein Fano manifolds
• Conical Kahler-Einstein metric revisited
• Bessel Functions, Heat Kernel and the Conical Kähler-Ricci Flow
• Remarks on logarithmic K-stability
• Alpha invariants and K-stability for general polarisations of Fano varieties
• On the Spectrum of weighted Laplacian operator and its application to uniqueness of Kähler Einstein metrics
• On the Geodesics in the space of Kähler metrics with prescribed singularities
• Edges, Orbifolds, and Seiberg-Witten Theory
• On the uniqueness of conical Kähler-Einstein metrics
• The space of Kähler potentials on an asymptotically locally euclidean Kähler manifold