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A thermodynamical f...
A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics
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- Berman, Robert, 1976 (author)
- Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper, matematik,Department of Mathematical Sciences, Mathematics,University of Gothenburg,Chalmers tekniska högskola,Chalmers University of Technology
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(creator_code:org_t)
- Elsevier BV, 2013
- 2013
- English.
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In: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 248, s. 1254-1297
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Abstract
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- 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.
Subject headings
- NATURVETENSKAP -- Matematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics (hsv//eng)
Keyword
- Monge-Ampere equation
- Kahler-Einstein manifolds
- Variational methods
- SCALAR CURVATURE
- HOLDER CONTINUITY
- COMPLEX-SURFACES
- K-ENERGY
- MANIFOLDS
- SPACE
- CONVERGENCE
- EXISTENCE
- FLOW
- EXISTENCE
Publication and Content Type
- ref (subject category)
- art (subject category)
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