A thermodynamical formalism for Monge-Amp¨¨re equations, Moser-Trudinger inequalities and K?hler-Einstein metrics

详细信息    查看全文

• 作者：Robert J. Berman
• 关键词：32Q20 ; 32W20 ; 35A15
• 刊名：Advances in Mathematics
• 出版年：2013
• 出版时间：25 November, 2013
• 年：2013
• 卷：248
• 期：Complete
• 页码：1254-1297
• 全文大小：468 K

文摘

We develop a variational calculus for a certain free energy functional on the space of all probability measures on a K?hler 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-Amp¨¨re equations of mean field type, twisted K?hler-Einstein metrics and Moser-Trudinger type inequalities on K?hler manifolds are given. Tian?s ¦Á-invariant is generalized to singular measures, allowing in particular a proof of the existence of K?hler-Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning K?hler 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 K?hler-Einstein metric, when a unique one exists, which is in line with a well-known conjecture.