Mathematics – Differential Geometry
Scientific paper
2010-11-17
Mathematics
Differential Geometry
v2: 51 pages. No figures. The main new feature in v.2 (apart from some stylistic changes is that some new developments in conn
Scientific paper
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-Amp\`ere equations of mean field type, twisted Kahler-Einstein metrics and Moser-Trudinger type inequalities on Kahler manifolds are given. Tian's \alpha- 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.
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