From MongeAmpere equations to envelopes and geodesic rays in the zero temperature limit
Abstract
Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of oneone type. By a wellknown envelope construction this data determines a canonical thetapsh function u which is not two times differentiable, in general. We introduce a family of regularizations of u, parametrized by a positive number beta, defined as the smooth solutions of complex MongeAmpere equations of AubinYau type. It is shown that, as beta tends to infinity, the regularizations converge to the envelope u in the strongest possible Holder sense. A generalization of this result to the case of a nef and big cohomology class is also obtained. As a consequence new PDE proofs are obtained for the regularity results for envelopes in [14] (which, however, are weaker than the results in [14] in the case of a nonnef big class). Applications to the regularization problem for quasipsh functions and geodesic rays in the closure of the space of Kahler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where beta appears as the inverse temperature. This point of view also leads to an interpretation of the regularizations as transcendental Bergman metrics.
 Publication:

arXiv eprints
 Pub Date:
 July 2013
 DOI:
 10.48550/arXiv.1307.3008
 arXiv:
 arXiv:1307.3008
 Bibcode:
 2013arXiv1307.3008B
 Keywords:

 Mathematics  Complex Variables;
 Mathematical Physics;
 Mathematics  Differential Geometry
 EPrint:
 28 pages. Version 2: 29 pages. Improved exposition, references updated. Version 3: 31 pages. A direct proof of the bound on the MongeAmp\`ere mass of the envelope for a general big class has been included and Theorem 2.2 has been generalized to measures satisfying a BernsteinMarkov property