2013/07/11 by Robert J. Berman, Berman, Robert J. · 7 citations
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.1307.3008
openalex publication_date 2013/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a compact complex manifold equipped with a smooth (but not\nnecessarily positive) closed form theta of one-one type. By a well-known\nenvelope construction this data determines a canonical theta-psh function u\nwhich is not two times differentiable, in general. We introduce a family of\nregularizations of u, parametrized by a positive number beta, defined as the\nsmooth solutions of complex Monge-Ampere equations of Aubin-Yau type. It is\nshown that, as beta tends to infinity, the regularizations converge to the\nenvelope u in the strongest possible Holder sense. A generalization of this\nresult to the case of a nef and big cohomology class is also obtained. As a\nconsequence new PDE proofs are obtained for the regularity results for\nenvelopes in [14] (which, however, are weaker than the results in [14] in the\ncase of a non-nef big class). Applications to the regularization problem for\nquasi-psh functions and geodesic rays in the closure of the space of Kahler\nmetrics are given. As briefly explained there is a statistical mechanical\nmotivation for this regularization procedure, where beta appears as the inverse\ntemperature. This point of view also leads to an interpretation of the\nregularizations as transcendental Bergman metrics.\n