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Let X be a compact complex manifold equipped with a smooth (but notnecessarily positive) closed form theta of one-one type. By a well-knownenvelope construction this data determines a canonical theta-psh function uwhich is not two times differentiable, in general. We introduce a family ofregularizations of u, parametrized by a positive number beta, defined as thesmooth solutions of complex Monge-Ampere equations of Aubin-Yau type. It isshown that, as beta tends to infinity, the regularizations converge to theenvelope u in the strongest possible Holder sense. A generalization of thisresult to the case of a nef and big cohomology class is also obtained. As aconsequence new PDE proofs are obtained for the regularity results forenvelopes in [14] (which, however, are weaker than the results in [14] in thecase of a non-nef big class). Applications to the regularization problem forquasi-psh functions and geodesic rays in the closure of the space of Kahlermetrics are given. As briefly explained there is a statistical mechanicalmotivation for this regularization procedure, where beta appears as the inversetemperature. This point of view also leads to an interpretation of theregularizations as transcendental Bergman metrics.

From Monge-Ampere equations to envelopes and geodesic rays in the zero temperature limit