1307.3008 (Robert J. Berman)

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

Robert J. Berman

Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of one-one type. By a well-known envelope construction this data determines a canonical theta-psh 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 Monge-Ampere equations of Aubin-Yau type. It is shown that, as beta tends to infinity, the regularizations converge to the envelope u in the strongest possible Holder sense. More generally, a generalization of this result to the case of a nef and big cohomology class is obtained. Application to the regularization problem for geodesic rays in 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.

View original: http://arxiv.org/abs/1307.3008