The Dirichlet problem for degenerate complex Monge-Ampere equations

TL;DR

This paper investigates the Dirichlet problem for degenerate complex Monge-Ampère equations on Kähler manifolds, establishing regularity estimates and constructing geodesic rays in the space of Kähler potentials.

Contribution

It provides new C^{1,α} estimates away from divisors and constructs geodesic rays for test configurations, advancing understanding of degenerate Monge-Ampère equations.

Findings

Established C^{1,α} regularity estimates away from divisors.

Constructed C^{1,α} geodesic rays in Kähler potential space.

Extended techniques of pluripotential theory to degenerate cases.

Abstract

The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,\alpha} estimates away from a divisor are obtained, by combining techniques of Blocki, Tsuji, Yau, and pluripotential theory. In particular, C^{1,\alpha} geodesic rays in the space of Kaehler potentials are constructed for each test configuration