Regularity of geodesic rays and Monge-Ampere equations

TL;DR

This paper proves that geodesic rays arising from Bergman geodesics are always of class C^{1,α} and can be extended as solutions to a Monge-Ampère Dirichlet problem on compact Kähler manifolds, advancing understanding of complex geometric analysis.

Contribution

It establishes the regularity of geodesic rays from Bergman geodesics and links them to solutions of Monge-Ampère equations on compact Kähler manifolds.

Findings

Geodesic rays are of class C^{1,α} for 0<α<1.

Geodesic rays can be extended as solutions to a Monge-Ampère Dirichlet problem.

Provides a new connection between geodesic rays and Monge-Ampère equations.

Abstract

It is shown that the geodesic rays constructed as limits of Bergman geodesics from a test configuration are always of class $C^{1,\alpha}, 0<\alpha<1$. An essential step is to establish that the rays can be extended as solutions of a Dirichlet problem for a Monge-Ampere equation on a Kaehler manifold which is compact.