On the optimal regularity of weak geodesics in the space of metrics on a polarized manifold

TL;DR

This paper proves that weak geodesics in the space of positively curved Hermitian metrics on an ample line bundle are C^{1,1}-smooth, improving previous regularity bounds and extending results to complex Monge-Ampère equations.

Contribution

It establishes C^{1,1} regularity of weak geodesics in the space of metrics, enhancing Chen's bounds and generalizing Blocki's results using envelope techniques.

Findings

Weak geodesics are C^{1,1}-smooth for fixed time t.

Improves the regularity bounds on the Laplacian of geodesics.

Provides a regularity result for complex Monge-Ampère equations.

Abstract

Let (X,L) be a polarized compact manifold, i.e. L is an ample line bundle over X and denote by H the infinite dimensional space of all positively curved Hermitian metrics on L equipped with the Mabuchi metric. In this short note we show, using Bedford-Taylor type envelope techniques developed in the authors previous work [ber2], that Chen's weak geodesic connecting any two elements in H are C^{1,1}-smooth, i.e. the real Hessian is bounded, for any fixed time t, thus improving the original bound on the Laplacins due to Chen. This also gives a partial generalization of Blocki's refinement of Chen's regularity result. More generally, a regularity result for complex Monge-Ampere equations over X\timesD, for D a pseudconvex domain in \C^{n} is given.