The conical complex Monge-Amp\`ere equations on K\"ahler manifolds

TL;DR

This paper establishes existence, uniqueness, and regularity results for conical complex Monge-Ampère equations on Kähler manifolds, including regularity of bounded solutions, using uniform gradient estimates for approximating equations.

Contribution

It introduces new uniform gradient estimates for approximating equations, enabling the proof of regularity and well-posedness of conical complex Monge-Ampère equations with weak initial data.

Findings

Proved existence and uniqueness of solutions.

Established $C^{2,eta}$ regularity for bounded solutions.

Developed uniform gradient estimates for approximation sequences.

Abstract

In this paper, by providing the uniform gradient estimates for a sequence of the approximating equations, we prove the existence, uniqueness and regularity of the conical parabolic complex Monge-Amp\`ere equation with weak initial data. As an application, we prove a regularity estimates, that is, any $L^{\infty}$-solution of the conical complex Monge-Amp\`ere equation admits the $C^{2,\alpha,\beta}$-regularity.