The $L^\infty$ estimates for parabolic complex Monge–Ampère and Hessian equations

Xiuxiong Chen (Institute of Geometry and Physics, University of Science and Technology of China, Hefei, Anhui, China; and Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Jingrui Cheng (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Abstract

In this paper, we consider a version of parabolic complex Monge–Ampère equations, and use a PDE approach similar to Phong et al. to establish $L^\infty$ and Hölder estimates. We also generalize the $L^\infty$ estimates to parabolic Hessian equations.

Keywords

$L^\infty$ estimate, complex Monge–Ampère equations, complex Hessian equations

2010 Mathematics Subject Classification

35K96

Full Text (PDF format)

The research of both authors is partially supported by the Simons Foundation

Received 7 March 2022

Received revised 30 August 2022

Accepted 26 October 2022

Published 30 January 2024