Power convexity of solutions to complex Monge-Amp\`ere equation in $\mathbb{C}^2$

TL;DR

This paper proves that solutions to the complex Monge-Ampère equation in two complex dimensions exhibit power convexity, advancing understanding of their geometric properties in complex analysis.

Contribution

It establishes the power convexity of solutions to the complex Monge-Ampère equation in 2, using the constant rank theorem and deformation techniques.

Findings

Solutions are power convex in 2

Method based on constant rank theorem and deformation process

Advances geometric understanding of complex Monge-Ampère solutions

Abstract

The convexity of solutions to boundary value problems for fully nonlinear elliptic partial differential equations (such as real or complex $k$-Hessian equations) is a challenging topic. In this paper, we establish the power convexity of solutions to the Dirichlet problem for the complex Monge-Amp\`ere equation on bounded, smooth, strictly convex domain in $\mathbb{C}^2$. Our approach is based on the constant rank theorem and the deformation process.