Liouville and Calabi-Yau type theorems for complex Hessian equations

TL;DR

This paper establishes Liouville and Calabi-Yau type theorems for complex Hessian equations, leading to a priori gradient estimates and solutions for these equations on compact Kähler manifolds.

Contribution

It proves a Liouville theorem for entire maximal m-subharmonic functions and derives gradient estimates, completing the solution to the non-degenerate Hessian equation on compact Kähler manifolds.

Findings

Liouville theorem for maximal m-subharmonic functions

A priori gradient estimate for complex Hessian equations

Existence of continuous weak solutions in degenerate cases

Abstract

We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact K\"ahler manifold. This terminates the program, initiated by Hou, Ma and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right hand side in some $L^p, $ with sharp bound on $p$.