On the Hessian-cscK equations

TL;DR

This paper introduces a new coupled system of complex Hessian equations that generalizes cscK metrics, develops a variational framework, and establishes a key a priori estimate based on entropy.

Contribution

It generalizes the cscK equation to a coupled Hessian system and formulates a variational approach using a Hessian Mabuchi K-energy.

Findings

The coupled system can be realized as an Euler-Lagrange equation.

The space of k-Hessian potentials has a negative sectional curvature.

An a priori C^0-estimate depending on entropy is established.

Abstract

In this paper, we propose a coupled system of complex Hessian equations which generalizes the equation for constant scalar curvature K\"ahler (cscK) metrics. We show this system can be realized variationally as the Euler-Lagrange equation of a Hessian version of the Mabuchi K-energy in an infinite dimensional space of $k$-Hessian potentials, which can be seen as an infinite dimensional Riemannian manifold with negative sectional curvature. Finally, we prove an a priori $C^0$-estimate for this system which depends on the Entropy, which generalizes a fundamental result of Chen and Cheng for cscK metrics.