On the Hessian-cscK equations

In this paper, we propose a coupled system of complex Hessian equations which generalizes the equation for constant scalar curvature Kähler (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 [ 1 ] for cscK metrics.