On the Hessian-cscK equations

DOI10.1007/S00209-022-03187-1arXiv2110.11169MaRDI QIDQ6380947

Publication date: 21 October 2021

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.

Mathematics Subject Classification ID

Global differential geometry of Hermitian and Kählerian manifolds (53C55) Kähler manifolds (32Q15) Complex Monge-Ampère operators (32W20) Monge-Ampère equations (35J96)

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