Communications in Analysis and Geometry

Volume 31 (2023)

Number 10

The Dirichlet principle for the complex $k$-Hessian functional

Pages: 2471 – 2506

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n10.a5

Authors

Yi Wang (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Hang Xu (Department of Mathematics, University of California at San Diego, La Jolla, Calif., U.S.A.)

Abstract

We study the variational structure of the complex $k$-Hessian equation on bounded domain $X \subset \mathbb{C}^n$ with boundary $M = \partial X$. We prove that the Dirichlet problem $\sigma_k (\partial \bar{\partial} u) = 0$ in $X$, and $u = f$ on $M$ is variational and we give an explicit construction of the associated functional $\mathcal{E}_k (u)$. Moreover we prove $\mathcal{E}_k (u)$ satisfies the Dirichlet principle. In a special case when $k = 2$, our constructed functional $\mathcal{E}_2 (u)$ involves the Hermitian mean curvature of the boundary, the notion first introduced and studied by X. Wang $\href{https://doi.org/10.1142/S0219199716500632}{[37]}$. Earlier work of J. Case and and the first author of this article $\href{https://doi.org/10.1016/j.jfa.2018.08.024}{[9]}$ introduced a boundary operator for the (real) $k$-Hessian functional which satisfies the Dirichlet principle. The present paper shows that there is a parallel picture in the complex setting.

Received 1 June 2021

Accepted 10 June 2021

Published 13 August 2024