Plurisubharmonicity of the Dirichlet energy and deformations of polarized manifolds

TL;DR

This paper proves that the Dirichlet energy of a family of pluriharmonic maps from polarized Kähler manifolds to a nonpositively curved Riemannian manifold varies subharmonically with the deformation parameter.

Contribution

It establishes the plurisubharmonicity of the Dirichlet energy under deformations of polarized Kähler manifolds for pluriharmonic maps, extending previous results in geometric analysis.

Findings

Dirichlet energy is subharmonic in deformations of polarized Kähler manifolds.

Pluriharmonic maps' energy behavior is governed by complex geometric structures.

Results apply to families over the unit disk with nonpositive curvature targets.

Abstract

We show that if $\{M_t\}_{t\in \Delta}$ is a polarized family of compact K\"ahler manifolds over the open unit disk $\Delta$, if $N$ is a Riemannian manifold of nonpositive complexified sectional curvature, and if $\{\phi_t:M_t\to N\}_{t\in \Delta}$ is a smooth family of pluriharmonic maps, then the Dirichlet energy $E(\phi_t)$ is a subharmonic function of $t\in\Delta$.