Liouville Action for Harmonic Diffeomorphisms

TL;DR

This paper introduces a Liouville action functional for harmonic diffeomorphisms between compact Riemann surfaces, deriving its variational formula as the source surface varies, contributing to the understanding of geometric functionals in complex analysis.

Contribution

It defines a new Liouville action for harmonic diffeomorphisms and derives its variational formula with respect to source surface variations.

Findings

Defined a Liouville action for harmonic diffeomorphisms

Derived the variational formula for the action

Applicable to surfaces of genus g ≥ 2

Abstract

In this paper, we introduce a Liouville action for a harmonic diffeomorphism from a compact Riemann surface to a compact hyperbolic Riemann surface of genus $g\ge 2$. We derive the variational formula of this Liouville action for harmonic diffeomorphisms when the source Riemann surfaces vary with a fixed target Riemann surface.