Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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D. Angella - N. Istrati - A. Otiman - N. Tardini

Variational problems in conformal geometry

created by tardini on 12 Sep 2019
modified on 30 Apr 2021


Published Paper

Inserted: 12 sep 2019
Last Updated: 30 apr 2021

Journal: J. Geom. Anal.
Volume: 31
Number: 3
Pages: 3230–3251
Year: 2021
Doi: 10.1007/s12220-020-00392-4

ArXiv: 1907.11189 PDF


We study the Euler-Lagrange equation for several natural functionals defined on a conformal class of almost Hermitian metrics, whose expression involves the Lee form $\theta$ of the metric. We show that the Gauduchon metrics are the unique extremal metrics of the functional corresponding to the norm of the codifferential of the Lee form. We prove that on compact complex surfaces, in every conformal class there exists a unique metric, up to multiplication by a constant, which is extremal for the functional given by the $L^2$-norm of $dJ\theta$, where $J$ denotes the complex structure. These extremal metrics are not the Gauduchon metrics in general, hence we extend their definition to any dimension and show that they give unique representatives, up to constant multiples, of any conformal class of almost Hermitian metrics.

Tags: SIR2014-AnHyC

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