Conformal energy in four dimension
type: Item
instance of: scholarly article
title: Conformal energy in four dimension
zbMATH Open document ID: 1026.53036
author: Xingwang Xu
author: Paul C. Yang
published in: Mathematische Annalen
publication date: 2003-03-27T00:00:00Z
review text: In [Invent. Math. 69, 269-291 (1982; Zbl 0503.53042)], \textit{P. Li} and \textit{S.-T. Yau} introduced the notion of conformal volume for compact Riemannian manifolds which admit conformal immersions into spheres. This conformal volume depends only on the conformal class of the metric. In the same spirit, the authors of this article restrict to four-dimensional compact Riemannian spaces and define their conformal energy using the Paneitz operator. This is a fourth order elliptic operator involving the Laplacian and the curvature of the manifold. Again, this notion only depends on the conformal structure. Two lower bounds for the conformal energy are derived: one in terms of the first non-zero eigenvalue of the Paneitz operator and another in terms of the conformal volume. Calculating the conformal energy of a given compact four-manifold is not an easy task. The authors succeed in doing this for \(S^2\times S^2\). Finally, they derive an obstruction to the existence of minimal immersions of compact four-manifolds into spheres.
reviewed by: Eric Boeckx
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 1889184
zbMATH Keywords: conformal energy
zbMATH Keywords: conformal volume
zbMATH Keywords: Paneitz operator
zbMATH Keywords: minimal immersions in spheres
MaRDI profile type: Publication
full work available at URL: s00208-002-0357-x
OpenAlex ID: W2080050062
DOI: 10.1007/S00208-002-0357-X
description: scientific article; zbMATH DE number 1889184