On the functional logdet and related flows on the space of closed embedded curves on \(S^ 2\)
type: Item
instance of: scholarly article
title: On the functional logdet and related flows on the space of closed embedded curves on \(S^ 2\)
zbMATH Open document ID: 0805.58062
author: Dan Burghelea
author: T. Kappeler
author: Patrick McDonald
author: Leonid Friedlander
published in: Journal of Functional Analysis
publication date: 1994-08-29T00:00:00Z
review text: The paper is devoted to the study of the critical points of a new functional defined on the space of closed, simple, nonparametrized curves on a two-dimensional Riemannian manifold. The definition of the new functional goes as follows. Any oriented nonparametrized embedded curve on the two dimensional sphere \(S^ 2\) divides it into two parts. Take the manifold with boundary which has as interior the points of the sphere not in the curve, and as boundary, the disjoint union of two copies of the curve with opposite orientation. The resulting compact manifold has two connected components, denoted \(D^{+}_{\Gamma}\) and \(D^ -_ \Gamma\) according to the orientation of \(\Gamma\). Moreover, any Riemannian structure on the sphere can be lifted to it. Thus, given a fixed Riemannian structure on the sphere, the authors consider the Laplacian operators for the lifted Riemannian metric on the component \(D^ +_ \Gamma\), and, following \textit{D. B. Ray} and \textit{I. Singer} [Adv. Math. 7, 145-210 (1971; Zbl 0239.58014)], they define the zeta- regularized determinant, \(\text{logdet}(\Delta_ g,D^ +_ \Gamma)\) of the Laplace-Beltrami operator. The new functional, \(h_ g\), is then the sum \(\text{logdet}(\Delta_ g,D^ +_ \Gamma) + \text{logdet}(\Delta_ g,D^ +_{\iota(\Gamma)})\), where \(\iota(\Gamma)\) denotes the same underlying curve, but with reversed orientation. Considered separately, the logdet functional of these zeta functionals have no critical points and are not bounded neither below nor above. But the sum, \(h_ g\), of both logdet functionals has interesting properties. The paper is devoted to the study of the critical points of this sum of logdet functionals. It is well known that in the conformal class of any Riemannian metric \(g\) on the two-dimensional sphere there is a unique metric of constant curvature \(1\). One of the main results of the paper is the characterization of the critical points of the new functional as conformal circles; i.e., curves of consta…
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 58D10
zbMATH DE Number: 599771
zbMATH Keywords: zeta-regularized determinant
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: conformal circles
zbMATH Keywords: embedded curves
reviewed by: Juan Monterde
MaRDI profile type: Publication
full work available at URL: jfan.1994.1038
OpenAlex ID: W2032126373
DOI: 10.1006/JFAN.1994.1038
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description: scientific article; zbMATH DE number 599771