Extremals of determinants of Laplacians
type: Item
instance of: scholarly article
title: Extremals of determinants of Laplacians
zbMATH Open document ID: 0653.53022
DOI: 10.1016/0022-1236(88)90070-5
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: Let M be a compact Riemann surface with smooth boundary dM. Let \(\Delta\) be the Laplacian with Dirichlet boundary condition. If \(\{\lambda_ n\}\) are the non-zero eigenvalues of \(\Delta\), then the zeta function \(\zeta (s,\Delta)=\sum_ n\lambda_ n^{-s}\) is holomorphic at \(s=0\) and - \(\zeta\) '(0,\(\Delta)\) is the functional determinant. This is a non-local spectral invariant. A metric g on M is said to be uniform if (i) \(dM=\emptyset\) and the metric g has constant curvature or (ii) dM\(\neq \emptyset\) and g is flat. The authors show that the uniform metric minimizes ths inducing normal variation. Then structures are preserved by invariant, isometric infinitesimal variations.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4065845
zbMATH Keywords: Riemann surface
zbMATH Keywords: Laplacian
zbMATH Keywords: uniform metric
zbMATH Keywords: infinitesimal variations
Wikidata QID: Q29040618
author: Brad G. Osgood
author: Ralph S. Phillips
author: Peter Sarnak
reviewed by: Yosio Mutō
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90070-5
OpenAlex ID: W2096898175
cites work: Meilleures constantes dans le théorème d'inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire
cites work: Riemannian structures of prescribed Gaussian curvature for compact 2- manifolds
cites work: Q3034341
cites work: A proof of the Bieberbach conjecture
cites work: Successive Coefficients of Univalent Functions
cites work: Three-manifolds with positive Ricci curvature
cites work: Can One Hear the Shape of a Drum?
cites work: Curvature functions for compact 2-manifolds
cites work: Curvature and the eigenvalues of the Laplacian
cites work: Q5614914
cites work: On the principal frequency of a membrane
cites work: On the positivity of the effective action in a theory of random surfaces
cites work: A generalization of Courant's nodal domain theorem.
cites work: Fredholm eigenvalues of multiply-connected domains
cites work: Connections and conformal mapping
cites work: Q3900552
cites work: Q5547919
cites work: Q3259806
cites work: Determinants of Laplacians and Multiple Gamma Functions
cites work: Q4079943
cites work: Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces
description: scientific article; zbMATH DE number 4065845