On the nodal count for flat tori
type: Item
instance of: scholarly article
title: On the nodal count for flat tori
zbMATH Open document ID: 1252.58014
author: Jochen Brüning
author: David Fajman
published in: Communications in Mathematical Physics
publication date: 2012-08-13T00:00:00Z
review text: Let \(\Delta\) be the scalar Laplacian on a compact Riemannian manifold \((M,g)\) of dimension \(m\). If \(\lambda\) is an eigenvalue of \(\Delta\), let \(E(\lambda,\Delta)\) be the corresponding eigenspace. If \(0\neq\psi\in E(\lambda,\Delta)\), the nodal count \(nc(\psi)\) is the number of connected components of \(\psi^{-1}(\mathbb{R}-\{0\})\). One has \(nc(\psi)\leq C(M)\lambda^{2/m}\) by a theorem of Courant. Thus \(nc_\Delta(\lambda):=\{nc(\psi):\Delta\psi=\lambda\psi\}\) is a finite subset of \(\mathbb{Z}\). We say \((M_1,g_1)\) is isospectral to \((M_2,g_2)\) if the eigenvalue multiplicities are the same. They are said to have the same nodal count if there is a bijection from the projective space \(PE(\lambda,\Delta_1)\) to \(PE(\lambda,\Delta_2)\) preserving the nodal count. The authors use the 4-parameter family of isospectral tori in dimension 4 of \textit{J. H. Conway} and \textit{N. J. A. Sloan} [Int. Math. Res. Not. 1992, No. 4, 93--96 (1992; Zbl 0770.11022)] to provide examples of non-isometric isospectral pairs with the same nodal count and examples of non-isometric isospectral pairs distinguished by their nodal count.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J53
zbMATH DE Number: 6065317
zbMATH Keywords: nodal domain
zbMATH Keywords: isospectral manifold
zbMATH Keywords: flat torus
zbMATH Keywords: Conway--Sloan family
MaRDI profile type: Publication
full work available at URL: s00220-012-1432-0
OpenAlex ID: W2125645603
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DOI: 10.1007/S00220-012-1432-0
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description: scientific article; zbMATH DE number 6065317