Nodal geometry on Riemannian manifolds
type: Item
instance of: scholarly article
title: Nodal geometry on Riemannian manifolds
zbMATH Open document ID: 0727.58048
author: Sagun Chanillo
author: Benjamin Muckenhoupt
published in: Journal of Differential Geometry
publication date: 1991-01-01T00:00:00Z
review text: Let M be a compact connected Riemannian manifold of dimension m without boundary. Let u be an eigenfunction of the Laplacian corresponding to an eigenvalue \(\lambda >1\). The authors show: Theorem: (a) \(\| \log | u| \|_{BMO}\leq c(M)\lambda^ m\cdot \log (\lambda).\) (b) Let B be a ball in M and let \(\Omega\subset B\) be one of the connected components of \(\{\) \(x\in B:\) u(x)\(\neq 0\}\). If \(\Omega\) intersects the middle half of B, \[ vol(\Omega)\geq c(M)\lambda^{-2m^ 2-m/2}(\log \lambda)^{-2m}vol(B). \] Remark: Similar theorems have been proved by H. Donnelly and C. Fefferman with \(\lambda^ m\cdot \log (\lambda)\) replaced by \(\lambda^{m(m+2)/4}\) in (a) and with the second bound replaced by \(\lambda^{-(m+m^ 2(m+2))/2}\) in (b).
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4201228
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: eigenfunction
zbMATH Keywords: Laplacian
Wikidata QID: Q115181589
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214446991
OpenAlex ID: W1543277952
DOI: 10.4310/JDG/1214446991
description: scientific article; zbMATH DE number 4201228