On the lowest eigenvalue of the Hodge Laplacian
type: Item
instance of: scholarly article
title: On the lowest eigenvalue of the Hodge Laplacian
zbMATH Open document ID: 0874.58087
author: Sagun Chanillo
author: François Treves
published in: Journal of Differential Geometry
publication date: 1997-10-01T00:00:00Z
review text: The article gives a lower bound for the least positive eigenvalue of the Hodge-Laplacian acting on \(p\)-forms in a compact \(C^\infty\) manifold \(\mathcal M\) without boundary. The bound is related to the parameters of a certain kind of coverings \(\{{\mathcal U}_i\}\) of \(\mathcal M\) by finitely many geodesically convex sets. The parameters are the number of sets \({\mathcal U}_i\), their geodesic radii and bounds on the differential of the exponential map in each \({\mathcal U}_i\). The estimate is combined with the Łojasiewicz inequality to show that the positive eigenvalues of the Hodge-Laplacian on the regular level sets of a \(C^\omega\) function \(\Phi:S^n\to\mathbb{R}\) are larger than a constant times a power of the distance between the value of \(\Phi\) and the set of the critical values of \(\Phi\).
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1052961
zbMATH Keywords: Riemannian metric
zbMATH Keywords: injectivity radius
zbMATH Keywords: analytic level sets
zbMATH Keywords: eigenvalues
zbMATH Keywords: Hodge-Laplacian
Wikidata QID: Q115175887
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214459798
OpenAlex ID: W1570501355
DOI: 10.4310/JDG/1214459798
description: scientific article; zbMATH DE number 1052961