An eigenvalue estimate for the \(\overline{\partial}\)-Laplacian
type: Item
instance of: scholarly article
title: An eigenvalue estimate for the \(\overline{\partial}\)-Laplacian
zbMATH Open document ID: 1042.58015
author: Bo Berndtsson
published in: Journal of Differential Geometry
publication date: 2004-03-08T00:00:00Z
review text: Let \(F\) be a holomorphic line bundle over an \(n\)-dimensional compact holomorphic manifold \(X\). Fix a Hermitian metric on \(X\) and a line bundle metric on \(F\). Let \(\square=\bar\partial \bar\partial^*+\bar\partial^*\bar\partial\) be the associated Laplacian. Let \(h_{\leq\lambda}^{(p,q)}(F)\) be the number of eigenvalues (counted with multiplicity) of \(\square\) on \((p,q)\) forms with values in \(F\) which are less than or equal to \(\lambda\). The author gives an asymptotic estimate for the distribution of eigenvalues: Theorem 1.1: Assume \(L\) is a line bundle with a metric of semipositive curvature. Let \(E\) be an arbitrary line bundle. Take \(q\geq1\). If \(0\leq\lambda\leq k\), then \(h_{\leq\lambda}^{n,q}(L^k\otimes E)\leq C(\lambda+1)^qk^{n-q}\). If \(1\leq k\leq\lambda\), then \(h_{\leq\lambda}^{n,q}(L^k\otimes E)\leq C\lambda^n\). Note: By replacing \(E\) by \(E\otimes K^{-1}\), a similar estimate holds for \(h_{\leq\lambda}^{0,q}\). One obtains thereby the estimate \(h^{0,q}_{\leq 0}(L^k\otimes E)\leq Ck^{n-q}\) for the dimensions of the Dolbeault cohomology groups.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 32W50
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32L10
zbMATH DE Number: 2052864
Wikidata QID: Q115201782
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090351103
OpenAlex ID: W1556456618
DOI: 10.4310/JDG/1090351103
Recommended article: Eigenvalue estimate of the \(p\)-Laplace operator
Recommended article: Q4892580
Recommended article: Q4825247
Recommended article: Estimates on eigenvalues of Laplacian
Recommended article: Q4872186
Recommended article: Estimates of the principal eigenvalue of the \(p\)-Laplacian
Recommended article: Eigenvalue estimates for Laplacians on measure spaces
Recommended article: Eigenvalue estimate for the weighted \(p\)-Laplacian
Recommended article: \(L_1\)-estimates for eigenfunctions of the Dirichlet Laplacian
Recommended article: Estimates for eigenvalues of Laplacian operator with any order
description: scientific article; zbMATH DE number 2052864