Harmonic sections of polynomial growth
type: Item
instance of: scholarly article
title: Harmonic sections of polynomial growth
zbMATH Open document ID: 0880.53039
author: Peter Li
published in: Mathematical Research Letters
publication date: 1998-01-11T00:00:00Z
review text: Starting from a conjecture of S.-T. Yau and a series of more general statements due to \textit{J. H. Colding} and \textit{W. P. Minicozzi} (see, for instance, [Math. Res. Lett. 3, 241-246 (1996; Zbl 0849.53034)]), the author considers a complete manifold \(M\) satisfying the weak volume comparison condition and the mean value inequality. For a vector bundle \(E\) over \(M\), let \(S_d(M,E)\) be a linear subspace of sections of \(E\), such that all \(u\in S_d(M,E)\) satisfy (a) \(\Delta |u|^2 \geq 0\); (b) \(|u|(x)\leq O(\rho^d(x))\) as \(x\to\infty\). Then the dimension of \(S_d(M,E)\) is finite. Moreover, a sharp estimate of the dimension of the space of harmonic functions which have at most polynomial growth of degree \(d\) is established. Interesting geometric applications are given. We want to mention estimates of the dimensions of the space of Killing vector fields of polynomial growth on a complete Ricci flat manifold and of the space of holomorphic sections of polynomial growth of a Hermitian vector bundle over a complete Kähler manifold, respectively.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 1019784
zbMATH Keywords: harmonic functions
zbMATH Keywords: polynomial growth
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: weak volume comparison condition
zbMATH Keywords: mean value inequality
zbMATH Keywords: Ricci curvature
zbMATH Keywords: Kähler manifold
reviewed by: Ion Mihai
MaRDI profile type: Publication
Wikidata QID: Q125376037
full work available at URL: mrl.1997.v4.n1.a4
OpenAlex ID: W1973561295
DOI: 10.4310/MRL.1997.V4.N1.A4
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description: scientific article; zbMATH DE number 1019784