The Kelvin-Nevanlinna-Royden criterion for \(p\)-parabolicity
type: Item
instance of: scholarly article
title: The Kelvin-Nevanlinna-Royden criterion for \(p\)-parabolicity
zbMATH Open document ID: 0939.31011
DOI: 10.1007/PL00004775
author: Marc Troyanov
author: Vladimir Gol'dshtein
published in: Mathematische Zeitschrift
publication date: 2000-01-17T00:00:00Z
review text: The authors prove two theorems generalizing results of \textit{H. L. Royden} [Trans. Am. Math. Soc. 73, 40-94 (1952; Zbl 0049.17806)] for surfaces. Theorem A: A manifold \(\mathbb{M}\) is \(p\)-hyperbolic if and only if there exists a vector field \(X\) on \(\mathbb{M}\) such that (i) \(x\in L^n\) subject to \(p^{-1}+ q^{-1}= 1\), (ii) \(\operatorname {div}X\in L_{\text{loc}}^1\) and \((\operatorname {div}X)^-:= \min\{ (\operatorname {div}X),0\}\), (iii) \(0< \int_{\mathbb{M}} \operatorname {div}X\leq \infty\). Theorem B: A manifold \(\mathbb{M}\) is \(p\)-hyperbolic if and only if the cohomology with compact support \(H_{\text{comp},q}^n (\mathbb{M})=0\) subject to \(p^{-1}+ q^{-1}=1\).
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 31C45
zbMATH DE Number: 1389846
zbMATH Keywords: hyperbolic manifolds
zbMATH Keywords: \(p\)-hyperbolic
reviewed by: Erik W. Grafarend
MaRDI profile type: Publication
full work available at URL: pl00004775
OpenAlex ID: W1977545633
description: scientific article; zbMATH DE number 1389846