Gol’dshtein, Vladimir; Troyanov, Marc The Kelvin-Nevanlinna-Royden criterion for \(p\)-parabolicity. (English) Zbl 0939.31011 Math. Z. 232, No. 4, 607-619 (1999). The authors prove two theorems generalizing results of 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 \(p^{-1}+ q^{-1}=1\). Reviewer: Erik W.Grafarend (Stuttgart) Cited in 1 ReviewCited in 18 Documents MSC: 31C15 Potentials and capacities on other spaces 53C20 Global Riemannian geometry, including pinching 31C12 Potential theory on Riemannian manifolds and other spaces 31C45 Other generalizations (nonlinear potential theory, etc.) Keywords:hyperbolic manifolds; \(p\)-hyperbolic Citations:Zbl 0049.17806 PDFBibTeX XMLCite \textit{V. Gol'dshtein} and \textit{M. Troyanov}, Math. Z. 232, No. 4, 607--619 (1999; Zbl 0939.31011) Full Text: DOI