The Kelvin-Nevanlinna-Royden criterion for \(p\)-parabolicity
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 \(p^{-1}+ q^{-1}=1\).
scientific article; zbMATH DE number 1389846