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Reverse Khas’minskii condition. (English) Zbl 1242.53040

The author presents some equivalent characterizations of \(p\)-parabolicity for complete Riemannian manifolds in terms of existence of special exhaustion functions. In particular, R. Z. Khas’minskiĭ in [Theor. Probab. Appl. 5, 179–196 (1961); translation from Teor. Veroyatn. Primen. 5, 196–214 (1960; Zbl 0093.14902)] proved that if there exists a 2-superharmonic function \(\mathcal{K}\) defined outside a compact set on a complete Riemannian manifold \(R\) such that \(\lim_{x\to\infty} \mathcal{K}(x) = \infty\), then \(R\) is 2-parabolic, and L. Sario and M. Nakai in [Classification theory of Riemann surfaces. Die Grundlehren der mathematischen Wissenschaften 164. Berlin etc.: Springer (1970; Zbl 0199.40603)] were able to improve this result by showing that \(R\) is 2-parabolic if and only if there exists an Evans potential, i.e., a 2-harmonic function \(E : R{\setminus}K \to {\mathbb{R}}^+\) with \(\lim_{x\to\infty} \mathcal{E}(x) = \infty\). In this paper, the author shows a reverse Khas’minskiĭ condition valid for any \(p>1\) and discusses the existence of Evans potentials in the nonlinear case.

MSC:

53C20 Global Riemannian geometry, including pinching
31C12 Potential theory on Riemannian manifolds and other spaces
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