Li, Peter; Tam, Luen-Fai Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set. (English) Zbl 0622.58001 Ann. Math. (2) 125, 171-207 (1987). This paper gives a contribution to the existence for positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set in higher dimensions. It turns out that S.-T. Yau’s nonexistence theorem [Commun. Pure Appl. Math. 28, 201-228 (1975; Zbl 0291.31002)] fails to hold. By the structure theorem of Cheeger-Gromoll, a complete manifold with nonnegative sectional curvature outside a compact set must have finitely many ends. At each end E, one can consider the volume growth of the set obtained by intersecting E with a geodesic ball of radius t, denoted by V(t). An end E is said to be large if \(\int^{\infty}_{1}\{t/V(t)\}dt<\infty\). Otherwise, E is a small end. The authors show that given any large or small end there exists a unique bounded harmonic function on M satisfying some properties. In the event that M has no large end, then all positive harmonic functions are constant. Reviewer: Y.B.Shen Cited in 4 ReviewsCited in 37 Documents MSC: 58C05 Real-valued functions on manifolds 53C20 Global Riemannian geometry, including pinching Keywords:barrier, Buseman function; positive harmonic functions on complete manifolds; nonnegative curvature; small end; large end Citations:Zbl 0297.31005; Zbl 0291.31002 PDF BibTeX XML Cite \textit{P. Li} and \textit{L.-F. Tam}, Ann. Math. (2) 125, 171--207 (1987; Zbl 0622.58001) Full Text: DOI OpenURL