Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set. (English) Zbl 0622.58001

Li, Peter; Tam, Luen-Fai

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

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