Denjoy-Ahlfors Theorem for Harmonic Functions on Riemannian Manifolds and External Structure of Minimal Surfaces
arXiv:0903.2326 · doi:10.4310/CAG.1996.v4.n4.a2
Abstract
We extend the well-known Denjoy-Ahlfors theorem on the number of different asymptotic tracts of holomorphic functions to subharmonic functions on arbitrary Riemannian manifolds. We obtain some new versions of the Liouville theorem for $\p$-harmonic functions without requiring the geodesic completeness requirement of a manifold. Moreover, an upper estimate of the topological index of the height function on a minimal surface in has been established and, as a consequence, a new proof of Bernstein's theorem on entire solutions has been derived. Other applications to minimal surfaces are also discussed.
30 pages