Denjoy-Ahlfors Theorem for Harmonic Functions on Riemannian Manifolds and External Structure of Minimal Surfaces

Details Denjoy-Ahlfors Theorem for Harmonic Functions on Riemannian Manifolds and External Structure of Minimal Surfaces Denjoy-Ahlfors Theorem for Harmonic Functions on Riemannian Manifolds and External Structure of Minimal Surfaces

2009-03-13

arxiv.org/abs/0903.2326v1

Comm. Anal. Geom., 4 (1996), no. 4, 547--587

Mathematics

Differential Geometry

30 pages

Scientific paper

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 $\R{n}$ 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.

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