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

TL;DR

This paper generalizes the Denjoy-Ahlfors theorem to subharmonic functions on Riemannian manifolds, provides new Liouville theorems for p-harmonic functions, and offers insights into the structure of minimal surfaces, including a new proof of Bernstein's theorem.

Contribution

It extends classical theorems to broader geometric contexts and introduces new estimates for minimal surface properties without requiring geodesic completeness.

Findings

Generalized Denjoy-Ahlfors theorem for subharmonic functions on Riemannian manifolds

Derived new Liouville theorems for p-harmonic functions without geodesic completeness

Provided an upper estimate of the topological index of the height function on minimal surfaces

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 $\R{n}$ 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.