Existence and non-existence of minimal graphic and $p$-harmonic functions

Abstract

We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold $M$ with only one end if $M$ has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and $p$-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.

The authors are grateful to Professor Luciano Mari who pointed out an error in the proof of the previous version of Proposition 3.1 To appear in Proc. Roy. Soc. Edinburgh Sect. A

References in corpus (4)

• The half-space property and entire positive minimal graphs in M x R
• Asymptotic Dirichlet problem for A-harmonic and minimal graph equations in Cartan-Hadamard manifolds
• Solvability of minimal graph equation under pointwise pinching condition for sectional curvatures
• Asymptotic Dirichlet problem for $\mathcal{A}$-harmonic functions on manifolds with pinched curvature

Cited by in corpus (6)

• Bernstein and half-space properties for minimal graphs under Ricci lower bounds
• Non-negative Ricci curvature and Minimal graphs with linear growth
• Recent rigidity results for graphs with prescribed mean curvature
• Survey on the asymptotic Dirichlet problem for the minimal surface equation
• Poincaré inequality on minimal graphs over manifolds and applications
• Liouville theorem for minimal graphs over manifolds of nonnegative Ricci curvature