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.
- Pub Date:
- January 2017
- Mathematics - Differential Geometry
- 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