Non-negative Ricci curvature and Minimal graphs with linear growth

Abstract

We study minimal graphs with linear growth on complete manifolds $M^m$ with $\mathrm{Ric}\ge 0$. Under the further assumption that the $(m-2)$-th Ricci curvature in radial direction is bounded below by $Cr(x{)}^{-2}$, we prove that any such graph, if non-constant, forces tangent cones at infinity of $M$ to split off a line. Note that $M$ is not required to have Euclidean volume growth. We also show that $M$ may not split off any line. Our result parallels that obtained by Cheeger, Colding and Minicozzi for harmonic functions. The core of the paper is a new refinement of Korevaar's gradient estimate for minimal graphs, together with heat equation techniques.

arXiv comment34 pages. Final version, some further comments are included. To appear on Anal. PDE

References in corpus (8)

• Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows
• On the Distributional Hessian of the Distance Function
• On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth
• Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations
• Bernstein and half-space properties for minimal graphs under Ricci lower bounds
• Liouville type theorems for minimal graphs over manifolds
• Existence and non-existence of minimal graphic and $p$-harmonic functions
• Recent rigidity results for graphs with prescribed mean curvature

Cited by in corpus (3)

• Half-space Liouville-type theorems for minimal graphs with capillary boundary
• Liouville theorem for minimal graphs over manifolds of nonnegative Ricci curvature
• On splitting complete manifolds via infinity harmonic functions