Non-negative Ricci curvature and Minimal graphs with linear growth

TL;DR

This paper investigates minimal graphs with linear growth on non-compact manifolds with non-negative Ricci curvature, revealing conditions under which tangent cones split off a line and refining gradient estimates using heat equation methods.

Contribution

It introduces a new gradient estimate refinement for minimal graphs and explores geometric splitting phenomena under specific Ricci curvature bounds.

Findings

Non-constant minimal graphs induce tangent cones to split off a line.

Manifolds may not necessarily split off any line.

Refined gradient estimates for minimal graphs are established.

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 $C r(x)^{-2}$, 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.