Nonnegative Ricci curvature and minimal graphs with linear growth

Abstract

We study minimal graphs with linear growth on complete manifolds M m with Ric ≥ 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 nonconstant, 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.