Nonnegative Ricci curvature and minimal graphs with linear growth (Q6612314)

The paper studies minimal graphs with linear growth on complete manifolds \(M\) of dimension \(m\) with \(\mathrm{Ric}\ge0\). Assuming that the \((m-2)\)-th Ricci curvature in radial direction is bounded below by \(Cr(x)^{-2}\), it is shown that any such graph, if nonconstant, forces tangent cones at infinity of \(M\) to split off a line. Here \(M\) is not required to have Euclidean volume growth. It is also proved that \(M\) may not split off any line. The obtained results parallel the ones obtained for harmonic functions in [\textit{J. Cheeger} et al., Geom. Funct. Anal. 5, No. 6, 948--954 (1995; Zbl 0871.53032)]. The core of the paper is a new refinement of the Korevaar gradient estimate for minimal graphs in [\textit{N. Korevaar}, Proc. Symp. Pure Math. 45, Pt. 2, 81--89 (1986)].