Calculus of Variations and Geometric Measure Theory

G. Colombo - E. S. Gama - L. Mari - M. Rigoli

Non-negative Ricci curvature and minimal graphs with linear growth

created by mari1 on 21 Dec 2021


Submitted Paper

Inserted: 21 dec 2021

Pages: 33
Year: 2021

ArXiv: 2112.09886 PDF

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.

Links: arXiv: