We study minimal graphs with linear growth on complete manifolds M with Ric≥0. Under the further assumption that the (dim 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.

Non-negative Ricci curvature and minimal graphs with linear growth / G. Colombo, E. Souza Gama, L. Mari, M. Rigoli. - (2021 Dec 18).

Non-negative Ricci curvature and minimal graphs with linear growth

G. Colombo
;
L. Mari;M. Rigoli
2021

Abstract

We study minimal graphs with linear growth on complete manifolds M with Ric≥0. Under the further assumption that the (dim 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.
Bernstein theorem; splitting; minimal graph; Ricci curvature; tangent cone
Settore MAT/03 - Geometria
Settore MAT/05 - Analisi Matematica
https://arxiv.org/abs/2112.09886
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/894002
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