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Nonnegative Ricci
curvature and minimal graphs with linear growth
Giulio Colombo, Eddygledson S. Gama, Luciano Mari and Marco Rigoli |
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Vol. 17 (2024), No. 7, 2275–2310
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Abstract |
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We study minimal graphs with linear growth on complete manifolds with . Under the further assumption that the -th Ricci curvature in radial direction is bounded below by , we prove that any such graph, if nonconstant, forces tangent cones at infinity of to split off a line. Note that is not required to have Euclidean volume growth. We also show that 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. |
Keywords
Bernstein theorem, splitting, minimal graph, Ricci
curvature, tangent cone
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Mathematical Subject Classification
Primary: 53C21, 53C42
Secondary: 31C12, 53C24, 58J65
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Milestones
Received: 21 December 2021
Revised: 28 February 2023
Accepted: 8 May 2023
Published: 21 August 2024
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Authors |
| Giulio Colombo | |
| Dipartimento di Matematica “F.
Enriques” Università degli Studi di Milano Milano Italy |
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| Eddygledson S. Gama | |
| Departamento de Matemática Universidade Federal de Pernambuco Pernambuco Brazil |
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| Luciano Mari | |
| Dipartimento di Matematica “G.
Peano” Università degli Studi di Torino Torino Italy |
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| Marco Rigoli | |
| Dipartimento di Matematica “F.
Enriques” Università degli Studi di Milano Milano Italy |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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