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
18-dic-2021
https://arxiv.org/abs/2112.09886
File in questo prodotto:
File Dimensione Formato  
2112.09886.pdf

accesso aperto

Tipologia: Pre-print (manoscritto inviato all'editore)
Dimensione 1.13 MB
Formato Adobe PDF
1.13 MB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/894002
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact