Asian Journal of Mathematics

Volume 22 (2018)

Number 1

Submanifolds with constant Jordan angles and rigidity of the Lawson–Osserman cone

Pages: 75 – 110

DOI: http://dx.doi.org/10.4310/AJM.2018.v22.n1.a3

Authors

Jürgen Jost (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany)

Yuanlong Xin (School of Mathematical Sciences, Fudan University, Shanghai, China)

Ling Yang (School of Mathematical Sciences, Fudan University, Shanghai, China)

Abstract

The Lawson-Osserman cone is a four dimensional coassociative submanifold in $\mathbb{R}^7$ in terms of Harvey-Lawson calibrated geometries and the basic counterexample for Bernstein type results for minimal graphs of higher codimension in Euclidean space. We shall explore the geometry of this cone in terms of its basic property of constant Jordan angles and show a rigidity result within the class of coassociative submanifolds with constant Jordan angles. This will also shed new light on the higher codimension Bernstein problem.

Keywords

Lawson–Osserman cone, constant Jordan angles, coassociative submanifolds

2010 Mathematics Subject Classification

53A10, 58E20

Full Text (PDF format)

The first author is supported by the ERC Advanced Grant FP7-267087. ’The second author is supported by NSFC at grant No.11531012 and the third author is supported by NSFC at grant No. 11471078, 11622103. They are grateful to the Max Planck Institute for Mathematics in the Sciences in Leipzig for its hospitality and continuous support.

Received 29 October 2015

Accepted 13 October 2016

Published 10 May 2018