• Corpus ID: 119579134

# The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2

```@article{Assimos2018TheGO,
title={The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2},
author={Renan Assimos and J{\"u}rgen Jost},
journal={arXiv: Differential Geometry},
year={2018}
}```
• Published 24 November 2018
• Mathematics
• arXiv: Differential Geometry
We develop a general method to construct subsets of complete Riemannian manifolds that cannot contain images of non-constant harmonic maps from compact manifolds. We apply our method to the special case where the harmonic map is the Gauss map of a minimal submanifold and the complete manifold is a Grassmannian. With the help of a result by Allard, we can study the graph case and have an approach to prove Bernstein-type theorems. This enables us to extend Moser's Bernstein theorem to codimension…
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## References

SHOWING 1-10 OF 22 REFERENCES
The regularity of harmonic maps into spheres and applications to Bernstein problems
• Mathematics
• 2009
We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of
Some rigidity theorems for minimal submanifolds of the sphere
I t is well known tha t the regularity of minimal submanifolds can be reduced to the study of minimal cones and hence to compact minimal submanifolds of the sphere. A phenomenon related to regularity
The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension
• Mathematics
• 2011
We identify a region \$\Bbb{W}_{\f{1}{3}}\$ in a Grassmann manifold \$\grs{n}{m}\$, not covered by a usual matrix coordinate chart, with the following important property. For a complete \$n-\$submanifold
Two Reports on Harmonic Maps
• Mathematics
• 1995
Harmonic maps between Riemannian manifolds are solutions of systems of nonlinear partial differential equations which appear in different contexts of differential geometry. They include holomorphic
Bernstein type theorems for higher codimension
• Mathematics
• 1999
Abstract. We show a Bernstein theorem for minimal graphs of arbitrary dimension and codimension under a bound on the slope that improves previous results and is independent of the dimension and
On the first variation of a varifold
Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional area integrand in M. Our main
The tension field of the Gauss map
• Mathematics
• 1970
In this paper it is shown that the tension field of the Gauss map can be identified with the covariant derivative of the mean curvature vector field. Since a map with vanishing tension field is
Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system
• Mathematics
• 1977
w 1. In t roduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 w 2. The minimal surface system . . . . . . . . . . . . . . . . . . . . . . . . . 2 w 3. A brief summary of
Some properties and applications of harmonic mappings
© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1978, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www.
Minimal cones and the Bernstein problem
• Mathematics, Computer Science
• 1969
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