• 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}
}
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… 

Figures from this paper

The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2
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
Spherical Bernstein theorems for codimension 1 and 2
A result of B.Solomon (On the Gauss map of an area-minimizing hypersurface. 1984. Journal of Differential Geometry, 19(1), 221-232.) says that a compact minimal hypersurface $M^k$ of the sphere
On the intersection of minimal hypersurfaces of $S^k$.
It is known since the work of Frankel that two compactly immersed minimal hypersurfaces in a manifold with positive Ricci curvature must have an intersection point. Several generalizations of this
Harmonic maps from surfaces of arbitrary genus into spheres
We relate the existence problem of harmonic maps into $S^2$ to the convex geometry of $S^2$. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces
Minimal graphs of arbitrary codimension in Euclidean space with bounded 2-dilation
For any Λ > 0, let Mn,Λ denote the space containing all locally Lipschitz minimal graphs of dimension n and of arbitrary codimension m in Euclidean space R with uniformly bounded 2-dilation Λ of
Rigidity Results for Self-Shrinking Surfaces in ℝ4
In this paper, we give some rigidity results for complete self-shrinking surfaces properly immersed in ℝ4 under some assumptions regarding their Gauss images. More precisely, we prove that this has
Graphical Mean Curvature Flow
  • A. Savas-Halilaj
  • Mathematics
    Nonlinear Analysis, Differential Equations, and Applications
  • 2021
In this survey article, we discuss recent developments on the mean curvature flow of graphical submanifolds, generated by smooth maps between Riemannian manifolds. We will see interesting
Mean curvature flow of area decreasing maps in codimension two
We consider the graphical mean curvature flow of strictly area decreasing maps f : M → N between a compact Riemannian manifold M of dimension m > 1 and a complete Riemannian surface N of bounded
Rigidity of the Hopf fibration
In this paper, we study minimal maps between euclidean spheres. The Hopf fibrations provide explicit examples of such minimal maps. Moreover, their corresponding graphs have second fundamental form

References

SHOWING 1-10 OF 22 REFERENCES
The regularity of harmonic maps into spheres and applications to Bernstein problems
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
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
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
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
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
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
TLDR
DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches and beschränktes Recht auf Nutzung dieses Dokuments, der Copyright bleibt bei den Herausgebern oder sonstigen Rechteinhaber vor.
...
1
2
3
...