The Gauss image of entire graphs of higher codimension and Bernstein type theorems

Jürgen Jost, Y. L. Xin, Ling Yang

Type: Article

Publication Date: 2012-06-06

Citations: 31

DOI: https://doi.org/10.1007/s00526-012-0533-0

Abstract

Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore enable us to improve substantially previous results for the Lawson–Osseman problem concerning the regularity of minimal submanifolds in higher codimension and to derive Bernstein type results.

Locations

Calculus of Variations and Partial Differential Equations - View - PDF

Similar Works

Action	Title	Year	Authors
+	The Gauss image of entire graphs of higher codimension and Bernstein type theorems	2010	Jürgen Jost Y. L. Xin Ling Yang
+	The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension	2011	Jürgen Jost Y. L. Xin Ling Yang
+ PDF Chat	The Geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension	2016	Jürgen Jost Yuanlong Xin Ling Yang
+	A spherical Bernstein theorem for minimal submanifolds of higher codimension	2014	Jürgen Jost Y. L. Xin Ling Yang
+ PDF Chat	A spherical Bernstein theorem for minimal submanifolds of higher codimension	2018	Jürgen Jost Y. L. Xin Ling Yang
+	The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2	2018	Renan Assimos Jürgen Jost
+	The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2	2019	Renan Assimos Martins
+	A Bernstein type theorem for minimal hypersurfaces via Gauss maps	2020	Qi Ding
+ PDF Chat	Bernstein and Half-Space Properties for Minimal Graphs Under Ricci Lower Bounds	2021	Giulio Colombo Marco Magliaro Luciano Mari Marco Rigoli
+	Curvature estimates for submanifolds with prescribed Gauss image and mean curvature	2008	Y. L. Xin
+	Curvature estimates for minimal submanifolds of higher codimension and small G-rank	2012	Jürgen Jost Y. L. Xin Ling Yang
+ PDF Chat	On Two-Dimensional Immersions of Prescribed Mean Curvature in $\mathbb R^n$	2008	Matthias Bergner Steffen Fröhlich
+	A Bernstein type theorem for minimal hypersurfaces via Gauss maps	2018	Qi Ding
+	Curvature estimates for minimal submanifolds of higher codimension	2007	Y. L. Xin Ling Yang
+ PDF Chat	A sharp Bernstein-type theorem for entire minimal graphs	2018	Alberto Farina
+	Harmonic Gauss maps of submanifolds of arbitrary codimension and applications	2019	Daniel Bustos Jaime Ripoll
+ PDF Chat	Curvature estimates for submanifolds with prescribed Gauss image and mean curvature	2009	Y. L. Xin
+	A sharp Bernstein-type theorem for entire minimal graphs	2017	Alberto Farina
+	A sharp Bernstein-type theorem for entire minimal graphs	2017	Alberto Farina
+	Submanifolds immersed in a warped product with density	2020	Jogli G. Araújo Henrique F. de Lima Wallace F. Gomes Marco Antonio L. Velásquez

Works That Cite This (30)

Action	Title	Year	Authors
+	Existence and non-existence of minimal graphs	2017	Qi Ding Jürgen Jost Y. L. Xin
+	Boundary regularity for minimal graphs and mean curvature flows with higher codimension	2017	Qi Ding Jürgen Jost Y. L. Xin
+	Dirichlet boundary values on Euclidean balls with infinitely many solutions for the minimal surface system	2019	Xiaowei Xu Ling Yang Yongsheng Zhang
+	Dirichlet boundary values on Euclidean balls with infinitely many solutions for the minimal surface system	2019	Xiaowei Xu Ling Yang Yongsheng Zhang
+	On Lawson-Osserman Constructions	2016	Xiaowei Xu Ling Yang Yongsheng Zhang
+	Self-expanders of the mean curvature flow	2020	Knut Smoczyk
+ PDF Chat	Translating solitons of the mean curvature flow	2015	Y. L. Xin
+ PDF Chat	The rigidity theorems of self shrinkers via Gauss maps	2016	Qi Ding Y. L. Xin Ling Yang
+	Singularities of mean curvature flow	2021	Yuanlong Xin
+ PDF Chat	Existence and non-existence of minimal graphs	2023	Qi Ding Jürgen Jost Y. L. Xin

Works Cited by This (22)

Action	Title	Year	Authors
+	Harmonic maps which solve a free-boundary problem.	1987	Robert Gulliver Jürgen Jost
+	A priori estimates for harmonic mappings.	1982	Mariano Giaquinta Stefan Hildebrandt
+ PDF Chat	An extrinsic rigidity theorem for minimal immersions from $S^2$ into $S^n$	1979	Jo ao Lucas Marquês Barbosa
+	Complete minimal surfaces in Euclideann-space	1967	Shiing-Shen Chern Robert Osserman
+ PDF Chat	On graphic Bernstein type results in higher codimension	2002	Mu‐Tao Wang
+ PDF Chat	A Bernstein result for minimal graphs of controlled growth	1990	Klaus Ecker Gerhard Huisken
+ PDF Chat	On the regularity of the minima of variational integrals	1982	Mariano Giaquinta Enrico Giusti
+	On Harnack's theorem for elliptic differential equations	1961	Jürgen Moser
+ PDF Chat	The tension field of the Gauss map	1970	Ernst A. Ruh Jaak Vilms
+ PDF Chat	Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system	1977	H. Blaine Lawson Robert Osserman