Journal of Differential Geometry

The regularity of harmonic maps into spheres and applications to Bernsteing problems

Jűrgen Jost, Yuanlong Xin, and Ling Yang

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Abstract

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 a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasilinear elliptic systems.

We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial.

Article information

Source
J. Differential Geom., Volume 90, Number 1 (2012), 131-176.

Dates
First available in Project Euclid: 23 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1335209491

Digital Object Identifier
doi:10.4310/jdg/1335209491

Mathematical Reviews number (MathSciNet)
MR2891479

Zentralblatt MATH identifier
1250.53061

Citation

Jost, Jűrgen; Xin, Yuanlong; Yang, Ling. The regularity of harmonic maps into spheres and applications to Bernsteing problems. J. Differential Geom. 90 (2012), no. 1, 131--176. doi:10.4310/jdg/1335209491. https://projecteuclid.org/euclid.jdg/1335209491


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