

Preprint 78/2009
The regularity of harmonic maps into spheres and applications to Bernstein problems
Jürgen Jost, Yuanlong Xin, and Ling Yang
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Submission date: 14. Dec. 2009
published in: Journal of differential geometry, 90 (2012) 1, p. 131-176
DOI number (of the published article): 10.4310/jdg/1335209491
Bibtex
MSC-Numbers: 58E20, 53A10
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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
quasi-linear 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.