Preprint 78/2009

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

Jürgen Jost, Yuanlong Xin, and Ling Yang

Contact the author: Please use for correspondence this email.
Submission date: 14. Dec. 2009
published in: Journal of differential geometry, 90 (2012) 1, p. 131-176 
MSC-Numbers: 58E20, 53A10
Download full preprint: PDF (444 kB), PS ziped (497 kB)

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.

20.03.2019, 02:13