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.
"The regularity of harmonic maps into spheres and applications to Bernsteing problems." J. Differential Geom. 90 (1) 131 - 176, January 2012. https://doi.org/10.4310/jdg/1335209491