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

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

The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension

We identify a region \Bbb{W}_{\f{1}{3}} in a Grassmann manifold \grs{n}{m}, not covered by a usual matrix coordinate chart, with the following important property. For a complete $n-$submanifold in \ir{n+m} \, (n\ge 3, m\ge2) with parallel mean curvature whose image under the Gauss map is contained in a compact subset K\subset\Bbb{W}_{\f{1}{3}}\subset\grs{n}{m}, we can construct strongly subharmonic functions and derive a priori estimates for the harmonic Gauss map. While we do not know yet how close our region is to being optimal in this respect, it is substantially larger than what could be achieved previously with other methods. Consequently, this enables us to obtain substantially stronger Bernstein type theorems in higher codimension than previously known.Comment: 36 page

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

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.Comment: 28 page