1,782 research outputs found
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 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
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