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 nβsubmanifold 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