Abstract
It is proven that under suitable curvature assumptions, the image of the spherical Gauss map of a constant mean curvature hypersurface in Rn cannot be contained in any closed hemisphere, unless it lies in a great sphere. The proof depends on a uniqueness result for non-negative solutions of the generalised Emden-Fowler equation δu + K(x)U + R(x)u ω = 0 on a complete Riemannian manifold (M,g). An application of the latter result to stable minimal hypersurfaces is also given. By applying a variant of the maximum principle developed in [9], it is also shown that the Gauss map of a parallel mean curvature immersion of arbitrary codimension, cannot be obtained in a spherical cap of specified radius.
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