UNIQUENESS FOR A CLASS OF EMBEDDED WEINGARTEN HYPERSURFACES IN Sn+1

We apply the ‘non-collapsing’ technique, previously applied by Brendle in the proof of the Lawson conjecture and by Andrews and Li in the proof of the Pinkall-Sterling conjecture, to higher dimensional hypersurfaces satisfying a linear relation between the principal curvatures, under the additional assumption that the hypersurface has two distinct principal curvatures at each point. As special cases, the result gives simple new proofs of results of Otsuki for minimal hypersurfaces, and of Li and Wei for hypersurfaces with vanishing mth mean curvature.