Some new characterizations of the Wulff shape

Abstract

We consider characterizations of a closed orientable connected hypersurface X : M → R n+1 in terms of anisotropic higher order mean curvature, which is an extension of the usual mean curvature. On the one hand, we improve the result of Leyla Onat (2010) and remove the convexity condition there. On the other hand, we generalize the results on rigidity of standard spheres to the anisotropic setting and show that if M satisfies some assumptions involving constant linear combinations of anisotropic mean curvatures or other relations, then it must be the Wulff shape, up to translations and homotheties.