STABILITY OF HYPERSURFACES WITH CONSTANT (r+1)-TH ANISOTROPIC MEAN CURVATURE

Given a positive function F on S-n which satisfies a convexity condition, we define the r-th anisotropic mean curvature function H-r(F) for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature function. Let X : M -> Rn+1 be an n-dimensional closed hypersurface with H-r+1(F) = constant, for some r with 0 <= r <= n - 1, which is a critical point for a variational problem. We show that X(M) is stable if and only if X(M) is the Wulff shape.