We give a new proof of Palmer's result [6] that the Wulff shapes are the only closed, oriented, stable hypersurfaces with constant anisotropic mean curvature. Our approach is based on the construction of a suitable testfunction in the anisotropic index form, thus generalizing the original proof of Barbosa, do Carmo [1].

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We consider immersed hypersurfaces in euclidean R n+1 which are stable with respect to an elliptic parametric functional with integrand F = F (N ) depending on normal directions only. We prove an integral curvature estimate provided that F is sufficiently close to the area integrand, extending the classical estimate of Schoen, Simon and Yau [19] for stable minimal hypersurfaces in R n+1 , as well as the pointwise estimate of Simon [22] for F -minimizing hypersurfaces. As a crucial point of our analysis we derive a generalized Simons inequality for the laplacian of the length of a weighted second fundamental form with respect to an abstract metric associated with F . As an application, we obtain a new Bernstein result for complete F -stable hypersurfaces of dimension n ≤ 5.

We consider immersed hypersurfaces in euclidean R n+1 which are stable with respect to an elliptic parametric functional of the form F (X) = M F (N) dµ. We prove a pointwise curvature estimate provided that n 5 and F is sufficiently close to the area integrand. This extends the pointwise curvature estimates of Schoen, Simon and Yau [Acta Math. 134 (1975) 275] for stable minimal hypersurfaces in R n+1 and of Simon [Math. Z. 154 (1977) 265] for minimizers of F . Our result follows from an integral curvature estimate and a generalized Simons inequality that were established recently [Calc. Var. Partial Differential Equations (2004),