Ja n 20 08 STABILITY OF HYPERSURFACES WITH CONSTANT R - TH ANISOTROPIC MEAN CURVATURE

Stability of Hypersurfaces with Constant R-th Anisotropic Mean Curvature

Given a positive function F on S which satisfies a convexity condition, we define the r-th anisotropic mean curvature function H r for hypersurfaces in R n+1 which is a generalization of the usual r-th mean curvature function. Let X : M → R be an n-dimensional closed hypersurface with H r+1 =constant, for some r with 0 ≤ r ≤ n− 1, which is a critical point for a variational problem. We show tha...

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

Given a positive function F on S which satisfies a convexity condition, we define the r-th anisotropic mean curvature function H r for hypersurfaces in R n+1 which is a generalization of the usual r-th mean curvature function. LetX : M → R n+1 be an n-dimensional closed hypersurface with H r+1 = constant, for some r with 0≤ r ≤ n− 1, which is a critical point for a variational problem. We show ...

Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

Given a positive function F on S which satisfies a convexity condition, for 1 ≤ r ≤ n, we define the r-th anisotropic mean curvature function H r for hypersurfaces in R which is a generalization of the usual r-th mean curvature function. We prove that a compact embedded hypersurface without boundary in R with H r = constant is the Wulff shape, up to translations and homotheties. In case r = 1, ...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

A NEW CHARACTERIZATION OF r-STABLE HYPERSURFACES IN SPACE FORMS

In this paper we study the r-stability of closed hypersurfaces with constant r-th mean curvature in Riemannian manifolds of constant sectional curvature. In this setting, we obtain a characterization of the r-stable ones through of the analysis of the first eigenvalue of an operator naturally attached to the r-th mean curvature.