Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

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, ...

Mean Value Representations and Curvatures of Compact Convex Hypersurfaces

It is shown that the kernels for mean value representations of points in R in terms of the integrals over piecewise smooth hypersurfaces are divergence free vector fields defined by homogeneous functions of degree −(n+1), whose restrictions to the unit sphere are positive and orthogonal to the first harmonics. By Minkowski problem, such a function is the reciprocal of the composition of the Gau...

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...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Spacelike hypersurfaces in de Sitter space with constant higher-order mean curvature

ing from (2.6), we obtain that ∫ M ( H1Hr −Hr+1 〈N ,a〉dV = 0. (3.1) We know from Newton inequality [2] that Hr−1Hr+1 ≤ H2 r , where the equality implies that k1 = ··· = kn. Hence Hr−1 ( H1Hr −Hr+1 ≥Hr ( H1Hr−1−Hr ) . (3.2) It derives from Lemma 2.1 that 0≤H1/r r ≤H1/r−1 r−1 ≤ ··· ≤H1/2 2 ≤H1. (3.3) Thus we conclude that Hr−1 ( H1Hr −Hr+1 ≥Hr ( H1Hr1 −Hr ≥ 0, (3.4) and if r ≥ 2, the equalities h...