Higher order mean curvature estimates for bounded complete hypersurfaces

Eigenvalue estimates for submanifolds with locally bounded mean curvature

We give lower bounds estimates for the first Dirichilet eigenvalues for domains Ω in submanifolds M ⊂ N with locally bounded mean curvature. These lower bounds depend on the local injectivity radius, local upper bound for sectional curvature of N and local bound for the mean cuvature of M . For sumanifolds with bounded mean curvature of Hadamard manifolds these lower bounds depends only on the ...

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...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

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, ...

Constant mean curvature hypersurfaces foliated by spheres ∗

We ask when a constant mean curvature n-submanifold foliated by spheres in one of the Euclidean, hyperbolic and Lorentz–Minkowski spaces (En+1, Hn+1 or Ln+1), is a hypersurface of revolution. In En+1 and Ln+1 we will assume that the spheres lie in parallel hyperplanes and in the case of hyperbolic space Hn+1, the spheres will be contained in parallel horospheres. Finally, Riemann examples in L3...