Constant mean curvature hypersurfaces foliated by spheres ∗

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Constant Mean

Let V be a maximal globally hyperbolic flat n+1–dimensional space–time with compact Cauchy surface of hyperbolic type. We prove that V is globally foliated by constant mean curvature hypersurfaces Mτ , with mean curvature τ taking all values in (−∞, 0). For n ≥ 3, define the rescaled volume of Mτ by H = |τ | Vol(M, g), where g is the induced metric. Then H ≥ nVol(M, g0) where g0 is the hyperbol...

Aleksandrov’s Theorem: Closed Surfaces with Constant Mean Curvature

We present Aleksandrov’s proof that the only connected, closed, ndimensional C hypersurfaces (in R) of constant mean curvature are the spheres.

A Gluing Construction for Prescribed Mean Curvature

The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is the vanishing of the sum of certain integral moments of the spheres with respe...

Linear Weingarten hypersurfaces in a unit sphere

In this paper, by modifying Cheng-Yau$'$s technique to complete hypersurfaces in $S^{n+1}(1)$, we prove a rigidity theorem under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related which improve the result of [H. Li, Hypersurfaces with constant scalar curvature in space forms, {em Math. Ann.} {305} (1996), 665--672].