Prescribed mean curvature flow of non-compact space-like Cauchy hypersurfaces

Uniqueness of Starshaped Compact Hypersurfaces With Prescribed m-th Mean Curvature in Hyperbolic Space

Let ψ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurfaceM for which ψ, when evaluated onM , coincides with them−th elementary symmetric function of principal curvatures of M for a given m? The corresponding existence and uniqueness problems in Euclidean space have been investigated by several authors in the mid 1980’s. Rec...

Starshaped Compact Hypersurfaces with Prescribed M-th Mean Curvature in Elliptic Space

We consider the problem of nding a compact starshaped hypresurface in a space form for which the normalized m-th elementary symmetric function of principal curvatures is a prescribed function. In this paper conditions for existence of at least one solution to a nonlinear second order elliptic equation of that problem are established in case of a space form with positive sectional curvature.

STARSHAPED COMPACT HYPERSURFACES WITH PRESCRIBED k-TH MEAN CURVATURE IN HYPERBOLIC SPACE

In this paper we consider the problem of finding a star-shaped compact hypersurface with prescribed k-th mean curvature in hyperbolic space. Under some sufficient conditions, we obtain an existence result by establishing a priori estimates and using degree theory arguments.

STARSHAPED COMPACT HYPERSURFACES WITH PRESCRIBED m−TH MEAN CURVATURE IN HYPERBOLIC SPACE

Let S be the unit sphere in the Euclidean space R, and let e be the standard metric on S induced from R. Suppose that (u, ρ) are the spherical coordinates in R, where u ∈ S, ρ ∈ [0,∞). By choosing the smooth function φ(ρ) := sinh ρ on [0,∞) we can define a Riemannian metric h on the set {(u, ρ) : u ∈ S, 0 ≤ ρ < ∞} as follows h = dρ + φ(ρ)e. This gives the space form R(−1) which is the hyperboli...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form