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

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 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...

Spacelike hypersurfaces in Riemannian or Lorentzian space forms satisfying L_k(x)=Ax+b

We study connected orientable spacelike hypersurfaces $x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ where $L_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a fixed integer $0leq k

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form