The curvature and topological properties of hypersurfaces with constant scalar curvature

Hypersurfaces with Constant Scalar Curvature

Let M be a complete two-dimensional surface immersed into the three-dimensional Euclidean space. Then a classical theorem of Hilbert says that when the curvature of M is a non-zero constant, M must be the sphere. On the other hand, when the curvature of M is zero, a theorem of Har tman-Nirenberg [4] says that M must be a plane or a cylinder. These two theorems complete the classification of com...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Hypersurfaces with Constant Scalar Curvature in a Hyperbolic Space Form

Let M be a complete hypersurface with constant normalized scalar curvature R in a hyperbolic space form H. We prove that if R̄ = R + 1 ≥ 0 and the norm square |h| of the second fundamental form of M satisfies nR̄ ≤ sup |h| ≤ n (n− 2)(nR̄− 2) [n(n− 1)R̄ − 4(n− 1)R̄ + n], then either sup |h| = nR̄ and M is a totally umbilical hypersurface; or sup |h| = n (n− 2)(nR̄− 2) [n(n− 1)R̄ − 4(n− 1)R̄ + n], and M i...

Constant Scalar Curvature Hypersurfaces in the Extended Schwarzschild Space-time

In this paper we study the spherically symmetric CSC hypersurfaces of the extended Schwarzschild space-time. Especially, we analyse the embedding equation and we find the family of solutions or slices that results varying a parameter c, for fixed CSC parameter a and fixed time-translation parameter t0. The parameter c represents the amount of variation of volume of the 3-geometry during the ‘ti...

06 Singular Minimal Hypersurfaces and Scalar Curvature

Finding obstructions to positive scalar curvature and getting structural insight is presently based on two competing approaches: one path which is most travelled works in the context of spin geometry and gives quite a direct link to topology (cf. [GL1-2] and [G]). The second, much less used but a priori more general method of attack analyzes minimal hypersurfaces within the manifold under consi...