SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A SPHERE

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

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 scalar or mean curvature in a unit sphere

Let M be an n(n ≥ 3)-dimensional complete connected hypersurface in a unit sphere S(1). In this paper, we show that (1) if M has non-zero mean curvature and constant scalar curvature n(n−1)r and two distinct principal curvatures, one of which is simple, then M is isometric to the Riemannian product S( √ 1− c2) × Sn−1(c), c = n−2 nr if r ≥ n−2 n−1 and S ≤ (n−1)n(r−1)+2 n−2 + n−2 n(r−1)+2 . (2) i...

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...