Hypersurfaces and nonnegative curvature

Hypersurfaces with nonegative Ricci curvature in $$\mathbb {H}^{n+1}$$ H n + 1

“…Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier in [2].…”

“…Moreover the presence of two points in the boundary at infinity is a rigidity condition that forces φ(M) to be an equidistant hypersurface. Recently, in [4] it is shown that the same conclusion as in [1,2] holds for immersed hypersurfaces.…”

“…In [10] it is shown that the asymptotic boundary of a complete proper embedding of R 2 into H 3 with nonnegative Gaussian curvature has a single point asymptotic boundary. In [1,2] it is shown that a 1 The author is the corresponding author and partially supported by NSFC 11571185 2 The author is partially supported by NSF DMS-1608782 complete, noncompact, embedded hypersurface φ : M n → H n+1 with nonnegative sectional curvature has at most two points in its asymptotic boundary. Moreover the presence of two points in the boundary at infinity is a rigidity condition that forces φ(M) to be an equidistant hypersurface.…”

“…To show that a connected asymptotic boundary can only be a single point, we pursue an avenue closely related to [1,2]. We observe that hypersurfaces embedded in hyperbolic space with nonnegative Ricci curvature give rise to height functions that are Euclidean n-subharmonic.…”

Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

“…(2) nonnegative Ricci curvature if κ i j =i κ j ≥ n − 2 for i = 1, · · · , n − 1; (3) nonnegative sectional curvature if κ i κ j ≥ 1 for 1 ≤ i < j ≤ n − 1; (4) horospherical convex (h-convex) if κ i ≥ 1 for i = 1, · · · , n − 1, where κ 1 , · · · , κ n−1 are the principal curvatures of the hypersurface (Σ, g) in H n , respectively. In fact, these convexity conditions are in strictly ascending order [1,2]. In [16], Ge, Wang and Wu investigated the k-th Gauss-Bonnet curvature L k on hypersurface (Σ, g) in H n , which is defined by…”