Some Sharp Estimates for Convex Hypersurfaces of Pinched Normal Curvature

Abstract

For a convex domain D bounded by the hypersurface ∂D in a space of constant curvature we give sharp bounds on the width R − r of a spherical shell with radii R and r that can enclose ∂D, provided that normal curvatures of ∂D are pinched by two positive constants. Furthermore, in the Euclidean case we also present sharp estimates for the quotient R/r. 1. Preliminaries and the main results In [1] A. Borisenko and V. Miquel proved that a closed hypersurface with normal curvatures kn satisfying the inequality kn > 1 in the Lobachevsky space H(−1) can be put into a spherical shell between two concentric spheres of radii R and r such that the width R−r of the shell satis esR−r 6 ln 2. A similar estimate holds in Hadamard manifolds (see [2]). In [3] these results were extended for Riemannian manifolds of constant-signed sectional curvatures and hypersurfaces with normal curvatures bounded below. In the present paper we re ne some results from [3]. For this purpose we consider hypersurfaces with normal curvatures at any point and in any direction pinched by two positive constants. Such restriction allows us to obtain sharper estimates for the width R− r than in [3]. Furthermore, for such surfaces we are able to derive an upper bound on the quotient R/r, which can be arbitrarily large for a hypersurface with normal curvatures just bounded below. Let us denote by M(c) a complete simply connected m-dimensional Riemannian manifold of constant sectional curvatures equal to c. In order to state the main results, we need the following de nition. De nition 1.1. A hypersurface F ⊂ M(c) is said to be κ1, κ2-convex (with κ2 > κ1, and for c = 0 we assume that κ1 > 0, for c > 0 we assume that κ1 > 0 and for c √ −c), if for any point P ∈ F there exist two nested geodesic spheres S2 ⊂ S1 ⊂ M(c) of constant normal curvatures equal to, respectively, κ1 and κ2, and passing through P such that locally near P the hypersurface F lies inside S1 and outside S2. 2010 Mathematics Subject Classi cation. 53C40.