A New Result About Almost Umbilical Hypersurfaces of Real Space Forms

Abstract

Abstract. In this short note, we prove that an almost umbilical compact hypersurface of a real space form with almost Codazzi umbilicity tensor is embedded, diffeomorphic and quasiisometric to a round sphere. Then, we derive a new characterization of geodesic spheres in space forms. Let (M n , g) be a connected and oriented compact Riemannian manifold isometrically immersed into the simply-connected real space form M n+1 (δ) of constant curvature δ. Let B be the second fundamental form of the hypersurface and H its mean curvature. Since we consider only hypersurfaces, we take B as the real-valued second fundamental form. We denote by τ = B − Hg the traceless part of the second fundamental form, also called umbilicity tensor. We say that M is totally umbilical if τ = 0.It is a well-known fact that a compact (without boundary) totally umbilical hypersurface of a simply connected real space form is a geodesic sphere. In the present note, we will investigate the natural question of the stability of this rigidity result. In other words, if a compact hypersurface of a real space form is almost umbilical, is this hypersurface close to a sphere? In what sense?Shiohama and Xu proved in [18,19] that if ||τ || n is small enough, then M n is homeomorphic to the sphere S n . Later on, we obtain quantitative results about the closeness of almost hypersurfaces to spheres in [6,16]. For instance, we prove in [16], always for hypersurfaces of 2010 Mathematics Subject Classification. 53A42, 53C20, 53C21.

Quantitative Oscillation Estimates for Almost-Umbilical Closed Hypersurfaces In euclidean Space

“…Note that a similar result by Roth has recently appeared in [13]. In more general ambient spaces, he proves quasi-isometry of hypersurfaces to the sphere under certain assumptions, including smallness of the gradient of the second fundamental form.…”

Stability from rigidity via umbilicity

“…The version for hyperbolic and spherical spaces was treated in [CZ14] and some optimality results in [CJ15]. Further results of almost umbilical type in terms of other L p -norms were deduced in [Per11,Rot13,Rot15,RS18,Sch15]. In particular we will make use of the following estimate from [RS18, Thm.…”

Diameter estimate for closed manifolds with positive scalar curvature

“…Recently, the second author [19] applied the Perelman's entropy functional to prove a sharp upper diameter bound for a compact shrinking Ricci soliton. In this direction, further development can be referred to [3,12,13,14] and references therein. Besides, we would like to mention that Bakry and Ledoux [5] applied a sharp Sobolev inequality to give an alternative proof of the Myers' diameter estimate.…”