Geodesic
Some Sharp Estimates for Convex Hypersurfaces of Pinched Normal Curvature
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2015) no. 2, pp. 111-122 
K. Drach. Some Sharp Estimates for Convex Hypersurfaces of Pinched Normal Curvature. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2015) no. 2, pp. 111-122. http://geodesic.mathdoc.fr/item/JMAG_2015_11_2_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

For a convex domain $D$ bounded by the hypersurface $\partial D$ 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 $\partial D$, provided that normal curvatures of $\partial D$ are pinched by two positive constants. Furthermore, in the Euclidean case we also present sharp estimates for the quotient $R/r$. From the obtained estimates we derive stability results for almost umbilical hypersurfaces in the constant curvature spaces.

[1] A. Borisenko, V. Miquel, “Total Curvatures of Convex Hypersurfaces in Hyperbolic Space”, Illinois J. Math., 43:1 (1999), 61–78

[2] A. Borisenko, V. Miquel, “Comparison Theorems for Convex Hypersurfaces in Hadamard Manifolds”, Ann. Glob. Anal. Geom., 21:2 (2002), 191–202 | DOI

[3] A. Borisenko, K. Drach, “Closeness to Spheres of Hypersurfaces with Normal Curvature Bounded Below”, Sb. Math., 204:11 (2013), 1565–1583 | DOI

[4] R. Schneider, “Closed Convex Hypersurfaces with Curvature Restrictions”, Proc. Amer. Math. Soc., 103:4 (1988), 1201–1204 | DOI

[5] M. Gromov, “Stability and Pinching”, Seminari di Geometria (Bologna, 1992), ed. Ferri M., 55–97

[6] A. V. Pogorelov, Extrinsic Geometry of Convex Surfaces, Amer. Math. Soc., Providence, 1973

[7] K. Leichtweiss, “Nearly Umbilical Ovaloids in the $n$-Space are Close to Spheres”, Res. Math., 36 (1999), 102–109 | DOI

[8] V. I. Diskant, “Certain Estimates for Convex Surfaces with a Bounded Curvature Functions”, Sib. Math. J., 12 (1971), 109–125

[9] J. Scheuer, Quantitative Oscillation Estimates for Almost-Umbilical Closed Hypersurfaces in Euclidean Space, 2014, arXiv: 1404.2525

[10] R. Walter, “Some Analytical Properties of Geodesically Convex Sets”, Abh. Math. Sem. Univ. Hamburg, 45 (1976), 263–282 | DOI

[11] W. Blaschke, Kreis und Kugel, de Gruyter, Berlin, 1956

[12] H. Karcher, “Umkreise und Inkreise konvexer Kurven in der Spharischen und der Hyperbolischen Geometrie”, Math. Ann., 177 (1968), 122–132 | DOI

[13] R. Howard, “Blaschke's Rolling Theorem for Manifolds with Boundary”, Manuscripta Math., 99:4 (1999), 471–483 | DOI

[14] A. D. Milka, “On a Theorem of Schur and Schmidt”, Ukrain. Geom. Sb., 8 (1970), 95–102 (Russian)

[15] A. A. Borisenko, K. D. Drach, “Comparison Theorems for Support Functions of Hypersurfaces”, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky, 2015, no. 3, 11–16; CRM Preprint Series, No 1186, 2014