Strictly convex submanifolds and hypersurfaces of positive curvature.
type: Item
instance of: scholarly article
title: Strictly convex submanifolds and hypersurfaces of positive curvature.
zbMATH Open document ID: 1068.53009
author: Mohammad Ghomi
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: The results in this paper involve the notion of strictly convex (embedded) submanifolds and the notion of ovaloid. A hyperplane \(H\) in \(\mathbb R^m\) with normal vector \(\xi\) is said to be a nonsingular support hyperplane of a \(C^2\)-submanifold \(M \subset \mathbb R^m\) at \(p \in M\) if \(H \cap M = \{ p\}\), \(M\) lies on one side of \(H,\) and \(H\) has contact of order one with \(M.\) The last condition means that for every \(C^2\)-curve \(\gamma\) through \(p,\) with nonzero tangent vector \(\gamma^\prime (0), \) that vector is perpendicular to \(\xi\) but \(\gamma^{\prime\prime}(0)\) is no longer so. If a nonsingular support hyperplane can be constructed at every point of \(M,\) then \(M\) is said to be strictly convex. An ovaloid is a closed convex hypersurface \(O\) with bounded radii of curvature, i.e. through every point of \(O\) there passes a ball containing \(O,\) and a ball contained in \(O.\) The main result of the paper is the following characterization of the strictly convex property: Let \(M \subset \mathbb R^m\) be a smooth compact embedded submanifold, possibly with boundary. Then \(M\) lies in a smooth ovaloid if and only if \(M\) is strictly convex. Furthermore, in that case, any finite number of nonsingular support hyperplanes at various points of \(M\) may be extended to a smooth distribution of nonsingular support hyperplanes along \(M.\) Moreover, for every smooth distribution of nonsingular support hyperplanes along \(M\) there exists a smooth ovaloid which contains \(M\) and is tangent to the given distribution. The main theorem is applied to supply a sufficient condition for a Jordan curve \(\Gamma \subset \mathbb R^3\) to bound a disk with a given metric of positive curvature (a problem posed by S. T. Yau). Namely, the author proves the following: Let \(\Gamma \subset \mathbb R^3\) be a smooth simple closed curve without inflection points. Suppose that through every point \(p\) of \(\Gamma\) there passes a plane \(H\) such that \…
reviewed by: Ivko Dimitrić
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 1782679
zbMATH Keywords: ovaloid
zbMATH Keywords: strictly convex submanifold
zbMATH Keywords: hypersurfaces of positive curvature
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090348111
OpenAlex ID: W1571627278
Wikidata QID: Q115202848
DOI: 10.4310/JDG/1090348111
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description: scientific article; zbMATH DE number 1782679