Strictly convex submanifolds and hypersurfaces of positive curvature.
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 \(H \cap \Gamma = \{ p \}.\) Then \(\Gamma\) bounds a smooh convex embedded surface of positive curvature. Further applications of the main result are found in boundary value problems via Monge-Ampère equations, smoothing of convex polytopes and an extension of Hadamard's ovaloid theorem to hypersurfaces with boundary.
scientific article; zbMATH DE number 1782679