Locally convex hypersurfaces of prescribed curvature and boundary
type: Item
instance of: scholarly article
title: Locally convex hypersurfaces of prescribed curvature and boundary
zbMATH Open document ID: 0951.53004
author: F. Tomi
author: N. M. Ivochkina
published in: Calculus of Variations and Partial Differential Equations
publication date: 2000-12-13T00:00:00Z
review text: Frmo the introduction: We prove an existence theorem for locally strictly convex hypersurfaces in \(\mathbb{R}^{n+1}\) (or some flat \((n+1)\)-dimensional manifold), prescribed the boundary of the hypersurface and the curvature quotient \(S_{n,l}(\kappa_1,\dots, \kappa_n)\) as a function of the position. Here, we use the notations \(S_{n,l}:= S_n/S_l\), \(S_k\) the order \(k\) elementary symmetric function of \(n\) variables, \(S_0\equiv 1\) and \(\kappa_1,\dots, \kappa_n\) the principle curvatures. The case of the Gauss curvature to \(l=0\) and is included in our discussion. It is one of our goals to restrict the global shape of the hypersurfaces under investigation as little as possible. In particular, they need not be globally convex. More precisely, our surfaces are the image of sections in some smooth but otherwise arbitrary Euclidean straight line bundle. This includes Cartesian and radial graphs as special cases.
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1234912
zbMATH Keywords: Plateau problem
zbMATH Keywords: prescribed curvature
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: elementary symmetric function
zbMATH Keywords: Gauss curvature
MaRDI profile type: Publication
full work available at URL: s005260050110
OpenAlex ID: W2031519512
DOI: 10.1007/S005260050110
description: scientific article; zbMATH DE number 1234912