On the problem of realization of a given Gaussian curvature function
type: Item
instance of: scholarly article
title: On the problem of realization of a given Gaussian curvature function
zbMATH Open document ID: 0935.53001
DOI: 10.12775/TMNA.1998.013
published in: Topological Methods in Nonlinear Analysis
publication date: 2000-05-03T00:00:00Z
review text: The author investigates the local realization problem for the Gaussian curvature of a surface: Given a germ of a smooth function of two variables, find a surface whose Gaussian curvature is the given function. It is well known that any function germ \(g(x,y)\) is realizable as the Gaussian curvature of a surface \(z=f(x,y)\) if the curvature value at the central point is not zero, or if the curvature is vanishing at the central point but its differential does not vanish. The main result is, that the problem can be solved whenever \(g\) has a critical point of finite multiplicity at the origin. Similar results are proved for the local realization problem for the Hessian. The proofs proceed by solving the problem in the setting of formal power series.
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 35A25
zbMATH DE Number: 1270098
zbMATH Keywords: Gaussian curvature
zbMATH Keywords: Hessian
zbMATH Keywords: critical point
zbMATH Keywords: finite determinacy
author: Vładimir I. Arnold
reviewed by: Thomas Hasanis
MaRDI profile type: Publication
full work available at URL: tmna.1998.013
OpenAlex ID: W1541904541
description: scientific article; zbMATH DE number 1270098