The space of parabolic affine spheres with fixed compact boundary
type: Item
instance of: scholarly article
title: The space of parabolic affine spheres with fixed compact boundary
zbMATH Open document ID: 0956.53011
author: N. Delaunay
published in: Monatshefte für Mathematik
publication date: 2001-03-07T00:00:00Z
review text: It is well-known that the graph of every solution \(f\) of the equation of Monge-Ampère type \(\text{det}({\partial^2f\over \partial x_i\partial x_j})= 1\) is an improper affine sphere. The authors investigate the moduli space \({\mathcal M}\) of all solutions of this equation on the exterior \(\Omega\) of a plane Jordan curve of class \({\mathcal C}^\infty\) which fulfill the boundary condition \(f|_{\partial\Omega}= \varphi\in{\mathcal C}^\infty(\partial\Omega)\) and are themselves of class \({\mathcal C}^{2,\alpha}(\overline\Omega)\). It turns out that \({\mathcal M}\) (if not empty) is a 5-dimensional differentiable manifold. In the proof, the main tool is the Lewy transformation \(L_f: z= L_f(w):= w+2{\partial f\over\partial\overline w}\) together with the (holomorphic) Lewy function \(F:F(z):= \overline w- 2{\partial f\over\partial w}\) (\(w:= x_1+ ix_2\)).
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1466224
zbMATH Keywords: Monge-Ampère type equation
zbMATH Keywords: improper affine sphere
zbMATH Keywords: module space
zbMATH Keywords: Lewy transformation
reviewed by: Kurt Leichtweiß
MaRDI profile type: Publication
full work available at URL: s006050050084
OpenAlex ID: W2076988775
DOI: 10.1007/S006050050084
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description: scientific article; zbMATH DE number 1466224