Conformal complete metrics with prescribed non-negative Gaussian curvature in \({\mathbb{R}}^ 2\)
type: Item
instance of: scholarly article
title: Conformal complete metrics with prescribed non-negative Gaussian curvature in \({\mathbb{R}}^ 2\)
zbMATH Open document ID: 0592.53034
DOI: 10.1007/BF01394420
author: Patricio Aviles
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143325
review text: The paper deals with the existence of (complete) Riemannian metrics g on \({\mathbb{R}}^ 2\), conformal to the Euclidean metric \(g_ 0\) (by \(g=e^{2u}g_ 0)\) and possessing a prescribed Gaussian curvature \(k: {\mathbb{R}}^ 2\to {\mathbb{R}}\). This problem is intimately related to the nonlinear elliptic PDE \(\Delta u+k(x)e^{2u}=0\). It is always assumed that k is Hölder-continuous and that k(x) for large \(| x|\) is asymptotically comparable to some negative power of \(| x|\), in one sense or the other. For example, if k is positive somewhere and, asymptotically, \(| k(x)| \leq M/| x|^ b\) for some constants \(M>0\), \(b\geq 2\) then there is a complete metric g, for which an asymptotic rule like \[ (b-c)\cdot \ln | x| -2\tilde C\leq 2u(x)\leq (b-c)\cdot \ln | x| +2\tilde C \] holds. In case \(k\geq 0\) the condition \(b\geq 2\) may be relaxed to \(b>0\), provided some other assumptions are correspondingly sharpened. This is the most complicated case considered here. Several situations are exhibited where every solution of the elliptic PDE has an asymptotic behaviour like this. Finally, it is shown in the present frame that Cohn-Vossen's inequality is necessary and sufficient for the metric g to become complete. The methods are mainly analytical in nature, including e.g. the Leray- Schauder fixed point theory and the capacity of planar sets.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 3951482
zbMATH Keywords: complete Riemannian metric
zbMATH Keywords: prescribed Gaussian curvature
zbMATH Keywords: nonlinear elliptic PDE
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: Cohn-Vossen's inequality
reviewed by: Rolf Walter
MaRDI profile type: Publication
cites work: Q5179675
cites work: Isoperimetric Inequalities for a Nonlinear Eigenvalue Problem
cites work: The Gauss-Bonnet inequality and almost-geodesic loops
cites work: On Ricci curvature and geodesics
cites work: Elliptic operators and the decomposition of tensor fields
cites work: Curvature Measures
cites work: On a class of conformal metrics, with application to differential geometry in the large
cites work: Elliptic Partial Differential Equations of Second Order
cites work: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds
cites work: Q5509729
cites work: On subharmonic functions and differential geometry in the large
cites work: On the isoperimetric inequality on surfaces of variable Gaussian curvature
cites work: Q5637171
cites work: Curvature functions for compact 2-manifolds
cites work: Curvature functions for open 2-manifolds
cites work: On the equation \(\Delta u+Ke^{2u}=f\) and prescribed negative curvature in \({\mathbb{R}}^ 2\)
cites work: Q3961209
cites work: On the elliptic equation \(\Delta \omega + K(x)e^{2\omega} = 0\) and conformal metrics with prescribed Gaussian curvatures
cites work: ON THE EQUATION Δu+k(x)eu= 0
cites work: Isoperimetric Inequalities and Their Applications
cites work: Q5646042
cites work: Local behavior of solutions of quasi-linear equations
cites work: Best constant in Sobolev inequality
cites work: Removable singularities in Yang-Mills fields
cites work: Isoperimetric constants and the first eigenvalue of a compact riemannian manifold
OpenAlex ID: W2070826406
description: scientific article; zbMATH DE number 3951482