Injectivity of differentiable maps R² --> R² at infinity

2006/01/13 by Carlos Gutiérrez, Carlos Gutierrez, Gutierrez, Carlos +2
Mathematics · #26B99 #37C10 (Secondary) #37E30 #58C25 (Primary) #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #math.DS #msc:26B99 #msc:37C10 #msc:37E30 #msc:58C25

paper · pdf · doi:10.48550/arxiv.math/0601325

to appear in Bulletin of the Brazilian Mathematical Society

openalex publication_date 2006/01/13 · arxiv created 2006/03/06 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

The main result given in Theorem~1.1 is a condition for a map X, defined on the complement of a disk D in R² with values in R², to be extended to a topological embedding of R², not necessarily surjective. The map X is supposed to be just differentiable with the condition that, for some e>0, at each point the eigenvalues of the differential do not belong to the real interval (-e,∞). The extension is obtained by restricting X to the complement of some larger disc. The result has important connections with the property of asymptotic stability at infinity for differentiable vector fields.

Related

• The Markus-Yamabe Conjecture for Differentiable vector fields of R²
• Global injectivity of differentiable maps via W-condition in R²
• Asymptotic stability at infinity for differentiable vector fields of the plane
• Examples of differentiable mappings into non-locally convex spaces
• Differential Calculus, Manifolds and Lie Groups over Arbitrary Infinite Fields