Global injectivity of differentiable maps via W-condition in R²

2019/06/25 by Wei Liu, Liu, Wei
Biochemistry, Genetics and Molecular Biology · Mathematics · #14E07 #14E09 #14R15 #Advanced Differential Equations and Dynamical Systems #Biochemical Acid Research Studies #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Sphingolipid Metabolism and Signaling

paper · pdf · doi:10.48550/arxiv.1906.10648

openalex publication_date 2019/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the intrinsic relation between the global injectivity of differentiable local homeomorphisms F and the rate that tends to zero of Spec(F) in ℝ², where Spec(F) denotes the set of all (complex) eigenvalues of DF(x), for all x∈ ℝ². This depends on the W-condition deeply, which extends the *-condition and B-condition. The W-condition reveals the rate that tends to zero of real eigenvalues of DF can not exceed O(xln x(ln (ln x)/(lnln x))²)^-1 by the half-Reeb component method. This improves the theorems of Gutiérrez-Nguyen \citeGN07 and Rabanal \citeRR10. The W-condition is optimal for the half-Reeb component method in this paper setting.

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