On a differential test of homeomorphism, found by N.V. Efimov

TL;DR

This paper reviews Efimov's 1968 theorem on homeomorphisms in the plane, exploring its generalizations, applications in surface theory, function theory, the Jacobian conjecture, and dynamical systems stability.

Contribution

It provides an overview of Efimov's theorem, its extensions, and its relevance across various mathematical fields and problems.

Findings

Efimov's theorem characterizes when a plane mapping is a homeomorphism.

Generalizations extend the theorem to broader classes of functions and surfaces.

Applications include insights into the Jacobian conjecture and stability analysis.

Abstract

In the year 1968 N.V. Efimov has proven the following remarkable theorem: \textit{Let $f:\mathbb R^2\to\mathbb R^2\in C^1$ be such that $\det f'(x)<0$ for all $x\in\mathbb R^2$ and let there exist a function $a=a(x)>0$ and constants $C_1\geqslant 0$, $C_2\geqslant 0$ such that the inequalities $|1/a(x)-1/a(y)|\leqslant C_1 |x-y|+C_2$ and $|\det f'(x)|\geqslant a(x)|{\rm curl\,}f(x)|+a^2(x)$ hold true for all $x, y\in\mathbb R^2$. Then $f(\mathbb R^2)$ is a convex domain and $f$ maps $\mathbb R^2$ onto $f(\mathbb R^2)$ homeomorhically.} Here ${\rm curl\,}f(x)$ stands for the curl of $f$ at $x\in\mathbb R^2$. This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of functions, as well as in the study of the Jacobian conjecture and global asymptotic stability of dynamical systems.