Around Efimov's differential test for homeomorphism

TL;DR

This paper reviews generalizations and applications of Efimov's differential test for homeomorphism, focusing on surface theory, inverse functions, the Jacobian Conjecture, and stability of dynamical systems.

Contribution

It provides an overview of the extensions of Efimov's theorem and explores their relevance in various mathematical fields and problems.

Findings

Efimov's conditions imply the convexity of the image domain.

Generalizations extend the theorem to broader classes of functions.

Applications include insights into the Jacobian Conjecture and dynamical system stability.

Abstract

In 1968, N.\,V.~Efimov proved 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 $a(x)>0$ 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 $|\det f'(x)|\geqslant a(x)|\operatorname{curl}f(x)|+a^2(x)$ 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 $f(\mathbb{R}^2)$ homeomorphically.} Here $\operatorname{curl}f(x)$ 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 global inverse functions, as well as in the study of the Jacobian Conjecture and the global asymptotic stability of dynamical systems.