Around Efimov’s differential test for homeomorphism

In 1968, Efimov proved the following remarkable theorem: Letf: R²→ R²∈ C¹be such thatdet f^′(x) < 0 for allx∈ R²and let there exist a function a(x) > 0 and constantsC₁⩾ 0 , C₂⩾ 0 such that the inequalities| 1 / a(x) - 1 / a(y) | ⩽ C₁| x- y| + C₂and| det f^′(x) | ⩾ a(x) | curl f(x) | + a²(x) hold true for allx, y∈ R². Thenf(R²) is a convex domain andf mapsR²ontof(R²) homeomorphically. Here curl f(x) stands for the curl of f at x∈ R². 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.