Mathematics – Differential Geometry
Scientific paper
2010-10-18
Contemporary Problems of Mathematics and Mechanics (Sovremennye Problemy Matematiki i Mekhaniki), Vol. 6, no. 2 (2011), 18-26.
Mathematics
Differential Geometry
In Russian, 10 pages
Scientific paper
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.
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