On global diffeomorphisms of euclidian space

Abstract

As IS WELL KNOWN, the first attempt at characterizing global diffeomorphisms of IR” goes back to the 1906 work of Hadamard [l]. Since that time, progress in the global diffeomorphism problem has been rather slow. Notable contributions are those of Levy [2] who generalized Hadamard’s theorem to Banach spaces and of Banach and Mazur [3] and Caccioppoli [4] who proved the so-called “properness criterion” saying that a proper local homeomorphism of a Banach space is a global homeomorphism. The standard proofs of the theorems of Hadamard and Levy and of Banach, Mazur and Caccioppoli rely upon ideas of what is now called covering space theory, via the important “homotopy lifting property” of coverings (see e.g. [5]). This approach makes it clear that it is only lack of compactness of IR” that allows for the existence of noninjective local homeomorphisms when n L 2, the complex exponential when n = 2 being the most famous example, The theorems mentioned above were found again by various authors, sometimes in special cases and sometimes in a different framework but always through some variant of the same argument. A synthesis was made by Rheinboldt [6] in 1969. Part of his results were rediscovered five years later by Plastock [7] and given a somewhat simpler exposition for the special case of Banach spaces. Other contributions were made by Meisters and Olech [8] (global homeomorphisms of bounded domains in I?“), Meyer [9] (complementing the work of Levy) and Radulescu and Radulescu [lo] (expanding upon Plastock’s results). Many other articles have been devoted to special cases, where extra hypotheses are introduced consistently with the problem at hand. For instance, polynomial mappings only are relevant to the Jacobian conjecture of algebraic geometry, and hypotheses about the spectrum of the derivative at each point come up in a natural way in global stability questions for differential equations. The new results presented in this paper have been obtained through arguments from critical point theory and differential topology. Here, the central notion is that of admissible flow. Given a mapping F: Ii?” + IR”, the main property of what we call an F-admissible flow q: IR x (Ii?” F-‘(O)) + I?” F-‘(O) is that, for every x E IR” F-‘(O), the function t jF(q(t, x))l (where 1.1 is the Euclidean norm) is strictly increasing and tends to 0 (resp. 00) as t tends to --co (resp. 00). The basic feature we prove is that a local diffeomorphism F is a global one if and only if F-'(O) # (ZI and there is an F-admissible flow. This statement makes it clear that 0 plays a special role regarding F-admissibility, and the condition F-'(O) # 0 cannot be eliminated (as examples will show). The introduction of F-admissibility followed the author’s remark that, in Hilbert space, Hadamard’s theorem could be derived from the mountain pass theorem of Ambrosetti and