Injectivity
of maps: connections with algebraic geometry, dynamical systems and geometries
of negative curvature
Motivated by our work on simply-connected embedded minimal surfaces we became interested in other injectivity questions, in special the problem of deciding when a local diffeomorphism of R^n is injective. Problems of this type occur in many branches of mathematics, ranging from dynamical systems, algebraic geometry, non-linear analysis (also in infinite dimensions), to applied areas like mathematical economics.
The solution by Gutierrez of the asymptotic stability conjecture in dynamical systems (solved also by Feller and Glutsuk) is a consequence of the following theorem of his: “a local diffeomorphism of the plane into itself is injective if the spectra of the linearizations miss the positive real axis". In [15] we use Pinchuk's counterexample to the real Jacobian conjecture to construct certain local diffeomorphisms which show that there is no higher dimensional version of the spectral theorem of Gutierrez. On the positive side, we prove in [15] a sharp injectivity theorem with "nearly spectral" hypothesis, similar in spirit to the one of Gutierrez that is valid in all dimensions. The key idea is the construction of a "non-linear adjoint" that allows for problems on injectivity to be converted into problems about surjectivity. One can then use degree theory to solve the associated existence problem. In [16] we extended a special but important case of the theorem of Gutierrez by geometrizing it to complete surfaces of non-positive curvature. The main result of [21] greatly improves the finite -- dimensional case of the well-known invertibility criterion of Hadamard, via the Palais-Smale condition. In our most recent works [22], [24], [25] our studies of global injectivity have become more topological. Using the fact that the Hopf map has no sections, as well as some classical estimates of Cronwall and Bieberbach in complex analysis, we study injectivity of local biholomorphisms. The main application states that a local biholomorphism of complex n - space into itself is injective if the pre-image of every complex line is a connected Riemann surface that can be uniquely embedded in CP^1, up to Mobius maps. This result was motivated by an earlier result whose proof uses the geodesic flow. In [24] and [25] we study the general topological set up underlying the Jacobian conjecture. We consider local diffeomorphisms of a manifold M onto R^n with the property that, outside of a stratified set in the target, the map is a cover map. Using topological and geometric ideas we prove that if the first homology of M vanishes then the degree of the cover is either 1 or $\infty$, provided that either the cover is regular or the stratified set is unknotted. We have discovered that several previously known results in the area of global injectivity, which were thought to be of an analytical nature, are in fact geometric/topological in nature. One of the best illustrations of this is a theorem that my student Cabral proved in his thesis: a self local diffeomorphism f of R^n is bijective iff the pre-image of every affine hyperplane is non-empty and acyclic: this result is logically stronger than the classical Hadamard theorem (if the norm of the inverse of the Jacobian matrix is uniformly bounded in R^n then the map is globally invertible). This is a thriving area of research and we are now exploring similar phenomena in infinite dimensions.
Related Publications
[12] F. Xavier, Invertibility of Bass-Connell-Wright polynomial maps, Math. Ann. 295, 163-166, (1993).
[15] F. J. Xavier (with B. Smyth), Injectivity of local diffeomorphisms from nearly spectral conditions, Jour. Diff. Equations., vol. 130, No. 2, 406-414, (1996).
[21] F. Xavier (with
[22] F. Xavier (with
[24] F. Xavier (with S. Nollet and L. Taylor) Birationality of étale maps via surgery (preprint)
[26] F. Xavier Using Gauss Maps to Detect Intersections (preprint)
[27] F. Xavier (with S. Nollet ) On Kulikov’s Problem (preprint)