Symplectic geometry is a marvelous thing, recently evolved into symplectic topology, and having deep connections to, among other prominent things, number theory and quantum mechanics (its place of origin, so to speak). One gets symplectic structure in connection with Heisenberg (or Weyl) commutation relations, or (ultimately equivalently) courtesy of the presence of a closed non-degenerate 2-form on a manifold of even dimension: the form’s non-degeneracy translates to a natural explicit isomorphism between tangent and cotangent spaces, locally, and this is of course a prelude to a great deal of differential geometry as well as analysis.
In the book under review, the connections with QM and its generalizations are absent (for these see, e.g., the fantastic book, Harmonic Analysis in Phase Space, by Gerald B. Folland, and the equally fantastic compendium, Symplectic Geometry and Topology, edited by Yakov Eliashberg and Lisa Traynor), and the emphasis is on analysis — with extreme prejudice, so to speak. The point is that “Stein manifolds have symplectic geometry built into them, which is responsible for many phenomena in complex geometry and analysis.” Cieliebak and Eliashberg set themselves the goals of exploring these symplectic geometric aspects, which they term “the road from Stein to Weinstein,” and the attendant applications of symplectic geometry to “the complex geometric world of Stein manifolds, [i.e.] ‘the road from Weinstein to Stein.’” The work they present accordingly sports as its Part 4 a major discourse titled “From Stein to Weinstein and Back,” taking the reader from Weinstein structure to deformations of Stein structures.” Evidently this is material addressed at the cognoscenti and this work is by no means for the non-initiated.
Nonetheless, for the curious non-initiated (like me) here are a few definitions, coupled with some hand-waving, to give an idea of the lay of the land. J-convexity, the focus of Part 1, has to do with almost complex structures on smooth manifolds (which are, by definition, endomorphisms J generalizing i in the sense that J2 = –id), and amounts to the condition that a function φ is J-convex iff, for all tangent vectors X, to the manifold, ωφ(X,JX) > 0, where ωφ is a certain special 2-form associated to φ which is responsible for the manifold’s symplectic structure. Stein manifolds are characterized by the condition that they admit so-called “exhausting” J-convex functions, where “exhausting” means bounded below and that pre-images of compacta are compact. There are other definitions, notably, that a C-manifold is Stein iff it’s holomorphically embeddable in some Cn (and in this connection, see pp. 93–94), but the preceding characterization directly brings out the symplectic angle.
Now, if we have an almost complex structure on a manifold of even dimension > 4, with our φ also Morse without critical points of index > ½(the manifold’s dimension), then one proves (Eliashberg, 1990) that there exists an integrable complex structure Jo on the manifold homotopic to J, with the property that relative to Jo the manifold is in fact Stein: this is part and parcel of the existence of Stein structures, and this theme is the focus of Part 2 of the book. With Morse functions having reared their heads we find that Part 3 is in fact devoted to Morse-Smale theory, and we come across Smale cobordisms and Morse and Smale homotopies in this part of the book, i.e. in Part 3.
With Part 4, then, and as already mentioned, it’s time to play Stein off against Weinstein (“and back again,” to steal from J. R. R. Tolkien), so it is proper to make some noises about Weinstein structures. A Weinstein structure on an even-dimensional manifold is the data (ω, Ξ, φ), where ω is a symplectic form, φ a Morse function, and Ξ a “complete Liouville vector field which is gradient-like for φ” (where “Liouville” means that the indicated Lie derivative agrees with ω). The thrust of the Stein-Weinstein interplay is that one can go back and forth from Stein structures to Weinstein structures. Moreover, quasi-conjecturally, there is an overarching ideology in place: on a compact smooth manifold with boundary, one navigates directly from so-called Stein domain structures to so-called Weinstein domain structures, and then one can navigate back again provided one factors things through generalized Morse functions. What are these domain structures? Well, let’s just say that generalized Morse functions drive the train. It is also clear that for a non-initiate like me, standing at the station, the train has begun to pull away, and it’s time to tie things up: the book finishes with Part 5, devoted to Stein manifolds and symplectic topology, which is of course red hot these days — another bit of rationale for this work of serious scholarship
The book sports an appendix containing biographical notes on the ‘main characters’ in the story: Hartogs, Levi, Oka, (Henri) Cartan, (Karl) Stein, Grauert, Morse, Whitney, Smale, Gromov, and Weinstein. These make interesting reading in themselves.
It’s not a textbook, it’s not for the newbie; instead it’s an important contribution to the literature in the interface between symplectic geometry/topology and analysis, being focused on two themes, namely, the interplay between Stein manifolds and Weinstein manifolds, as already discussed above, and to explore “the extent to which these notions are flexible,” in the sense of satisfying h-principles (for the precise meaning of which you’ll have to read Chapter 7).
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.