1.1. From the Publisher.

This book presents the first steps of a theory of confoliations designed to link geometry and topology of three-dimensional contact structures with the geometry and topology of codimension-one foliations on three-dimensional manifolds. Developing almost independently, these theories at first glance belonged to two different worlds: The theory of foliations is part of topology and dynamical systems, while contact geometry is the odd-dimensional "brother" of symplectic geometry.

However, both theories have developed a number of striking similarities. Confoliations - which interpolate between contact structures and codimension-one foliations - should help us to understand better links between the two theories. These links provide tools for transporting results from one field to the other.

Features:

A unified approach to the topology of codimension-one foliations and contact geometry.

Insight on the geometric nature of integrability.

New results, in particular on the perturbation of confoliations into contact structures.

Readership

Graduate students and research mathematicians working in differential and symplectic geometry, low-dimensional topology, the theory of foliations and several complex variables; some physicists and engineers.

1.2. From the Introduction.

In these notes we present the first steps of a theory of confoliations which is designed to link the geometry and topology of 3-dimensional contact structures with the geometry and topology of codimension 1 foliations on 3-manifolds. A confoliation is a mixed structure which interpolates between a codimension 1 foliation and a contact structure. This object (without the name) first appeared in work of Steve Altschuler. In this paper we will be concerned exclusively with the 3-dimensional case, although confoliations can be defined on higher dimensional manifolds as well.

Foliations and contact structures have been studied practically independently. Indeed at the first glance, these objects belong to two different worlds. The theory of foliations is a part of topology and dynamical systems while contact geometry is an odd-dimensional brother of symplectic geometry. However, the two theories have developed a number of striking similarities. In each case, an understanding developed that additional restrictions are important on foliations and contact structures to make them interesting and useful for applications, for otherwise , these structures are so flexible that they fit anywhere, hence produce no information about the topology of the underlying manifold. This extra flexibility is caused by the appearance of Reeb components in the case of foliations, and by the overtwisting phenomenon in the case of contact structures. In order to make the structures more rigid in the context of foliations the theory of taut foliations was developed, as well as the related theory of essential laminations. In the parallel world of contact geometry, the theory of tight contact structures were developed for similar purposes.

The theory of confoliations should help us to better understand links between the two theories and should provide an instrument for transporting the results from
one field to the other.

1.3. Review by Hansjörg Geiges.
Mathematical Reviews MR1483314 (98m:53042).

As this monograph shows and, one can expect future research will confirm, this unifying approach to the hitherto seemingly independent theories of foliations and contact structures is extremely fruitful. In a first preparatory chapter the authors give an extensive discussion of the geometric nature of integrability (of plane fields), in particular for plane fields transverse to 1-dimensional bundles. The second chapter studies the possibility of perturbing a (con)foliation into a contact structure. ... In the third chapter the authors concentrate their attention on taut foliations and tight contact structures (the complementary classes - foliations with generalised Reeb components [resp. overtwisted contact structures] - being more or less completely understood). ... This chapter, much as the preceding ones, is a veritable cornucopia of ideas and surprising links between contact geometry and the theory of foliations.

1.4. Review by C B Thomas.
Bulletin of the London Mathematical Society 31 (1999), 636-637.

... a confoliation is intermediate between a foliation (equal to zero everywhere) and a contact structure (strictly positive everywhere). The purpose of this monograph is to introduce the reader to confoliations, and to use their properties to show how the 1-form defining a codimension-one foliation on M can be continuously deformed to a 1-form defining a contact structure. ... Go out and buy this book before the pound weakens; the American Mathematical Society will sell it to you for at most sixteen dollars.

2.1. From the Publisher.

In differential geometry and topology one often deals with systems of partial differential equations, as well as partial differential inequalities, that have infinitely many solutions whatever boundary conditions are imposed. It was discovered in the fifties that the solvability of differential relations (i.e. equations and inequalities) of this kind can often be reduced to a problem of a purely homotopy-theoretic nature. One says in this case that the corresponding differential relation satisfies the $h$-principle. Two famous examples of the $h$-principle, the Nash-Kuiper $C^{1}$-isometric embedding theory in Riemannian geometry and the Smale-Hirsch immersion theory in differential topology, were later transformed by Gromov into powerful general methods for establishing the $h$-principle.

The authors cover two main methods for proving the $h$-principle: holonomic approximation and convex integration. The reader will find that, with a few notable exceptions, most instances of the $h$-principle can be treated by the methods considered here. A special emphasis in the book is made on applications to symplectic and contact geometry.

Gromov's famous book "Partial Differential Relations", which is devoted to the same subject, is an encyclopaedia of the h-principle, written for experts, while the present book is the first broadly accessible exposition of the theory and its applications. The book would be an excellent text for a graduate course on geometric methods for solving partial differential equations and inequalities. Geometers, topologists and analysts will also find much value in this very readable exposition of an important and remarkable topic.

Readership

Graduate students and research mathematicians interested in global analysis and analysis on manifolds.

2.2. From the Preface.

A partial differential relation $R$ is any condition imposed on the partial derivatives of an unknown function. A solution of $R$ is any function which satisfies this relation.

The classical partial differential relations, mostly rooted in Physics, are usually described by (systems of) equations. Moreover, the corresponding systems of equations are mostly determined: the number of unknown functions is equal to the number of equations. Given appropriate boundary conditions, such a differential relation usually has a unique solution. In some cases this solution can be found using certain analytical methods (potential theory, Fourier method and so on).

In differential geometry and topology one often deals with systems of partial differential equations, as well as partial differential inequalities, which have infinitely many solutions whatever boundary conditions are imposed. Moreover, sometimes solutions of these differential relations are $C^{0}$-dense in the corresponding space of functions or mappings. The systems of differential equations in question are usually (but not necessarily) underdetermined. We discuss in this book homotopical methods for solving this kind of differential relations. Any differential relation has an underlying algebraic relation which one gets by substituting derivatives by new independent variables. A solution of the corresponding algebraic relation is called a formal solution of the original differential relation $R$. Its existence is a necessary condition for the solvability of $R$, and it is a natural starting point for exploring $R$. Then one can try to deform the formal solution into a genuine solution. We say that the $h$-principle holds for a differential relation $R$ if any formal solution of $R$ can be deformed into a genuine solution.
...
There is an opinion that "the $h$-principle is the hardest part of Gromov's work to popularise." We have written our book in order to improve the situation. We consider here two geometrical methods: holonomic approximation, which is a version of the method of continuous sheaves, and convex integration. We do not pretend to cover here the content of Gromov's book, but rather want to prepare and motivate the reader to look for hidden treasures there. On the other hand, the reader interested in applications will find that with a few notable exceptions (e.g. Lohkamp's theory of negative Ricci curvature and Donaldson's theory of approximately holomorphic sections) most instances of the h-principle which are known today can be treated by the methods considered in the present book.
...
Acknowledgements. The book was partially written while the second author visited the Department of Mathematics of Stanford University, and the first author visited the Mathematical Institute of Leiden University and the Institute for Advanced Study at Princeton. The authors thank the host institutions for their hospitality. While writing this book the authors were partially supported by the National Science Foundation. The first author also acknowledges the support of The Veblen Fund during his stay at the IAS.

2.3. Table of Contents.

Part 1. Holonomic approximation.

Chapter 1. Jets and holonomy.
Chapter 2. Thom transversality theorem.
Chapter 3. Holonomic approximation.
Chapter 4. Applications.

Part 2. Differential relations and Gromov's $h$-principle.

Chapter 5. Differential relations.
Chapter 6. Homotopy principle.
Chapter 7. Open Diff $V$-invariant differential relations.
Chapter 8. Applications to closed manifolds.

Part 3. The homotopy principle in symplectic geometry.

Chapter 9. Symplectic and contact basics.
Chapter 10. Symplectic and contact structures on open manifolds.
Chapter 11. Symplectic and contact structures on closed manifolds.
Chapter 12. Embeddings into symplectic and contact manifolds.
Chapter 13. Microflexibility and holonomic $R$-approximation.
Chapter 14. First applications of microflexibility.
Chapter 15. Microflexible $U$-invariant differential relations.
Chapter 16. Further applications to symplectic geometry.

Part 4. Convex integration.

Chapter 17. One-dimensional convex integration.
Chapter 18. Homotopy principle for ample differential relations.
Chapter 19. Directed immersions and embeddings.
Chapter 20. First order linear differential operators.
Chapter 21. Nash-Kuiper theorem.

2.4. Review by: John B Etnyre.
Mathematical Reviews MR1909245 (2003g:53164).

The book under review, as the title implies, is an introduction to the h-principle. Anyone who has spent time trying to mine the beautiful depths of Gromov's book [M L Gromov, Partial differential relations, 1986] or the research literature on the h-principle will certainly welcome this book.

The basic idea of the $h$-principle is as follows: any differential equation (or relation) can be interpreted as a subset $S$ of an appropriate jet space and a solution is simply an appropriate function whose jet lies in S Gromov's strategy for solving differential relations was to first find a section of the jet bundle whose image is in $S$ and then try to show that there is a function whose jet agrees with this section. The first part of this program frequently has an algebraic (or sometime geometric) flavour, while the second part is usually more analytic. A differential equation (or relation) satisfies an $h$-principle if the second part of the above strategy follows automatically (though not necessarily easily) from the first part. Said another way, an equation (or relation) satisfies an $h$-principle if its solvability is determined by some algebraic (or geometric) data associated to the problem.

There are many ways to try to show that a problem satisfies an h-principle; this book considers two: holonomic approximations and convex integration. Holonomic approximations are a relatively recent variant of Gromov's continuous sheaves methods developed by the authors in [Essays on geometry and related topics, 2001].

The $h$-principle has been a useful way to prove, or interpret prior proofs of, results in topology and geometry. This book describes many of these applications with a specific emphasis on symplectic and contact geometry and various embedding and immersion theorems. In addition one can find a good introduction to the literature.

3.1. From the Publisher.

This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine complex (a.k.a. Stein) manifolds have canonically built into them symplectic geometry which is responsible for many phenomena in complex geometry and analysis. The goal of the book is the exploration of this symplectic geometry (the road from "Stein to Weinstein") and its applications in the complex geometric world of Stein manifolds (the road "back"). This is the first book which systematically explores this connection, thus providing a new approach to the classical subject of Stein manifolds. It also contains the first detailed investigation of Weinstein manifolds, the symplectic counterparts of Stein manifolds, which play an important role in symplectic and contact topology.

Assuming only a general background from differential topology, the book provides introductions to the various techniques from the theory of functions of several complex variables, symplectic geometry, $h$-principles, and Morse theory that enter the proofs of the main results. The main results of the book are original results of the authors, and several of these results appear here for the first time. The book will be beneficial for all students and mathematicians interested in geometric aspects of complex analysis, symplectic and contact topology, and the interconnections between these subjects.

Readership

Graduate students and research mathematicians interested in functions in several complex variables and symplectic and contact topology.

3.2. Contents.

Chapter 1. Introduction.

Part 1. $J$-convexity.

Chapter 2. $J$-convex functions and hypersurfaces.
Chapter 3. Smoothing.
Chapter 4. Shapes for $i$-convex hypersurfaces.
Chapter 5. Some complex analysis.

Part 2. Existence of Stein structures.

Chapter 6. Symplectic and contact preliminaries.
Chapter 7. The $h$-principles.
Chapter 8. The existence theorem.

Part 3. Morse-Smale theory for $J$-convex functions.

Chapter 9. Recollections from Morse theory.
Chapter 10. Modifications of $J$-convex Morse functions.

Part 4. From Stein to Weinstein and back.

Chapter 11. Weinstein structures.
Chapter 12. Modifications of Weinstein structures.
Chapter 13. Existence revisited.
Chapter 14. Deformations of flexible Weinstein structures.
Chapter 15. Deformations of Stein structures.

Part 5. Stein manifolds and symplectic topology.

Chapter 16. Stein manifolds of complex dimension two.
Chapter 17. Exotic Stein structures.

Appendix A. Some algebraic topology.
Appendix B. Obstructions to formal Legendrian isotopies.
Appendix C. Biographical notes on the main characters.

3.3. From the Preface.

In Spring 1996 Yasha Eliashberg gave a Nachdiplomvorlesung (a one semester graduate course) "Symplectic geometry of Stein manifolds" at ETH Zürich. Kai Cieliebak, at the time a graduate student at ETH, was assigned the task to take notes for this course, with the goal of having lecture notes ready for publication by the end of the course. At the end of the semester we had some 70 pages of typed notes, but they were nowhere close to being publishable. So we buried the idea of ever turning these notes into a book.

Seven years later Kai spent his first sabbatical at the Mathematical Sciences Research Institute (MSRI) in Berkeley. By that time, through work of Donaldson and others on approximately holomorphic sections on the one hand, and gluing formulas for holomorphic curves on the other hand, Weinstein manifolds had been recognised as fundamental objects in symplectic topology. Encouraged by the increasing interest in the subject, we dug out the old lecture notes and began turning them into a monograph on Stein and Weinstein manifolds.

Work on the book has continued on and off since then, with most progress happening during Kai's numerous visits to Stanford University and another sabbatical 2009 that we both spent at MSRI. Over this period of almost 10 years, the content of the book has been repeatedly changed and its scope significantly extended. Some of these changes and extensions were due to our improved understanding of the subject (e.g., a quantitative version of $J$-convexity which is preserved under approximately holomorphic diffeomorphisms), others due to new developments such as the construction of exotic Stein structures by Seidel and Smith, McLean, and others since 2005, and Murphy's h-principle for loose Legendrian knots in 2011. In fact, the present formulation of the main theorems in the book only became clear about a year ago. As a result of this process, only a few lines of the original lecture notes have survived in the final text (in Chapters 2-4).

The purpose of the book has also evolved over the past decade. Our original goal was a complete and detailed exposition of the existence theorem for Stein structures in [Yakov Eliashberg, Topological characterisation of Stein manifolds of dimension > 2]. While this remains an important goal, which we try to achieve in Chapters 2-8, the book has evolved around the following two broader themes: The first one, as indicated by the title, is the correspondence between the complex analytic notion of a Stein manifold and the symplectic notion of a Weinstein manifold. The second one is the extent to which these structures are flexible, i.e., satisfy an h-principle. In fact, until recently we believed the border between flexibility and rigidity to run between subcritical and critical structures, but Murphy's h-principle extends flexibility well into the critical range.

The book is roughly divided into "complex" and "symplectic" chapters. Thus Chapters 2-5 and 8-10 can be read as an exposition of the theory of $J$-convex functions on Stein manifolds, while Chapters 6-7, 9 and 11-14 provide an introduction to Weinstein manifolds and their deformations. However, our selection of material on both the complex and symplectic side is by no means representative for the respective fields. Thus on the complex side we focus only on topological aspects of Stein manifolds, ignoring most of the beautiful subject of several complex variables. On the symplectic side, the most notable omission is the relationship between Weinstein domains and Lefschetz fibrations over the disc.

Over the past 16 years we both gave many lecture courses, seminars, and talks on the subject of this book not only at our home institutions, Ludwig-Maximilians-Universität München and Stanford University, but also at various other places such as the Forschungsinstitut für Mathematik at ETH Zürich, University of Pennsylvania in Philadelphia, Columbia University in New York, the Courant Institute of Mathematical Sciences in New York, University of California in Berkeley, Washington University in St Louis, the Mathematical Sciences Research Institute in Berkeley, the Institute for Advanced Study in Princeton, and the Alfréd Rényi Institute of Mathematics in Budapest. We thank all these institutions for their support and hospitality.

3.4. Review by: Chris M Wendl.
Mathematical Reviews MR3012475.

In 1990, Y M Eliashberg published the startling result that a manifold of even dimension greater than four admits the structure of a Stein manifold (i.e. a complex manifold that can be properly and holomorphically embedded into some $\mathbb{C}^{N}$) if and only if certain obviously necessary homotopy-theoretic conditions are satisfied. The proof was based on a combination of Morse-theoretic ideas with several powerful h-principles emerging from the "soft" side of symplectic geometry, and it thus initiated the study of Stein manifolds from a symplectic topological perspective. In subsequent years, work of Eliashberg and M Gromov [Several complex variables and complex geometry, Part 2 (1989)], A D Weinstein [Hokkaido Math. J. 20 (1991)], R E Gompf [Ann. of Math. (2) 148 (1998)] and others revealed further deep insights into the symplectic nature of Stein manifolds: in particular, it became a well-known folk theorem that the essentially complex geometric objects we call "Stein structures" are in some sense equivalent to certain Morse-theoretic symplectic data called "Weinstein structures", so that the two notions can be (and in the symplectic literature often are) used interchangeably for most purposes. The impact that these results have had on symplectic topology over the last 20 years cannot be overstated, though their proofs have until now seemed largely inaccessible to all but a few experts. Much more recently, the exciting discovery by E Murphy ["Loose Legendrian embeddings in high dimensional contact manifolds"] of a flexible class of Legendrian submanifolds has brought the subject to maturity, leading in particular to a dramatically improved understanding of flexibility in Stein manifolds.

The book under review provides the first comprehensive account of this story, including detailed proofs of several important results that have been considered "standard" among symplectic topologists for many years but were only fully understood by a very few people, as well as new results about flexible Stein structures that have quickly come to be regarded as fundamental in the subject.
...
I would like to offer a tip for first-time readers: the authors have produced two shorter expository articles ["Stein structures: existence and flexibility"; "Flexible Weinstein manifolds"] that sketch the main ideas of the proofs in the book, and it is worth reading through these to get the big picture before delving into the (often quite intricate) details. The first, based on two lecture series given by K Cieliebak in 2012 at the IAS and at a summer school in Budapest, is a very digestible explanation of the existence theorem and flexibility in the subcritical case, while the second goes into more detail about loose Legendrians and flexible Weinstein structures.
...
The book concludes with three appendices, of which the first two review material from algebraic topology, and the third provides biographical notes on several of the mathematicians who have played major roles in the study of Stein manifolds. Unlike Chapters 2 through 17, Appendix C can easily be read just before going to sleep for the night, and I highly recommend it.

3.5. Review by: Michael Berg.
Mathematical Association of America (15 March 2013).
https://maa.org/press/maa-reviews/from-stein-to-weinstein-and-back-symplectic-geometry-of-affine-complex-manifolds

Symplectic geometry is a marvellous 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 generalisations 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 $J$ generalising $i$ in the sense that $J^{2} = -id$), and amounts to the condition that a function $\phi$ is $J$-convex iff, for all tangent vectors $X$, to the manifold, $\omega_{\phi}(X, JX) > 0$, where $\omega_{\phi}$ is a certain special 2-form associated to $\phi$ which is responsible for the manifold's symplectic structure. Stein manifolds are characterised 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 $\mathbb{C}$-manifold is Stein iff it's holomorphically embeddable in some $\mathbb{C}^{n}$, but the preceding characterisation directly brings out the symplectic angle.

Now, if we have an almost complex structure on a manifold of even dimension > 4, with our $\phi$ also Morse without critical points of index > $\large\frac{1}{2}\normalsize$ (the manifold's dimension), then one proves (Eliashberg, 1990) that there exists an integrable complex structure $J^{0}$ on the manifold homotopic to $J$, with the property that relative to $J^{0}$ 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 $(\omega, \xi, \phi)$, where $\omega$ is a symplectic form, $\phi$ a Morse function, and $\xi$ a "complete Liouville vector field which is gradient-like for $\phi$" (where "Liouville" means that the indicated Lie derivative agrees with $\omega$). 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 generalised Morse functions. What are these domain structures? Well, let's just say that generalised 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).

3.6. Review by: Alexandru Oancea.
Bulletin of the American Mathematical Society 52 (3) (2015), 521-530.

The book under review is a landmark piece of work that establishes a fundamental bridge between complex geometry and symplectic geometry. It is both a research monograph of the deepest kind and a panoramic companion to the two fields. The main characters are, respectively, Stein manifolds on the complex side and Weinstein manifolds on the symplectic side. The connection between the two is established by studying the corresponding geometric structures up to homotopy, or deformation, a context in which the h-principle plays a fundamental role.
...
As the authors mention in the introduction to the book, their "original goal was a complete and detailed exposition of the existence theorem for Stein structures in [Yakov Eliashberg, Topological characterisation of Stein manifolds of dimension > 2]." The outcome is a fascinating exploration and recasting of half a century of mathematics, embedded into a landmark research monograph. We can only be grateful to the authors, and perhaps the best way to express this gratitude is to read the book.