Celebratio Mathematica

[1] J. M. Èliašberg: “Singularities of folding type,” Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), pp. 1110–1126. In Russian. Translated in Mathematics of the USSR-Izvestiya 4:5 (1970), 1119–1134. MR 278321 Zbl 0226.57012 article

[2] M. L. Gromov and J. M. Èliašberg: “Nonsingular mappings of Stein manifolds,” Funkcional. Anal. i Priložen. 5 : 2 (1971), pp. 82–83. In Russian. Translated in Funct. Anal. Its Appl. 5:2 (1971), 156–157. MR 301236 Zbl 0234.32011 article

[3] M. L. Gromov and J. M. Èliašberg: “Elimination of singularities of smooth mappings,” Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), pp. 600–626. In Russian. Translated in Mathematics of the USSR-Izvestiya 5:3 (1971), 615–639. MR 301748 Zbl 0221.58009 article

[4] M. L. Gromov and J. M. Èliašberg: “Construction of nonsingular isoperimetric films,” Trudy Mat. Inst. Steklov. 116 (1971), pp. 18–33, 235. In Russian. Translated in Proc. Steklov Inst. Math. 116 (1971), 13–18. MR 388445 Zbl 0277.53031 article

A mapping \( f: M\to \mathbf{R}^N \) of a smooth \( n \)-dimensional manifold \( M \) with boundary \( \partial M \) is said to be isoperimetric if \[ V_n(f) \leqq C_N V_{n-1} (f|_{\partial M}) ,\] where the constant \( C_N \) depends only on \( N \), and \( V_k(g) \) denotes the \( k \)-dimensional volume of the map \( g \). In this paper there is given a necessary and sufficient condition for the existence of an isoperimetric imbedding or immersion \( f: M^n \to \mathbf{R}^N \) which extends the given imbedding or immersion \( g: \partial M \to \mathbf{R}^N \) of the boundary \( \partial M \) of the manifold \( M \).

Besides the result just mentioned, the paper proves certain approximation theorems of the following type. Suppose that \( M \) is a closed \( n \)-dimensional manifold, \( k > 0 \) and \( f: M\to \mathbf{R}^{n+k} \) is a smooth mapping. Then, if there exists an immersion \( g: M\to \mathbf{R}^{n+k} \), there also exists a sequence of immersions \( f_i: M\to \mathbf{R}^{n+k} \) which approximate the mapping \( f \) in the norms of the spaces \( W^l_p \), provided that either \( (l-1)p \lt k \), or \( (l-1) p=k \) and \( p \gt 1 \).

[5] J. M. Èliašberg: “Mappings with given singularities,” Uspehi Mat. Nauk 26 : 6(162) (1971), pp. 255–256. In Russian. MR 461559 Zbl 0223.57017 article

[6] J. M. Èliašberg: “Surgery of singularities of smooth mappings,” Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), pp. 1321–1347. In Russian. Translated in Mathematics of the USSR-Izvestiya 6:6 (1972), 1302–1326. MR 339261 Zbl Zbl 0254.57019 article

[7] M. L. Gromov and J. M. Èliašberg: “Construction of a smooth mapping with a prescribed Jacobian, I,” Funkcional. Anal. i Priložen. 7 : 1 (1973), pp. 33–40. In Russian. Translated in Functional Analysis and Its Applications 7:1 (1973), 27–33. MR 353357 Zbl 0285.57015 article

1.1.1. Let \( M \) and \( N \) be smooth \( n \)-dimensional manifolds, the second being provided with nondegenerate (i.e., nonvanishing) \( n \)-form \( \omega \). We ask when a given (possibly degenerate) \( n \)-form can be induced from \( \omega \) onto \( M \) by a smooth mapping \( M\to N \).

1.1.2. The appearance of the present paper is motivated by a question raised by V. I. Arnol’d (see [1]): Can an exact 2-form be induced into a two-dimensional sphere by a mapping onto a plane? Later we learned from V. I. Arnol’d that A. B. Katok has an affirmative answer to this problem and then D. V. Anasov informed us that A. B. Krygin has carried Katok’s results over to other orientable surfaces.

1.1.3. In the first part of our paper, published here, it is shown that every exact \( n \)-form can be induced onto a stably parallelizable manifold \( M \) by a mapping \( M\to R^n \).

This proposition is generalized and refined by the formulations in Sec. 1.2, which are proved in Sections 2.3 and 2.4. One necessary condition for inducibility is formulated in Sec. 1.3 and is demonstrated in Sec. 2.5.

In the second part of the paper inducibility conditions will be analyzed in detail, the case where \( M \) and \( N \) are of different dimensionality will be considered, and applications to other geometrical problems will be given.

[8] V. A. Zalgaller: Theory of envelopes. Nauka (Moscow), 1975. in Russian; with contributions from Ya. M. Eliashberg: Sections 14 (“Singularities of smooth mappings”) and 15 (”Envelopes from the point of view of singularity theory”). book

[9] N. M. Mišačev and J. M. Èliašberg: “Surgery of singularities of foliations,” Funkcional. Anal. i Priložen. 11 : 3 (1977), pp. 43–53, 96. In Russian. Translated in Functional Analysis and Its Applications 11:3 (1977), 197–206. MR 474326 Zbl 0449.57006 article

[10] Ya. M. Eliashberg: Lectures on global analysis: Topological methods for solving differentials. Equations and inequalities. State University named after A. M. Gorky; Syktyvkar State University named after 50th anniversary of the USSR (Perm; Syktyvkar), 1977. In Russian. book

[11] Y. Eliashberg: Esimates of the number of fixed points of area-preserving transformations of surfaces. Preprint, Syktyvkar University, 1979. In Russian. techreport

[12] J. M. Èliašberg: “Complex structures on a manifold with boundary,” Funktsional. Anal. i Prilozhen. 14 : 1 (1980), pp. 89–90. In Russian. Translated in Funct. Anal. Its Appl. 14 (1980), 75–76. MR 565117 Zbl 0479.32003 article

[13] Y. Eliashberg: Rigidity of symplectic surfaces. Preprint, 1981. techreport

[14] Ya. M. Eliashberg: “Rigidity of symplectic and contact structures,” Abstracts of Reports to the 7th Leningrad International Topology Conference (1982). article

[15] Y. Eliashberg and V. M. Kharmalov: “On the number of complex points of a real surface in a complex surface,” pp. 143–148 in Proc. Leningrad Int. Topol. Conf. (Leningrad, 1982). Edited by V. N. Faddeeva and A. A. Ivanova. Nauka (Leningrad), 1983. In Russian. inproceedings

[16] Y. Eliashberg: “Cobordisme des solutions de relations différentielles,” pp. 17–31 in Séminaire sud-rhodanien de géométrie [South Rhone seminar on geometry] (Université Claude Bernard, Lyon, France, 14–17 June 1983), vol. 1: Géométrie symplectique et de contact [Symplectic and contact geometry]. Edited by P. Dazord and N. Desolneux-Moulis. Travaux en Cours (Works in Progress). Hermann (Paris), 1984. MR 753850 Zbl 0542.57024 incollection

P. Dazord	Related
Nicole Desolneux-Moulis	Related

[17] Ya. M. Eliashberg: “Complexification of contact structures on 3-dimensional manifolds,” Uspekhi Mat. Nauk 40 : 6(246) (1985), pp. 161–162. In Russian. Translated in Russian Mathematical Surveys 40: 123 (1985), 123–124. MR 815508 article

[18] Ya. M. Eliashberg: “A theorem on the structure of wave fronts and its application in symplectic topology,” Funktsional. Anal. i Prilozhen. 21 : 3 (1987), pp. 65–72. In Russian; translated in Fuct. Anal. Appl. 21:3 (1987), 227–232. MR 911776 article

[19] Ya. M. Eliashberg: “Combinatorial methods in symplectic geometry,” pp. 531–539 in Proceedings of the International Congress of Mathematicians (University of California, Berkeley, California, 3–11 August 1986), vol. I. Edited by A. M. Gleason. American Mathematical Society (Providence, RI), 1987. MR 934253 inproceedings

[20] Ya. M. Eliashberg: “The structure of 1-dimensional wave fronts, nonstandard Legendrian loops and Bennequin’s theorem,” pp. 7–12 in Topology and geometry — Rohlin Seminar. Edited by O. Ya. Viro. Lecture Notes in Math. 1346. Springer (Berlin), 1988. MR 970069 Zbl 0662.58004 incollection

[21] Y. Eliashberg: “Three lectures on symplectic topology in Cala Gonone: Basic notions, problems and some methods,” pp. 27–49 in Conference on Differential Geometry and Topology (Sardinia, 1988), published as Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988). MR 1122856 incollection

[22] Y. Eliashberg: “Classification of overtwisted contact structures on 3-manifolds,” Invent. Math. 98 : 3 (1989), pp. 623–637. MR 1022310 Zbl 0684.57012 article

[23] Y. Eliashberg: “Topological characterization of Stein manifolds of dimension \( > 2 \),” Internat. J. Math. 1 : 1 (1990), pp. 29–46. MR 1044658 article

[24] R. Brooks, Y. Eliashberg, and C. McMullen: “The spectral geometry of flat disks,” Duke Math. J. 61 : 1 (1990), pp. 119–131. MR 1068382 article

Robert Brooks	Related
Curtis Tracy McMullen	Related

[25] Y. Eliashberg: “Filling by holomorphic discs and its applications,” pp. 45–67 in Geometry of low-dimensional manifolds (Durham, UK, 1989), vol. II. Edited by S. K. Donaldson and C. B. Thomas. London Math. Soc. Lecture Note Ser. 151. Cambridge University Press (Cambridge, UK), 1990. MR 1171908 Zbl 0731.53036 incollection

Charles B. Thomas	Related
S. K. Donaldson	Related

[26] Y. Eliashberg: “Existence and nonexistence of a Stein complex structure on differentiable manifolds,” pp. 61–69 in Seminars in complex analysis and geometry (Arcavacata, Italy, 1988). Edited by J. Guenot and D. Struppa. Sem. Conf. 4. EditEl (Rende, Italy), 1990. Papers from the seminar held at the University of Calabria. MR 1222249 incollection

[27] Y. Eliashberg and T. Ratiu: “The diameter of the symplectomorphism group is infinite,” Invent. Math. 103 : 2 (1991), pp. 327–340. MR 1085110 Zbl 0725.58006 article

[28] Y. Eliashberg: “On symplectic manifolds with some contact properties,” J. Differential Geom. 33 : 1 (1991), pp. 233–238. MR 1085141 Zbl 0735.53021 article

[29] Y. Eliashberg: “New invariants of open symplectic and contact manifolds,” J. Amer. Math. Soc. 4 : 3 (1991), pp. 513–520. MR 1102580 article

[30] Y. Eliashberg and T. Ratiu: “On the diameter of the symplectomorphism group of the ball,” pp. 169–172 in Symplectic geometry, groupoids, and integrable systems (Berkeley, California, 22 May–2 June 1989). Edited by P. Dazord and A. Weinstein. Math. Sci. Res. Inst. Publ. 20. Springer (New York), 1991. MR 1104925 Zbl 0729.58016 incollection

It is known (Arnold [1966], Ebin and Marsden [1970]) that the group of symplectomorphisms carries a natural weak Riemannian metric. In this paper the underlying manifold will always be the closed unit ball \( B \) in \( \mathbb{R}^{2n} \) endowed with the canonical symplectic structure. We will show here that the diameter (in the metric discussed below) of the group of symplectomorphisms of \( B \) which leaves the boundary \( \partial B \) pointwise fixed is infinite. In another paper (see Eliashberg and Ratiu [1989]) we extend the result to the group of all symplectomorphisms of any compact exact symplectic manifold. However, the proof in the case of the unit ball simplifies considerably and contains already the key ingredient of topological nature needed in the general case.

Tudor Ștefan Rațiu	Related
P. Dazord	Related
Alan Weinstein	Related

[31] Y. Eliashberg and M. Gromov: “Convex symplectic manifolds,” pp. 135–162 in Several complex variables and complex geometry: Part 2. Edited by E. Bedford, J. P. D’Angelo, R. E. Greene, and S. G. Krantz. Proc. Sympos. Pure Math. 52. American Mathematical Society, 1991. Proceedings of the 37th Annual Summer Research Institute (University of California, Santa Cruz, California, 10–30 July 1989). MR 1128541 Zbl 0742.53010 incollection

John P. D’Angelo	Related
Robert Everist Greene	Related
Steven Krantz	Related
Mikhael Leonidovich Gromov	Related

[32] Y. Eliashberg and H. Hofer: “Towards the definition of symplectic boundary,” Geom. Funct. Anal. II : 2 (1992), pp. 211–220. MR 1159830 Zbl 0756.53016 article

Let \( (W, \omega) \) be a compact symplectic manifold with smooth boundary. The restriction wlow degenerates along a 1-dimensional foliation \( \mathcal{L}_{\partial\omega} \) on \( \partial W \) which is called the characteristic foliation. The foliation \( \mathcal{L}_{\partial\omega} \) together with the transversal symplectic structure defined by \( \omega \) is a symplectic invariant of the compact manifold \( (W, \omega) \). The topology of \( \mathcal{L}_{\partial\omega} \) is very sensitive to deformation of \( \omega|_{\partial W} \) and, therefore, provides a lot of symplectic information about \( W \). If some additional conditions are met, then one can extract from \( \omega|_{\partial W} \) numerical invariants (see §1.2 below). The knowledge of the symplectic boundary \( (\partial W, \omega|_{\partial W}) \) allows for effectively distinguishing between non-symplectomorphic manifolds and sometimes gives the complete information about the symplectic manifold \( (W,\omega) \) (see §4.2 below.)

[33] Y. Eliashberg: “Contact 3-manifolds twenty years since J. Martinet’s work,” Ann. Inst. Fourier (Grenoble) 42 : 1–2 (1992), pp. 165–192. MR 1162559 article

[34] Y. Eliashberg and M. Gromov: “Embeddings of Stein manifolds of dimension \( n \) into the affine space of dimension \( 3n/2+1 \),” Ann. of Math. (2) 136 : 1 (1992), pp. 123–135. MR 1173927 Zbl 0758.32012 article

[35] Y. Eliashberg: “Legendrian and transversal knots in tight contact 3-manifolds,” pp. 171–193 in Topological methods in modern mathematics. Publish or Perish (Houston, TX), 1993. MR 1215964 Zbl 0809.53033 incollection

[36] Y. Eliashberg: “Classification of contact structures on \( \mathbb{R}^3 \),” Internat. Math. Res. Notices 3 (1993), pp. 87–91. MR 1208828 Zbl 0784.53022 article

[37] Y. Eliashberg and L. Polterovich: “Bi-invariant metrics on the group of Hamiltonian diffeomorphisms,” Internat. J. Math. 4 : 5 (1993), pp. 727–738. MR 1245350 Zbl 0795.58016 article

[38] Y. Eliashberg and L. Polterovich: “Unknottedness of Lagrangian surfaces in symplectic 4-manifolds,” Internat. Math. Res. Notices 11 (1993), pp. 295–301. MR 1248704 Zbl 0808.57021 article

1.1. Lagrangian surfaces in cotangent bundles. Let \( (M, \omega) \) be a symplectic 4-manifold. Do there exist two homotopic but nonisotopic embeddings of a closed surface into \( M \) which are Lagrangian with respect to \( \omega \)? This so-called Lagrangian knots problem has been known for a while and was considered, in particular, in Arnold’s survey [A]. To date, two results in this direction are known. It was discovered by K. Luttinger [Lu] that certain knot types of embeddings \( \mathbb{T}^2 \to \mathbb{R}^4 \) cannot be represented by Lagrangian ones. Further, the first author proved [E] that any two Lagrangian embeddings \[ S^1 \times [0, 1] \to \mathbb{R}^3 \times [0; 1] \] with standard boundary conditions are isotopic.

In the present paper we study this problem for Lagrangian surfaces in cotangent bundles.

[39] Y. Eliashberg and H. Hofer: “An energy-capacity inequality for the symplectic holonomy of hypersurfaces flat at infinity,” pp. 95–114 in Symplectic geometry. Edited by D. Salamon. London Math. Soc. Lecture Note Ser. 192. Cambridge University Press, 1993. MR 1297131 Zbl 0807.58014 incollection

Dietmar Arno Salamon	Related
Helmut Hofer	Related

[40] Y. Eliashberg and H. Hofer: “A Hamiltonian characterization of the three-ball,” Differential Integral Equations 7 : 5–6 (1994), pp. 1303–1324. MR 1269658 Zbl 0803.58045 article

In this paper we show that the closed three-ball can be distinguished from all other orientable three manifolds with a boundary diffeomorphic to \( S^2 \) via Hamiltonian mechanics. This characterization is built on the following fact: any orientable three manifold with boundary diffeomorphic to \( S^2 \) can be equipped with a contact form. Such a contact form defines a dynamical system. This dynamical system can always be considered as the isoenergetic flow for a suitable Hamiltonian system. We study such flows, which are standard near the boundary in a suitable sense. We show using nonlinear first order elliptic systems that on manifolds different form the three-ball, the flow must necessarily have periodic orbits, in contrast to the three-ball case.

[41] Y. Eliashberg and L. Polterovich: “New applications of Luttinger’s surgery,” Comment. Math. Helv. 69 : 4 (1994), pp. 512–522. MR 1303225 Zbl 0853.57012 article

[42] Y. Eliashberg, H. Hofer, and D. Salamon: “Lagrangian intersections in contact geometry,” Geom. Funct. Anal. 5 : 2 (1995), pp. 244–269. MR 1334868 Zbl 0844.58038 article

It is well-known that all problems of Contact geometry can be reformulated as problems of Symplectic geometry. This can be done via symplectization (see 2.1 below). In particular, the problem of Lagrangian intersections naturally arises in connection with several contact geometric questions (see 2.5 example, and below). However, there is one major difficulty when one tries to realize this approach: the symplectizations of contact manifolds are non-compact and, what is even worse, non-convex (see [EGrl]). This leads to the loss of compactness for the spaces of holomorphic curves and thus creates serious difficulties for the traditional Floer homology approach. The goal of this paper is to show that this problem can be successfully overcome by using an idea from [H].

Dietmar Arno Salamon	Related
Helmut Hofer	Related

[43] Y. Eliashberg: “Topology of 2-knots in \( \mathbf{ R}^4 \) and symplectic geometry,” pp. 335–353 in The Floer memorial volume. Edited by H. Hofer, C. H. Taubes, A. Weinstein, and E. Zehnder. Progr. Math. 133. Birkhäuser (Basel), 1995. MR 1362834 Zbl 0863.57023 incollection

Helmut Hofer	Related
Eduard J. Zehnder	Related
Clifford Henry Taubes	Related
Alan Weinstein	Related

[44] Y. Eliashberg: “Unique holomorphically fillable contact structure on the 3-torus,” Internat. Math. Res. Notices 2 (1996), pp. 77–82. MR 1383953 Zbl 0852.58034 article

[45] Y. M. Eliashberg and W. P. Thurston: “Contact structures and foliations on 3-manifolds,” Turkish J. Math. 20 : 1 (1996), pp. 19–35. MR 1392660 article

Theory of foliations and contact geometry were developed independently, despite that the both theories exhibited a lot of striking similarities. In particular, it was understood in both cases that additional restrictions on foliations and contact structures are needed in order to make them interesting for applications. In the context of foliations this led to the theory of taut foliations ([22],[10]) and the related theory of essential laminations ([11]). In contact geometry there were studied tight contact structures ([2],[5],[12]) which exhibited a lot of similar properties. In this paper we study relations and analogies between the two structures. Proofs of the main results only sketched here. More detailed account of the subject can be found in our paper [9].

[46] Y. Eliashberg and L. Polterovich: “Local Lagrangian 2-knots are trivial,” Ann. of Math. (2) 144 : 1 (1996), pp. 61–76. MR 1405943 Zbl 0872.57030 article

[47] Y. Eliashberg and H. Hofer: “Unseen symplectic boundaries,” pp. 178–189 in Manifolds and geometry (Pisa, 1993). Edited by P. de Bartolomeis, F. Tricerri, and E. Vesentini. Sympos. Math. 36. Cambridge University Press (Cambridge, UK), 1996. MR 1410072 incollection

Edoardo Vesentini	Related
Helmut Hofer	Related

[48] C. B. Thomas, Y. Eliashberg, and E. Giroux: “3-dimensional contact geometry,” pp. 48–65 in Contact and symplectic geometry. Edited by C. B. Thomas. Publ. Newton Inst. 8. Cambridge University Press, 1996. MR 1432458 Zbl 0852.00028 incollection

Charles B. Thomas	Related
Emmanuel Giroux	Related

[49] Y. Eliashberg and M. Gromov: “Lagrangian intersections and the stable Morse theory,” Boll. Un. Mat. Ital. B (7) 11 : 2 (1997), pp. 289–326. MR 1456266 Zbl 0964.58009 article

[50] Y. Eliashberg: “Symplectic geometry of plurisubharmonic functions,” pp. 49–67 in Gauge theory and symplectic geometry (Montreal, PQ, 1995). Edited by J. Hurtubise, F. Lalonde, and G. Sabidussi. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 488. Kluwer (Dordrecht), 1997. Notes by Miguel Abreu. MR 1461569 Zbl 0881.32010 incollection

In these lectures we describe symplectic geometry related to the notion of pseudo-convexity (or \( J \)-convexity). The notion of \( J \)-convexity, which is a complex analog of convexity, is one of the basic mathematical notions. Symplectic geometry built-in into this notion is essential for understanding the structure of affine (or Stein) complex manifolds. It plays also a major role in the classification of Stein complex structures up to deformation. Plurisubharmonic (or \( J \)-convex) functions on complex manifolds are analagous to convex functions on Riemannian manifolds but their theory is much richer. Symplectic geometry is crucial for understanding Morse-theoretic properties of \( J \)-convex functions.

Our goal is just the general picture of the subject. The proofs are only indicated or sketched here.

François Lalonde	Related
Jacques Hurtubise	Related
Miguel Tribolet Abreu	Related

[51] Y. Eliashberg and L. Polterovich: “The problem of Lagrangian knots in four-manifolds,” pp. 313–327 in Geometric topology (Athens, GA, 1993), vol. 1: 1993 Georgia International Topology Conference. Edited by W. H. Kazez. AMS/IP Stud. Adv. Math. 2.1. American Mathematical Society (Providence, RI), 1997. MR 1470735 Zbl 0889.57036 incollection

Leonid V. Polterovich	Related
William H. Kazez	Related

[52] Y. Eliashberg and N. M. Mishachev: “Wrinkling of smooth mappings and its applications, I,” Invent. Math. 130 : 2 (1997), pp. 345–369. MR 1474161 Zbl 0896.58010 article

This paper, together with its sequel papers [EM1] and [EM2], contains a reexposition of the authors’ theory of mappings with simple singularities (see [E1], [E2], [M1], [M2]). In addition, we give here new applications of the theory. The method of wrinkling of smooth mappings described in these papers provides an easier path to theorems from [E1] and [E2] about construction of mappings with prescribed singularities. It gives an alternative proof of Thurston’s theorem about foliations of codimension \( > 1 \) (see [Th] and [ME]). The method gives a new and, we think, simpler proof of K. Igusa’s theorem on mappings without higher singularities (see [Ig]). Moreover, it allows us to remove all dimensional restrictions in this theorem.

[53] Y. Eliashberg: “Invariants in contact topology,” pp. 327–338 in Proceedings of the International Congress of Mathematicians, II (Berlin, 1998), published as Doc. Math. Extra Vol. II (1998). MR 1648083 Zbl 0913.53010 inproceedings

[54] Y. Eliashberg and M. Fraser: “Classification of topologically trivial Legendrian knots,” pp. 17–51 in Geometry, topology, and dynamics. CRM Proc. Lecture Notes 15. Amer. Math. Soc. (Providence, RI), 1998. MR 1619122 Zbl 0907.53021 incollection

[55] Y. M. Eliashberg and W. P. Thurston: Confoliations. University Lecture Series 13. American Mathematical Society (Providence, RI), 1998. MR 1483314 book

The theory of contact structures and the theory of foliations have developed rather independently. They come from separate traditions, and have different flavors. However, in dimension 3, they have evolved in parallel directions that have powerful topological applications involving tight contact structures on the one hand and taut foliations on the other. The present work develops the foundations for a theory of confoliations to link these two theories, with the aim of developing a combined toolkit that includes both the strongly geometric constructions characteristic of foliation theory and the analytic tools, the connections to four-dimensional topology, and the flexibility characteristic of the theory of contact structures. In particular, we prove that every \( C^2 \) taut foliation can be \( C^0 \)-perturbed to give a tight contact structure.

[56] Y. Eliashberg: “Symplectic topology in the nineties,” Differential Geom. Appl. 9 : 1–2 (1998), pp. 59–88. MR 1636301 article

[57] Y. Eliashberg and N. M. Mishachev: “Wrinkling of smooth mappings, III: Foliations of codimension greater than one,” Topol. Methods Nonlinear Anal. 11 : 2 (1998), pp. 321–350. MR 1659446 Zbl 0927.58022 article

This is the third paper in our Wrinkling saga (see [EM1], [EM2]). The first paper [EM1] was devoted to the foundations of the method. The second paper [EM2], as well as the current one are devoted to the applications of the wrinkling process. In [EM2] we proved, among other results, a generalized Igusa’s theorem about functions with moderate singularities.

The current paper is devoted to applications of the wrinkling method in the foliation theory. The results of this paper essentially overlap with our paper [ME], which was written twenty years ago, soon after Thurston’s remarkable discovery (see [Th1]) of an \( h \)-principle for foliations of codimension greater than one on closed manifolds. The paper [ME] contained an alternative proof of Thurston’s theorem from [Th1], and was based on the technique of surgery of singularities which was developed in [E2]. The proof presented in this paper is based on the wrinkling method. Although essentially similar to our proof in [ME], the current proof is, in our opinion, more transparent and easier to understand. Besides Thurston’s theorem we prove here a generalized version of our results from [ME] related to families of foliations.

[58] Y. Eliashberg and M. Gromov: “Lagrangian intersection theory: Finite-dimensional approach,” pp. 27–118 in Geometry of differential equations. Edited by A. Khovanskiĭ, A. Varchenko, and V. Vassiliev. Amer. Math. Soc. Transl. Ser. 2 186. American Mathematical Society (Providence, RI), 1998. MR 1732407 Zbl 0919.58015 incollection

[59] Y. M. Eliashberg and N. M. Mishachev: “Wrinkling of smooth mappings, II: Wrinkling of embeddings and K. Igusa’s theorem,” Topology 39 : 4 (2000), pp. 711–732. MR 1760426 Zbl 0964.58028 article

[60] Y. Eliashberg and L. Polterovich: “Partially ordered groups and geometry of contact transformations,” Geom. Funct. Anal. 10 : 6 (2000), pp. 1448–1476. MR 1810748 Zbl 0986.53036 article

[61] Y. Eliashberg, A. Givental, and H. Hofer: “Introduction to symplectic field theory,” pp. 560–673 in Visions in Mathematics. Edited by N. Alon, J. Bourgain, A. Connes, M. Gromov, and V. Milman. 2000. Special volume, GAFA2000, of Geometric and Functional Analysis. MR 1826267 Zbl 0989.81114 incollection

Helmut Hofer	Related
Alain Connes	Related
Alexander Borisovich Givental	Related
Mikhael Leonidovich Gromov	Related
Jean Bourgain	Related

[62] Y. Eliashberg: “Topology of Lagrangian submanifolds,” pp. 125–133 in Proceedings of the Workshop “Algebraic Geometry and Integrable Systems related to String Theory” (Kyoto, 2000), published as Sūrikaisekikenkyūsho Kōkyūroku 1232 (2001). MR 1905888 Zbl 1322.53083 inproceedings

[63] Y. M. Eliashberg and N. M. Mishachev: “Holonomic approximation and Gromov’s \( h \)-principle,” pp. 271–285 in Essays on geometry and related topics. Edited by É. Ghys, P. de la Harpe, V. F. R. Jones, V. Sergiescu, and T. Tsuboi. Monogr. Enseign. Math. 38. Enseignement Math. (Geneva), 2001. In 2 volumes. MR 1929330 Zbl 1025.58001 incollection

In 1969 M. Gromov in his PhD thesis greatly generalized Smale–Hirsch–Phillips immersion-submersion theory (see [Sm], [Hi], [Ph]) by proving what is now called the \( h \)-principle for invariant open differential relations over open manifolds. Gromov extracted the original geometric idea of Smale and put it to work in the maximal possible generality. Gromov’s thesis was brought to the West by A. Phillips and was popularized in his talks. However, most western mathematicians first learned about Gromov’s theory from A. Haefliger’s article [H]. The current paper is devoted to the same subject as the papers of Gromov and Haefliger. It seems to us that we further purified Smale–Gromov’s original idea by extracting from it a simple but very general theorem about holonomic approximation of sections of jet-bundles (see Theorem 1.2.1 below). We show below that Gromov’s theorem as well as some other results in the \( h \)-principle spirit are immediate corollaries of Theorem 1.2.1.

Takashi Tsuboi	Related
Nikolai M. Mishachev	Related

[64] Y. M. Eliashberg and N. M. Mishachev: Introduction to the \( h \)-principle. Graduate Studies in Mathematics 48. American Mathematical Society (Providence, RI), 2002. MR 1909245 Zbl 1008.58001 book

[65] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder: “Compactness results in symplectic field theory,” Geom. Topol. 7 (2003), pp. 799–888. MR 2026549 Zbl 1131.53312 article

Helmut Hofer	Related
Eduard J. Zehnder	Related
Frederic Bourgeois	Related
Krzysztof Wysocki	Related

[66] Y. Eliashberg: “A few remarks about symplectic filling,” Geom. Topol. 8 (2004), pp. 277–293. MR 2023279 article

[67] Y. Eliashberg, S. S. Kim, and L. Polterovich: “Geometry of contact transformations and domains: orderability versus squeezing,” Geom. Topol. 10 (2006), pp. 1635–1747. MR 2284048 Zbl 1134.53044 article

Gromov’s famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding is that the answer depends on the sizes of the domains in question: We establish contact non-squeezing on large scales, and show that it disappears on small scales. The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a contact manifold. In contrast to our earlier beliefs, we show that the answer to this question is very sensitive to the topology of the manifold. For instance, we prove that the standard contact sphere is non-orderable while the real projective space is known to be orderable. Our methods include a new embedding technique in contact geometry as well as a generalized Floer homology theory which contains both cylindrical contact homology and Hamiltonian Floer homology. We discuss links to a number of miscellaneous topics such as topology of free loops spaces, quantum mechanics and semigroups.

Sang Seon Kim	Related
Leonid V. Polterovich	Related

[68] Y. Eliashberg: “Symplectic field theory and its applications,” pp. 217–246 in International Congress of Mathematicians, vol. 1. Edited by M. Sanz-Solé, J. Soria, J. L. Varona, and J. Verdera. European Mathematical Society (Zürich), 2007. MR 2334192 Zbl 1128.53059 incollection

Marta Sanz-Solé	Related
Juan Luis Varona	Related
Javier Soria	Related
Joan Manuel Verdera Melenchón	Related

[69] Y. Eliashberg, K. Fukaya, V. Muñoz, and F. Presas: “Foreword,” pp. iii–iv in Proceedings of the 7th Workshop on Symplectic and Contact Topology (Madrid, Spain, 16–19 August 2006), published as Geom. Dedicata 132 (2008). MR 2396905 Zbl 1136.53300 incollection

[70] Y. Eliashberg and M. Fraser: “Topologically trivial Legendrian knots,” J. Symplectic Geom. 7 : 2 (2009), pp. 77–127. MR 2496415 Zbl 1179.57040 article

[71] Y. Eliashberg, S. S. Kim, and L. Polterovich: “Erratum to ‘Geometry of contact transformations and domains: Orderability versus squeezing’ \( [ \)MR2284048\( ] \),” Geom. Topol. 13 : 2 (2009), pp. 1175–1176. MR 2491659 Zbl 1161.53071 article

Sang Seon Kim	Related
Leonid V. Polterovich	Related

[72] Y. M. Eliashberg and N. M. Mishachev: “Wrinkled embeddings,” pp. 207–232 in Foliations, geometry, and topology. Edited by N. C. Saldanha, L. Conlon, R. Langevin, T. Tsuboi, and P. Walczak. Contemp. Math. 498. American Mathematical Society (Providence, RI), 2009. MR 2664601 Zbl 1194.57036 incollection

Rémi Langevin	Related
Takashi Tsuboi	Related
Paweł Grzegorz Walczak	Related
Nikolai M. Mishachev	Related

[73] Y. Eliashberg and L. Polterovich: Symplectic quasi-states on the quadric surface and Lagrangian submanifolds. Preprint, 2010. ArXiv 1006.2501 techreport

[74] Y. Eliashberg, S. Galatius, and N. Mishachev: “Madsen–Weiss for geometrically minded topologists,” Geom. Topol. 15 : 1 (2011), pp. 411–472. MR 2776850 Zbl 1211.57012 article

Soren Galatius	Related
Nikolai M. Mishachev	Related

[75] F. Bourgeois, T. Ekholm, and Y. Eliashberg: “Symplectic homology product via Legendrian surgery,” Proc. Natl. Acad. Sci. USA 108 : 20 (2011), pp. 8114–8121. MR 2806647 Zbl 1256.53049 article

Tobias Ekholm	Related
Frederic Bourgeois	Related

[76] Y. M. Eliashberg and N. M. Mishachev: “Topology of spaces of \( S \)-immersions,” pp. 147–167 in Perspectives in analysis, geometry, and topology. Edited by I. Itenberg, B. Jöricke, and M. Passare. Progr. Math. 296. Springer (New York), 2012. MR 2884035 Zbl 1267.57033 incollection

[77] F. Bourgeois, T. Ekholm, and Y. Eliashberg: “Effect of Legendrian surgery,” Geom. Topol. 16 : 1 (2012), pp. 301–389. With an appendix by Sheel Ganatra and Maksim Maydanskiy. MR 2916289 article

Tobias Ekholm	Related
Sheel Ganatra	Related
Frederic Bourgeois	Related

[78] Y. M. Eliashberg and N. M. Mishachev: “The space of framed functions is contractible,” pp. 81–109 in Essays in mathematics and its applications. Edited by P. M. Pardalos and T. M. Rassias. Springer (Heidelberg), 2012. MR 2975585 incollection

According to Igusa [Ann Math 119:1–58, 1984] a generalized Morse function on \( M \) is a smooth function \( M\to\mathbb{R} \) with only Morse and birth-death singularities and a framed function on \( M \) is a generalized Morse function with an additional structure: a framing of the negative eigenspace at each critical point of \( f \). In [Trans. Am. Math. Soc. 301(2):431–477, 1987] Igusa proved that the space of framed generalized Morse functions is \( (\dim M-1) \)-connected. Lurie gave in [arXiv:0905.0465] an algebraic topological proof that the space of framed functions is contractible. In this paper we give a geometric proof of Igusa–Lurie’s theorem using methods of our paper [Eliashberg and Mishachev, Topology 39:711–732, 2000].

[79] K. Cieliebak and Y. Eliashberg: From Stein to Weinstein and back: Symplectic geometry of affine complex manifolds. American Mathematical Society Colloquium Publications 59. American Mathematical Society (Providence, RI), 2012. MR 3012475 Zbl 1262.32026 book

[80] Y. Eliashberg and E. Murphy: “Lagrangian caps,” Geom. Funct. Anal. 23 : 5 (2013), pp. 1483–1514. MR 3102911 Zbl 1308.53121 article

[81] T. Ekholm, Y. Eliashberg, E. Murphy, and I. Smith: “Constructing exact Lagrangian immersions with few double points,” Geom. Funct. Anal. 23 : 6 (2013), pp. 1772–1803. MR 3132903 Zbl 1283.53074 article

Emmy Murphy	Related
Tobias Ekholm	Related
Ivan Smith	Related

[82] K. Cieliebak and Y. Eliashberg: “Stein structures: existence and flexibility,” pp. 357–388 in Contact and symplectic topology. Edited by F. Bourgeois, V. Colin, and A. Stipsicz. Bolyai Soc. Math. Stud. 26. János Bolyai Math. Soc. (Budapest), 2014. MR 3220946 Zbl 1335.32007 incollection

Frederic Bourgeois	Related
Kai Cieliebak	Related

[83] Y. Eliashberg: “Recent progress in symplectic flexibility,” pp. 3–18 in The influence of Solomon Lefschetz in geometry and topology. Edited by L. Katzarkov, E. Lupercio, and F. J. Turrubiates. Contemp. Math. 621. American Mathematical Society (Providence, RI), 2014. MR 3289318 Zbl 1330.57001 incollection

Ludmil Katzarkov	Related
Solomon Lefschetz	Related

[84] K. Cieliebak and Y. Eliashberg: “Flexible Weinstein manifolds,” pp. 1–42 in Symplectic, Poisson, and noncommutative geometry. Edited by T. Eguchi, Y. Eliashberg, and Y. Maeda. Math. Sci. Res. Inst. Publ. 62. Cambridge University Press (New York), 2014. MR 3380672 Zbl 1338.53003 ArXiv 1305.1635 incollection

Kai Cieliebak	Related
Yoshiaki Maeda	Related
T. Eguchi	Related

[85] Y. Eliashberg: “Recent advances in symplectic flexibility,” Bull. Amer. Math. Soc. (N.S.) 52 : 1 (2015), pp. 1–26. MR 3286479 Zbl 1310.53001 article

[86] K. Cieliebak and Y. Eliashberg: “The topology of rationally and polynomially convex domains,” Invent. Math. 199 : 1 (2015), pp. 215–238. MR 3294960 Zbl 1310.32014 article

[87] M. S. Borman, Y. Eliashberg, and E. Murphy: “Existence and classification of overtwisted contact structures in all dimensions,” Acta Math. 215 : 2 (2015), pp. 281–361. MR 3455235 Zbl 1344.53060 article

Emmy Murphy	Related
Matthew Strom Borman	Related

[88] Y. Eliashberg: “Book review of Leonid Polterovich and Daniel Rosen, ‘Function theory on symplectic manifolds’,” Bull. Amer. Math. Soc. (N.S.) 54 : 1 (2017), pp. 135–140. MR 3686324 Zbl 1352.00013 article

[89] K. Cieliebak, Y. Eliashberg, and L. Polterovich: “Contact orderability up to conjugation,” Regul. Chaotic Dyn. 22 : 6 (2017), pp. 585–602. MR 3736463 Zbl 1390.53096 article

Kai Cieliebak	Related
Leonid V. Polterovich	Related

[90] Y. Eliashberg: “Weinstein manifolds revisited,” pp. 59–82 in Modern geometry: a celebration of the work of Simon Donaldson. Edited by V. Muñoz, I. Smith, and R. P. Thomas. Proc. Sympos. Pure Math. 99. American Mathematical Society (Providence, RI), 2018. MR 3838879 Zbl 1448.53083 incollection

[91] Y. Eliashberg: “Rigid and flexible facets of symplectic topology,” pp. 493–514 in Geometry in history. Edited by S. G. Dani and A. Papadopoulos. Springer (Cham, Switzerland), 2019. MR 3965772 Zbl 1454.53002 incollection

[92] Y. Eliashberg, S. Ganatra, and O. Lazarev: “Flexible Lagrangians,” Int. Math. Res. Not. 2020 : 8 (2020), pp. 2408–2435. MR 4090744 Zbl 1437.53067 article

We introduce and discuss notions of regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains. There is a surprising abundance of flexible Lagrangians. In turn, this leads to new constructions of Legendrians submanifolds and Weinstein manifolds. For instance, many closed \( n \)-manifolds of dimension \( n > 2 \) can be realized as exact Lagrangian submanifolds of \( T^{\ast} S^n \) with possibly exotic Weinstein symplectic structures. These Weinstein structures on \( T^{\ast} S^n \), infinitely many of which are distinct, are formed by a single handle attachment to the standard 2\( n \)-ball along the Legendrian boundaries of flexible Lagrangians. We also formulate a number of open problems.

Sheel Ganatra	Related
Oleg Lazarev	Related

[93] Y. Eliashberg and N. Mishachev: The space of tight contact structures on \( {\mathbb R}^3 \) is contractible. Preprint, 2021. ArXiv 2108.09452 techreport

[94] K. Cieliebak and Y. Eliashberg: “New applications of symplectic topology in several complex variables,” J. Geom. Anal. 31 : 3 (2021), pp. 3252–3271. MR 4225841 Zbl 1461.53001 article

[95] Y. Eliashberg, N. Ogawa, and T. Yoshiyasu: “Stabilized convex symplectic manifolds are Weinstein,” Kyoto J. Math. 61 : 2 (2021), pp. 323–337. MR 4342379 Zbl 1470.53069 article

[96] D. Álvarez-Gavela, Y. Eliashberg, and D. Nadler: Positive arborealization of polarized Weinstein manifolds. Preprint, 2022. ArXiv 2011.08962 techreport

[97] D. Álvarez-Gavela, Y. Eliashberg, and D. Nadler: “Geomorphology of Lagrangian ridges,” J. Topol. 15 : 2 (2022), pp. 844–877. MR 4441606 Zbl 07738220 article

[98]Y. M. Eliashberg and W. P. Thurston: “Contact structures and foliations on 3-manifolds,” pp. 257–273 in Collected works of William P. Thurston with commentary, vol. 1: Foliations, surfaces and differential geometry. Edited by B. Farb, D. Gabai, and S. P. Kerckhoff. American Mathematical Society (Providence, RI), 2022. MR 4554445 incollection

Theory of foliations and contact geometry were developed independently, despite that the both theories exhibited a lot of striking similarities. In particular, it was understood in both cases that additional restrictions on foliations and contact structures are needed in order to make them interesting for applications. In the context of foliations this led to the theory of taut foliations ([22],[10]) and the related theory of essential laminations ([11]). In contact geometry there were studied tight contact structures ([2],[5],[12]) which exhibited a lot of similar properties. In this paper we study relations and analogies between the two structures. Proofs of the main results only sketched here. More detailed account of the subject can be found in our paper [9].

[99] Y. M. Eliashberg and W. P. Thurston: “Confoliations,” pp. 281–351 in Collected works of William P. Thurston with commentary, vol. 1: Foliations, surfaces and differential geometry. Edited by B. Farb, D. Gabai, and S. P. Kerckhoff. American Mathematical Society (Providence, RI), 2022. MR 4554446 incollection

[100] Y. Eliashberg and E. Murphy: “Making cobordisms symplectic,” J. Amer. Math. Soc. 36 : 1 (2023), pp. 1–29. MR 4495837 Zbl 1509.57022 article

Symplectic cobordisms. We say that \( (W, \omega, \xi_{-}, \xi_{+}) \) is a symplectic cobordism between contact manifolds \( (\partial_{\pm} W, \xi_{\pm}) \) if

\( W \) is a smooth cobordism between \( \partial_{-}W \) and \( \partial_{+}W \), and
\( \omega \) is a symplectic form which admits a Liouville vector field \( Z \) near \( \partial W \) such that \( Z \) is inwardly transverse to \( \partial_{-}W \), outwardly transverse to \( \partial_{+}W \) and the contact forms \( \lambda_{\pm} = \iota (Z) \omega |_{\partial_{\pm}W} \) define the contact structures \( \xi_{\pm} \).

All cobordisms we consider in this paper are assumed to be connected while their boundaries \( \partial_{\pm}W \) could be disconnected. We recall that a vector field \( Z \) is called Liouville for a symplectic form \( \omega \) if \( d(\iota (Z) \omega)=\omega \).

[101] Y. Eliashberg and D. M. Pancholi: “Honda–Huang’s work on contact convexity revisited,” pp. 453–492 in Essays in geometry — dedicated to Norbert A’Campo. Edited by A. Papadopoulos. IRMA Lect. Math. Theor. Phys. 34. EMS Press (Berlin), 2023. MR 4631278 incollection

[102] D. Álvarez-Gavela, Y. Eliashberg, and D. Nadler: “Arboreal models and their stability,” J. Symplectic Geom. 21 : 2 (2023), pp. 331–381. MR 4653157 Zbl 07744114 article

The main result of this paper is the uniqueness of local arboreal models, defined as the closure of the class of smooth germs of Lagrangian submanifolds under the operation of taking iterated transverse Liouville cones. A parametric version implies that the space of germs of symplectomorphisms that preserve the local model is weakly homotopy equivalent to the space of automorphisms of the corresponding signed rooted tree. Hence the local symplectic topology around a canonical model reduces to combina- torics, even parametrically. This paper can be read independently, but it is part of a series of papers by the authors on the arborealization program

Yakov M. Eliashberg

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