From Stein to Weinstein and Back

Cieliebak, Kai; Eliashberg, Yakov

doi:10.1090/coll/059

Cited by 200 publications

(346 citation statements)

References 0 publications

Supporting

Mentioning

344

Contrasting

Unclassified

Order By: Relevance

“…We can only be grateful to the authors, and perhaps the best way to express this gratitude is to read the book. Here is a tip borrowed from Chris Wendl's thorough review on MathSciNet: the two expository articles by the authors [6,7] can serve as points of entry. Finally, we refer to the very recent survey paper by Eliashberg [10] for a much broader perspective on the topic of symplectic flexibility.…”

Section: Modern Symplectic Geometry Was Born In a Longmentioning

confidence: 99%

Book Review: From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds

Oancea¹

2015

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Section: Modern Symplectic Geometry Was Born In a Longmentioning

confidence: 99%

Book Review: From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds

Oancea¹

2015

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

“…The stabilization construction for Legendrian submanifolds, first introdiced in [24] and also described in [8,67], can be informally defined as follows. In an appropriate Darboux coordinate system near a point on a Legendrian submanifold Λ, its front projection has a cuspidal edge; see Figure 5.1.…”

Section: Loose Legendrian Knotsmentioning

confidence: 99%

“…is Stein, then for any exhausting J-convex function φ : V → R the vector field X J,φ can be made complete by composing φ with any function h : R → R with positive first and second derivatives; see [8]. Assuming that this was already done, we associate with a Stein complex manifold (V, J) together with an exhausting J-…”

Section: The Quadruple (V ω X φ) Is Then Called a Weinstein Manifomentioning

confidence: 99%

“…W ⊂ C n admits a defining i-convex function, so in particular it admits a defining Morse function without critical points of index > n (see, e.g., [8]). It follows that any holomorphically, rationally, or polynomially convex domain has the same property.…”

Section: Topology Of Polynomially and Rationally Convex Domains Any mentioning

confidence: 99%

“…Further analysis of condition (T) yields We note that the proofs of all the above results concerning polynomial and rational convexity are based, in addition to the theory of flexible Weinstein manifolds and analytic techniques from the book [8], on the following complex analytic characterization of the polynomial and rational convexity.…”

Section: Theorem 517 ([9]) a Compact Domain W ⊂ C N N ≥ 3 Is Smomentioning

confidence: 99%

See 2 more Smart Citations

Untitled

2015

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Flexible and rigid methods coexisted in symplectic topology from its inception. While the rigid methods dominated the development of the subject during the last three decades, the balance has somewhat shifted to the flexible side in the last three years. In the talk we survey the recent advances in symplectic flexibility in the work of S. Borman, K. Cieliebak, T. Ekholm, E. Murphy, I. Smith, and the author.This survey article is an expanded version of author's article [27] and his talk at the Current Events Bulletin at the AMS meeting in Baltimore in January 2014. It also uses materials from the papers [6,9,20,32] and the book [8]. The h-principleMany problems in mathematics and its applications deal with partial differential equations, partial differential inequalities or, more generally, with partial differential relations (see [48] and also [31]), i.e., any conditions imposed on partial derivatives of an unknown function. A solution of such a partial differential relation R is any function which satisfies this relation.With any differential relation one can associate the underlying algebraic relation by substituting all the derivatives entering the relation with new independent functions. The existence of a solution of the corresponding algebraic relation, called a formal solution of the original differential relation R, is a necessary condition for the solvability of R. Though it seems that this necessary condition should be very far from being sufficient, it was a surprising discovery in the 1950s of geometrically interesting problems where existence of a formal solution is the only obstruction for the genuine solvability. Two of the first such non-trivial examples were the C 1 -isometric embedding theorem of J. Nash and N. Kuiper [57,68] and the immersion theory of S. Smale and M. Hirsch [55,77]. After Gromov's remarkable series of papers, beginning with his paper [47] and culminating in his book [48], the area crystallized as an independent subject, called the h-principle.Rigid and flexible results coexist in many areas of geometry, but nowhere else do they come so close to each other, as in symplectic topology, which serves as a rich source of examples on both sides of the flexible-rigid spectrum. Flexible and rigid problems and the development of each side toward the other shaped and continue to shape the subject of symplectic topology from its inception.

show abstract

Rigid and Flexible Facets of Symplectic Topology

Eliashberg

2019

Geometry in History

View full text Add to dashboard Cite

From Stein to Weinstein and Back

Cited by 200 publications

References 0 publications

Book Review: From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds

Book Review: From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds

Untitled

Rigid and Flexible Facets of Symplectic Topology

Contact Info

Product

Resources

About