Introduction to the ℎ-Principle

Eliashberg, Yakov; Mishachev, N.

doi:10.1090/gsm/048

Cited by 274 publications

(404 citation statements)

References 0 publications

Supporting

Mentioning

400

Contrasting

Unclassified

Order By: Relevance

“…Indeed, in the terminology of Gromov [6], see also Eliashberg-Mishachev [1] or Spring [11], we show that curves of constant curvature satisfy the relative C 1 -dense h-principle in the space of C α≥2 immersions. Yet we do not appeal to, nor are we aware of, any general h-principle theorems which may be immediately relevant; instead, we give a direct proof which is for the most part constructive.…”

Section: Introductionmentioning

confidence: 93%

“…By Thom's transversality theorem [1], the first n derivatives of a C n -dense set of mappings f ∈ C n ([a, b], R n ) are linearly independent everywhere except at a finite set of points. We will quickly verify this fact in the next paragraph.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…The standard norm on C α (Γ, R n ) is denoted by (1) f α := sup f (i) j (t) t ∈ Γ, 1 ≤ i ≤ α, 1 ≤ j ≤ n , is the tantrix of f . The space of embeddings Emb α (Γ, R n ) ⊂ Imm α (Γ, R n ) is the collection of those curves which are one-to-one.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

h-Principles for curves and knots of constant curvature

Ghomi

2007

Geom Dedicata

View full text Add to dashboard Cite

Abstract. We prove that C ∞ curves of constant curvature satisfy, in the sense of Gromov, the relative C 1 -dense h-principle in the space of immersed curves in Euclidean space R n≥3 . In particular, in the isotopy class of any given C 1 knot f there exists a C ∞ knot e f of constant curvature which is C 1 -close to f . More importantly, we show that if f is C 2 , then the curvature of e f may be set equal to any constant c which is not smaller than the maximum curvature of f . We may also require that e f be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory, and a sharp estimate for the position of the average value of a parameterized curve within its convex hull.

show abstract

Section: Introductionmentioning

confidence: 93%

Section: Proof Of Theorem 11mentioning

confidence: 99%

See 1 more Smart Citation

h-Principles for curves and knots of constant curvature

Ghomi

2007

Geom Dedicata

View full text Add to dashboard Cite

show abstract

“…Another very interesting topic is Gromov's h-principle, for which we refer the reader to [152], and [74]. Perhaps not only the author is tired at this point.…”

Section: Theorem 235 Let X Be a Compact Kähler Manifold And Let K Bementioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

View full text Add to dashboard Cite

One of the main themes of this long article is the study of projective varieties which are K(H,1)'s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)'s, through Teichmüller theory. The main trhust of the paper is to show how in the case of K(H,1)'s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lönne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an Communicated by Efim Zelmanov.The present work took place in the realm of the DFG Forschergruppe 790 "Classification of algebraic surfaces and compact complex manifolds". Part of the article was written when the author was visiting KIAS, Seoul, as KIAS research scholar.

show abstract

“…We now require 2(n + 1 − r) > n. Thus the codimension of S in X (1) is greater than the dimension of M . It follows from a basic result of differential topology that any map of M into X (1) may be perturbed so as to not intersect S. The (simplest case of) the Thom Transversality Theorem is more precise.…”

Section: Lemma 21 S Is a Stratified Subset Of Xmentioning

confidence: 99%