Eliashberg simplifies singularities in geometry.
problem Complex singularities in geometric structures.
method Philosophy of the h-principle and simplification techniques.
result Simplified understanding of singularities in geometry.
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48 results for “Eliashberg's h-principle”
Extends h-principle to stratified spaces using sheaf and jet theories.
problem Applying h-principle to stratified spaces.
method Developed new sheaf and bundle theories for stratified spaces, and proved the h-principle.
result Stratified continuous sheaves and homotopy fiber sheaves lead to the parametric h-principle.
Paper proves h-principles for symplectic structures and foliations.
problem Existence of conformal symplectic structures and foliations.
method Application of h-principles and foliated Morse theory.
result Linear deformation of foliations to contact structures.
We extend Y.Eliashberg's -principle to smooth maps of surfaces which are allowed to have cusp singularities, as well as folds. More precisely, we prove a necessary and sufficient condition for a given map of surfaces to be homotopic to one with given loci of folds and cusps. Then we use these results to obtain a nec…
Study shows how to section map between holonomic and formal solutions.
problem Relative -principle for overtwisted contact structures.
method Apply relative -principle to contact geometry.
result Find infinite cyclic subgroups in contactomorphism group.
Anti-self-dual blowups IImath.DG
Study shows how to create special metrics on 4-manifolds with certain spheres.
problem Creating metrics with specific properties on 4-manifolds with embedded spheres.
method Using Eliashberg's h-principle for overtwisted contact structures to construct self-dual harmonic forms.
result Proves existence of metrics on 4-manifolds with anti-self-dual harmonic forms for certain spheres.
Paper generalizes wrinkled embedding concept to jet spaces.
problem Approximating homotopies of embeddings with singularities.
method Defines wrinkled embeddings for differential information in jet spaces.
result Holonomic approximation theorem holds for multi-valued sections with simple singularities.
Let M and N be closed n-dimensional manifolds, and equip N with a volume form σ. Let μbe an exact n-form on M. Arnold then asked the question: When can one find a map f:;N such that f*σ=μ. In 1973 Eliashberg and Gromov showed that this problem is, in a deep sense, trivial: It satisfies an h-principle, and whenever one …
Classifies convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
problem Classifying convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
method Contact isotopy classification, h-principle, fundamental groups, contact mapping class group.
result Establishes an h-principle for convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
Analytic curves have infinite codimension of singular germs.
problem Understanding the codimension of singular tangent curves in analytic distributions.
method Formalizing asymptotic statements about finite jets of tangent curves and applying the h-principle.
result The subspace of singular germs has infinite codimension within smooth curves.
The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working on…
We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashberg's h-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected a…
In this paper we extend Y.Eliashberg's -principle to arbitrary generic smooth maps of smooth manifolds. Namely, we prove a necessary and sufficient condition for a continuous map of smooth manifolds of the same dimension to be homotopic to a generic map with a prescribed Thom-Boardman singularity at each point…
This paper generalizes wrinkling techniques to Haefliger structures, linking them to foliations.
problem Proving h-principles for partial differential relations with controlled singularities.
method Generalizing wrinkled embeddings to Haefliger structures and interpreting them as holonomic approximations.
result Haefliger structures provide a framework for making general wrinkling statements and imply connectivity results.
This paper solves a complex differential relation using a novel 'avoidance trick'.
problem Classifying tangent distributions satisfying non-involutivity conditions.
method Convex integration with an 'avoidance trick'.
result First example of a differential relation that is ample in some directions but not all.
Shows flexible sheaves as fibrant objects for Gromov's h-principle.
problem Applying the h-principle to partial differential relations.
method Interprets flexible sheaves as fibrant objects in a model structure.
result Flexible sheaves can be understood as fibrant objects.
New method proves -principles for stable forms on manifolds.
problem Proving -principles for stable forms on manifolds.
method Convex integration applied to stable forms.
result Proved -principles for 4 classes of stable forms.
In this paper we give three applications of a method to prove h-principles on closed manifolds. Under weaker conditions this method proves a homological h-principle, under stronger conditions it proves a homotopical one. The three applications are as follows: a homotopical version of Vassiliev's h-principle, the contra…
Maps to manifolds transverse to certain distributions satisfy an -principle.
problem Maps to manifolds transverse to certain distributions.
method Proving the complete -principle for transverse maps.
result Maps to manifolds transverse to certain distributions satisfy the -principle.
Study h-principles for non-integrable distributions on manifolds.
problem Existence and classification of maximally non-integrable distributions of derived length one.
method Introduced formal structures and used h-principles to discuss existence and classification.
result Discussed existence and classification of maximally non-integrable distributions of derived length one.
Proves h-principle for loose Legendrian embeddings in contact topology.
problem Existence of non-loose Legendrian embeddings.
method h-principle from microlocal sheaf theory.
result Existence of non-loose Legendrian embeddings.
Bing's house-like spines approximate all PL manifolds.
problem Understanding spines of PL manifolds.
method Approximation rigidity and h-principle.
result PL manifolds have spines similar to Bing's house.
The paper extends local h-principles to complex structures on Stein manifolds.
problem Existence of local h-principles for complex structures on Stein manifolds.
method Introducing realifications of partial holomorphic relations and proving h-principles for them.
result Local h-principles can be extended to complex structures on Stein manifolds.
The h-principle helps solve complex geometric problems.
problem Solving complex geometric problems using the h-principle.
method Developed from the Oka-Grauert principle and Gromov's theory, the h-principle is applied to Oka manifolds and maps.
result Recent developments and applications of the h-principle in complex analysis and geometry.
Study proves h-principles for curves in bracket-generating distributions.
problem Proving h-principles for curves in higher-dimensional bracket-generating distributions.
method Proves complete h-principles for embedded regular horizontal and transverse curves.
result Contrasts with 3D contact case, where full h-principle for transverse/legendrian knots does not hold.
Following a line of reasoning suggested by Eliashberg, we prove Cerf's theorem that any diffeomorphism of the 3-sphere extends over the 4-ball. To this end we develop a moduli-theoretic version of Eliashberg's filling-with-holomorphic-discs method.
The study proves a strong parametric h-principle for minimal surfaces.
problem Proving a parametric h-principle for minimal surfaces.
method Using a parametric h-principle due to Forstneric and Larusson.
result The space of complete nonflat conformal minimal immersions has the same homotopy type as the space of continuous maps.
Proves bijection between smooth conformal immersions and immersions.
problem Finding conformal immersions of closed Riemannian surfaces.
method Reformulated using -principle and proved bijection on path connected components.
result Induces a bijection between smooth conformal immersions and immersions.
A bypass in the middlemath.GT
New proofs of h-principles in contact 3-manifolds.
problem Characterizing contractible bypasses in contact 3-manifolds.
method Topological characterization and disjoint bypass construction.
result New proofs of h-principles in overtwisted contact 3-manifolds.
Proves existence of strongly overtwisted contact structures on 3-manifolds.
problem Existence of overtwisted contact structures on 3-manifolds.
method Complete h-principle for overtwisted disks and Legendrians.
result Homotopy type of overtwisted disk embeddings is determined.
We prove -principle for locally conformal symplectic foliations and contact foliations on open manifolds. We interpret the result on principle of contact foliations in terms of the regular Jacobi structures.
Curves with constant torsion can be deformed arbitrarily.
problem Deforming curves of constant torsion in Euclidean space.
method Convex integration and degree theory.
result Existence of knots with constant torsion in each isotopy class.
This paper formalizes the h-principle and sphere eversion in differential topology.
problem Formalizing the h-principle and sphere eversion in differential topology.
method Lean formalization of the local h-principle for first-order partial differential relations, using convex integration.
result Reproves Smale's sphere eversion theorem and formalizes advanced mathematics.
Proves the relative h-principle for SL(3,R)^2 3-forms on 6-manifolds.
problem Proving the relative h-principle for specific 3-forms on 6-manifolds.
method Uses convex integration with avoidance and transversality arguments.
result Proves the h-principle for SL(3,R)^2 3-forms on oriented 6-manifolds.
In this note we survey some recent results for the Euler equations in compressible and incompressible fluid dynamics. The main point of all these theorems is the surprising fact that a suitable variant of Gromov's -principle holds in several cases.
We apply the barcodes of persistent homology theory to the Chekanov-Eliashberg algebra of a Legendrian submanifold to deduce displacement energy bounds for arbitrary Legendrians. We do not require the full Chekanov-Eliashberg algebra to admit an augmentation as we linearize the algebra only below a certain action level…
We present a new, completely three-dimensional proof of the fact, due to Gabai-Eliashberg-Thurston, that every closed, oriented, irreducible 3-manifold with nonzero second homology carries a universally tight contact structure.
Smooth curves with specific curvature can be closely approximated.
problem Approximating smooth curves with prescribed curvature.
method Application of -principle to -dense approximation of curves.
result Existence of knots with prescribed curvature.
The Gromov-Eliashberg theorem says that the group of symplectomorphisms of a symplectic manifold is C^0-closed in the group of diffeomorphisms. This can be translated into a statement about the Lagrangian submanifolds which are graphs of symplectomorphisms. It is also known that such Lagrangian submanifolds are locally…
Generalizes Thurston's jiggling lemma for piecewise smooth solutions.
problem Creating piecewise smooth solutions of differential relations without homotopical assumptions.
method Jiggling arbitrary sections of to construct solutions of .
result Generalization of Thurston's lemma for piecewise smooth solutions of differential relations.
The paper studies how Lagrangian cobordisms affect DGAs of Legendrian ends.
problem Understanding how Lagrangian cobordisms impact DGAs of Legendrian ends.
method Adapting the map induced by cobordisms on DGAs to linearizations using augmentations, and showing invariance under Lagrangian isotopy.
result The induced map on linearized Legendrian contact homology is invariant under Lagrangian isotopy under mild hypotheses.
Maximally non-integrable almost complex structures: an -principle and cohomological propertiesmath.DG
Study on almost complex structures with maximal Nijenhuis tensor rank and cohomological properties.
problem Maximally non-integrable almost complex structures and their cohomological properties.
method h-principle and topological invariants characterization.
result Existence of almost complex structures with maximal Nijenhuis tensor rank on parallelizable and certain manifolds.
This paper presents a natural extension to foliated spaces of the following result due to Gromov : the h-principle for open, invariant differential relations is valid on open manifolds. The definition of openness for foliated spaces adopted here involves a certain type of Morse functions. Consequences concerning the pr…
We give a proof of, for the case of contact structures defined by global contact 1-forms, a Theorem stated by Eliashberg that for any overtwisted contact structure on a closed 3-manifold, its contact homology is 0. A different proof is also outlined in the appendix by Yakov Eliashberg.
We link Ginzburg algebras to Weinstein manifolds and Legendrian knots.
problem Understanding the relationship between Ginzburg algebras and Weinstein manifolds.
method Associated a stopped Weinstein manifold to a quiver and subquiver, proving quasi-isomorphism of relative Ginzburg algebra and Chekanov-Eliashberg dg-algebra.
result Relative Ginzburg algebra is quasi-isomorphic to Chekanov-Eliashberg dg-algebra of a singular Legendrian unknot link.
We prove an analogue of Thurston's h-principle for -dimensional foliations on manifolds of dimension bigger or equal to , in the presence of a fiber-wise non-degenerate -form. This helps us understand the flexibility of rank regular Poisson structures on open manifolds with dimension bigger or equal to …
We show that a classical result of Gromov in symplectic geometry extends to the context of symplectic foliations, which we regard as a -principle for (regular) Poisson geometry. Namely, we formulate a sufficient cohomological criterion for a regular bivector to be homotopic to a regular Poisson structure, in the spi…
We prove an h-principle for poisson structures on closed manifolds.