arXiv research

Shows flexible sheaves as fibrant objects for Gromov's h-principle.

Extends h-principle to stratified spaces using sheaf and jet theories.

In 1969 M. Gromov in his PhD thesis greatly generalized Smale-Hirsch-Phillips immersion-submersion theory 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 generali…

The h-principle helps solve complex geometric problems.

Maps to manifolds transverse to certain distributions satisfy an $h$-principle.

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 $h$-principle holds in several cases.

Proves h-principle for loose Legendrian embeddings in contact topology.

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 show that a classical result of Gromov in symplectic geometry extends to the context of symplectic foliations, which we regard as a $h$-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…

Analytic curves have infinite codimension of singular germs.

The article proves the existence of horizontal immersions into fat distributions and contact structures.

This paper formalizes the h-principle and sphere eversion in differential topology.

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 …

We show that the 2-jet bundle of local Riemannian metrics on an arbitrary differentiable manifold admits a section which pointwise fulfills the curvature relation sec(g)=a for any real number a. It follows by Gromov's h-principle for open, invariant differential relations that every noncompact differentiable manifold c…

The first part of this article intends to present the role played by Thom in diffusing Smale's ideas about immersion theory, at a time (1957) where some famous mathematicians were doubtful about them: it is clearly impossible to make the sphere inside out! Around a decade later, M. Gromov transformed Smale's idea in wh…

On a Weinstein manifold, we define a constructible co/sheaf of categories on the skeleton. The construction works with arbitrary coefficients, and depends only on the homotopy class of a section of the Lagrangian Grassmannian of the stable symplectic normal bundle. The definition is as follows. Take any, possibly high …

The immersions of a smooth manifold $M$ in a symplectic manifold $(N,σ)$ inducing a given closed form $ω$ on $M$ satisfy the $C^0$-dense $h$-principle in the space of all continuous maps which pull back the deRham cohomology class of $σ$ onto that of $ω$. In this paper we prove a foliated version of this result due to …

The paper solves symplectic embedding problems in higher dimensions, proving new embedding conditions.

The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the $h-$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…

The Nash-Kuiper Theorem states that the collection of $C^1$-isometric embeddings from a Riemannian manifold $M^n$ into $\mathbb{E}^N$ is $C^0$-dense within the collection of all smooth 1-Lipschitz embeddings provided that $n < N$. This result is now known to be a consequence of Gromov's more general $h$-principle. Ther…

Let $Σ$ be a codimension one submanifold of an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We give a necessary condition for an isometric immersion of $Σ$ into $\mathbb R^q$ equipped with the standard Euclidean metric, $q\geqslant n+1$, to be locally isometrically $C^1$-extendable to $M$. Even if this cond…

The paper solves a 25-year-old problem about maximal growth distributions on manifolds.

New method proves $h$-principles for stable forms on manifolds.

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…

Paper proves h-principles for symplectic structures and foliations.

For some geometries including symplectic and contact structures on an n-dimensional manifold, we introduce a two-step approach to Gromov's h-principle. From formal geometric data, the first step builds a transversely geometric Haefliger structure of codimension n. This step works on all manifolds, even closed. The seco…

Study h-principles for non-integrable distributions on manifolds.

Bing's house-like spines approximate all PL manifolds.

The paper extends local h-principles to complex structures on Stein manifolds.

Study proves h-principles for curves in bracket-generating distributions.

Study shows how to section map between holonomic and formal solutions.

The study proves a strong parametric h-principle for minimal surfaces.

Proves bijection between smooth conformal immersions and immersions.

New proofs of h-principles in contact 3-manifolds.

Proves existence of strongly overtwisted contact structures on 3-manifolds.

We prove $h$-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We interpret the result on $h$ principle of contact foliations in terms of the regular Jacobi structures.

Curves with constant torsion can be deformed arbitrarily.

Eliashberg simplifies singularities in geometry.

Proves the relative h-principle for SL(3,R)^2 3-forms on 6-manifolds.

Smooth curves with specific curvature can be closely approximated.

Generalizes Thurston's jiggling lemma for piecewise smooth solutions.

This paper solves a complex differential relation using a novel 'avoidance trick'.

Study on almost complex structures with maximal Nijenhuis tensor rank and cohomological properties.

In the mid-1980's, M. Gromov used his machinery of the $h$-principle to prove that there exists totally real embeddings of $S^3$ into $\mathbb{C}^3$. Subsequently, Patrick Ahern and Walter Rudin explicitly demonstrated such a totally real embedding. In this paper, we consider the generic situation for such embeddings, …

Totally real immersions $f$ of a closed real surface $Σ$ in an almost complex surface $M$ are completely classified, up to homotopy through totally real immersions, by suitably defined homotopy classes $\frak{M}(f)$ of mappings from $Σ$ into a specific real 5-manifold $E(M)$, while $\frak{M}(f)$ themselves are subject …

We prove an analogue of Thurston's h-principle for $2$-dimensional foliations on manifolds of dimension bigger or equal to $4$, in the presence of a fiber-wise non-degenerate $2$-form. This helps us understand the flexibility of rank $2$ regular Poisson structures on open manifolds with dimension bigger or equal to $4$…

We prove an h-principle for poisson structures on closed manifolds.

We prove that every immersed $C^2$-curve $γ$ in $\mathbb R^n$, $n\geqslant 3$ with curvature $k_γ$ can be $C^1$-approximated by immersed $C^2$-curves having prescribed curvature $k>k_γ$. The approximating curves satisfy a $C^1$-dense $h$-principle. As an application we obtain the existence of $C^2$-knots of arbitrary p…