Questions tagged [h-principle]

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Recognizing sections up to isotopy

Let $E$, $B$ be smooth manifolds, $\pi\colon E\to B$ be a smooth fiber bundle, and $h:B\to E$ be a smooth embedding. I would like to learn what is known about the following Question. When does there ...
5 votes
1 answer
275 views

Making a submanifold transverse to a vector field by an isotopy

Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known ...
6 votes
1 answer
322 views

Can the methods of algebra characterize nonlinear PDE blow-ups?

Consider 2 simple differential equations: $x'(t)=x(t)^2, x(0)=1$ and $x'(t)=-x(t)^2, x(0)=1$. As $t$ goes from $0$ to $\infty$, the first equation ($x(t)=1/(1-t)$) will lead to a finite-time blow-up, ...
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0 votes
0 answers
77 views

relationship between "linear approximation" to immersions and formal immersions

I'm reading these notes Here, I am regarding $\mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$ If we take $\mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element ...
5 votes
0 answers
185 views

h-principle for pairs

Let $A,B$ be complex analytic spaces. Suppose that $[A,B]$ satisfies the h-principle: i.e. every class of continuous function $f:A \to B$ up to homotopy, contains a holomorphic representative. Let $C \...
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7 votes
1 answer
205 views

Short embeddings for open manifolds and dimension reduction of sets

The question is maybe a bit technical, but I find the related construction very beautiful. In the very famous work - "$C^1$-isometric imbeddings" by J.Nash (1954) the author presented the ...
6 votes
1 answer
219 views

Immersions of manifolds with boundary (regular homotopy classes, h-principle)

Have regular homotopy classes of immersions of manifolds with boundary been studied? i.e. immersions $(M, \partial M) \looparrowright (N, \partial N)$. The references I've looked at from this question ...
2 votes
0 answers
108 views

Wrinkling smooth functions

I am interested in applying a result from the work by Eliashberg and Mishachev on wrinkling. Namely, in their first paper on wrinkling, they prove Theorem 1.6 B (Theorem 1.6 A is a non-parameterized ...
4 votes
0 answers
240 views

H-principle for smoothing

I'm trying to find algebraic embedding of singular curves into projective varieties that can be smoothed symplectically but not algebraically. It's not hard (e.g. using the methods in Hartshorne-...
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5 votes
1 answer
233 views

Almost complex structures on a 4-ball that are not tamed

Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called tamed if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $...
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5 votes
0 answers
153 views

h-principle on Hilbert manifolds

Gromov's h-principle is a powerful tool in studying various geometric structure on open, finite-dimensional manifolds. Is there any generalization of h-principle to (necessarily open) infinite-...
2 votes
1 answer
426 views

Question about the h-principle

So generally we define a differential relation to be $\mathcal{R} \subset X^{(r)}.$ In the case that $X=M\times N$ is it possible to have $\mathcal{R}=X^{(1)}$? So in this case the formal solutions ...
3 votes
0 answers
238 views

The relative h-principle and extension problems

As a beginner for h-principles, I want to know why the relative h-principle cannot imply a positive solution to the problems for extending symplectic structures. The following is a relative h-...
26 votes
1 answer
980 views

Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?

Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
31 votes
5 answers
6k views

H-principle and PDE's

According to Wikipedia: "In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations ...
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1 vote
1 answer
362 views

Holomorphic h-principle for compact manifolds

The Oka principle for Stein manifolds says (roughly) that the only obstructions for "things" are topological obstructions (for instance every smooth complex vector bundle admits a holomorphic ...
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