# Questions tagged [h-principle]

The h-principle tag has no usage guidance.

13
questions

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### 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 ...

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### 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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152 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**

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**1**answer

143 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 ...

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98 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 ...

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217 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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votes

**1**answer

203 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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143 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-...

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**1**answer

390 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 ...

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220 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-...

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704 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 ...

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**5**answers

5k 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**answer

334 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 ...