# Questions tagged [h-principle]

The h-principle tag has no usage guidance.

14
questions

**6**

votes

**1**answer

236 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, ...

**0**

votes

**0**answers

70 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

178 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 \...

**7**

votes

**1**answer

171 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

171 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

106 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

228 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-...

**5**

votes

**1**answer

222 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 $...

**5**

votes

**0**answers

146 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

418 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

227 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-...

**25**

votes

**0**answers

766 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 ...

**29**

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 ...

**1**

vote

**1**answer

348 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 ...