Questions tagged [h-principle]
The h-principle tag has no usage guidance.
16
questions
7
votes
1
answer
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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 ...
- 483
5
votes
1
answer
388
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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 ...
- 483
6
votes
1
answer
399
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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, ...
- 179
0
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0
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85
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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 ...
- 131
5
votes
0
answers
188
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 $C \...
- 1,379
7
votes
1
answer
240
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Short embeddings for open manifolds and dimension reduction of sets
The question is maybe a bit technical, but I find the related construction very beautiful.
In the very famous work - "$C^1$-isometric imbeddings" by J.Nash (1954) the
author presented the ...
- 537
6
votes
1
answer
253
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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 ...
- 176
2
votes
0
answers
140
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 ...
- 383
4
votes
0
answers
243
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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-...
- 1,971
5
votes
1
answer
246
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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 $...
- 14k
5
votes
0
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164
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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-...
- 606
2
votes
1
answer
450
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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 ...
- 23
3
votes
0
answers
262
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-...
- 31
26
votes
1
answer
1k
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 ...
- 27.2k
31
votes
5
answers
7k
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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 ...
- 375
1
vote
1
answer
378
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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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