Problems on Fredholm sections of polyfold bundles, perturbations, and implicit function theorems
To simplify notation, we will study the Cauchy-Riemann equation for maps from the torus
to with a nonconstant almost complex structure .
The advanced versions of the following problems study maps on cylinders
,
,
.
Contents
- 1 strong scale-smooth bundles
- 2 strong M-polyfold bundles
- 3 regularizing and sections
- 4 sections of M-polyfold bundles
- 5 linear sc-Fredholm property
- 6 linear sc-Fredholm property on M-polyfold
- 7 nonlinear sc-Fredholm property
- 8 nonlinear sc-Fredholm property on M-polyfold
- 9 implicit function theorem
- 10 transverse perturbations
- 11
- 12
- 13
- 14
strong scale-smooth bundles
Understand in what sense forms a strong bundle over .
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strong M-polyfold bundles
Understand in what sense
forms a strong bundle over the M-polyfold
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regularizing and sections
Verify that is scale smooth and regularizing but not .
Then find some examples of sections of . (Hint: Think about classical geometric perturbations of the Cauchy-Riemann equation.)
Are perturbations of the almost complex structure -perturbations of ?
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sections of M-polyfold bundles
In what sense is the following section scale-smooth and regularizing?
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linear sc-Fredholm property
Compute the linearization of at a constant map , and show that it is a linear sc-Fredholm operator.
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linear sc-Fredholm property on M-polyfold
Compute the linearization of at points and , and show that both are linear sc-Fredholm operators.
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nonlinear sc-Fredholm property
Show that is an sc-Fredholm section. (Hint: Use the fact that it is continuously differentiable, and the linearizations are Fredholm.)
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nonlinear sc-Fredholm property on M-polyfold
Compute the expression for the section in the M-polyfold and bundle charts arising from plus-minus-(pre)gluing.
To simplify the technical setting in the following, restrict to functions/sections in .
Then find a filled section for by using the invertible operator .
Finally, sketch out a proof for this filled section being sc-Fredholm.
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implicit function theorem
Explain how the contraction germ form of an sc-Fredholm section is used in the proof that transverse sc-Fredholm sections have a smooth, finite dimensional, zero set.
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transverse perturbations
Suppose that an sc-Fredholm section of an M-polyfold bundle (modeled on scale-Hilbert spaces) is transverse except for a compact subset . Then, given any open neighbourhood Failed to parse (unknown function "\matcal"): \matcal{U}\subset\mathcal{B}
of , find a perturbation
supported in Failed to parse (unknown function "\matcal"): \matcal{U}
such that ,
by working through the following steps.
- For any find a stabilization Failed to parse (syntax error): P_z: B_\epsilon(z) \times O_z \to \mathcal{E}|_{B_\epsilon(z)
on a neighborhood of by a finite dimensional vector space ,
so that is a transverse section over .
- Use compactness of and the existence of smooth cutoff functions (on scale-Hilbert spaces) to construct a global section
that is transverse after stabilization with a finite dimensional vector space .
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