View source for 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 <math>T^2 = \{(s,t)\in \R^2 \}/ (s,t)\sim (s+1,t)\sim (s,t+1)</math> 

to <math>\R^{2n}</math> with a nonconstant almost complex structure <math>J : \R^{2n} \to \R^{2n\times 2n}</math>. 

The advanced versions of the following problems study maps on cylinders 

<math>Z_R = \{(s,t)\in \R^2 \,|\, -R \leq s\leq R \} /(s,t)\sim (s,t+1)</math>, 

<math>Z^- = \{(s,t)\in \R^2 \,|\, s\leq 0 \} /(s,t)\sim (s,t+1)</math>, 

<math>Z^+ = \{(s,t)\in \R^2 \,|\, s\geq 0 \} /(s,t)\sim (s,t+1)</math>. 



== strong scale-smooth bundles  ==
Understand in what sense <math>\mathcal{E}=\textstyle\bigcup_{u\in \mathcal{B}} H^2(T^2, u^*{\rm T}\R^{2n})</math> 
forms a strong bundle over <math>\mathcal{B}=H^3(T^2, \R^{2n})</math>.



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== strong M-polyfold bundles  ==

Understand in what sense 
<center>
<math>
\textstyle\mathcal{E}_Z=\bigcup_{u\in \mathcal{B}_Z} H^2(Z_R, u^*{\rm T}\R^{2n}) \;\cup\;  
\bigcup_{(u^-,u^+)\in \mathcal{B}_Z} H^2(Z^-, u_-^*{\rm T}\R^{2n}) \times H^2(Z^+, u_+^*{\rm T}\R^{2n})
</math> 
</center>
forms a strong bundle over the M-polyfold 
<math>\textstyle
\mathcal{B}_Z=  \bigcup_{R\geq 42} H^3(Z_R, \R^{2n}) \;\cup\;  H^3(Z^-, \R^{2n})\times H^3(Z^+, \R^{2n})
</math>



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== regularizing and <math>sc^+</math> sections  ==

Verify that <math>\sigma: \mathcal{B}\to \mathcal{E}, \; u \mapsto (u, \partial_s u + J(u)\partial_t u )</math> is scale smooth and regularizing but not <math>sc^+</math>. 

Then find some examples of <math>sc^+</math> sections of <math>\mathcal{E}\to \mathcal{B}</math>. (Hint: Think about classical geometric perturbations of the Cauchy-Riemann equation.) 

Are perturbations of the almost complex structure <math>sc^+</math>-perturbations of <math>\sigma</math>?



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== sections of M-polyfold bundles  ==

In what sense is the following section scale-smooth and regularizing?
<center>
<math>\sigma_Z: \mathcal{B}_Z\to \mathcal{E}_Z, \;  b \mapsto 
\begin{cases}
\bigl(u, \partial_s u + J(u)\partial_t u \bigr) &; b=u \\
\bigl( (u_-,u_+) , \bigl( \partial_s u_- + J(u_-)\partial_t u_-, \partial_s u_+ + J(u_+)\partial_t u_+  \bigr) \bigr) &; b=(u_-,u_+)
\end{cases}
</math>
</center>



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== linear sc-Fredholm property  ==

Compute the linearization of <math>\sigma</math> at a constant map <math>u(s,t)\equiv p \in \R^{2n}</math>, and show that it is a linear sc-Fredholm operator. 

<div class="toccolours mw-collapsible mw-collapsed">
Solution
<div class="mw-collapsible-content">
Note that, since <math>u(s,t)\equiv p</math>, we have that <math>\mathcal{E}_u\cong H^2(T^2,T_p\mathbb{R}^{2n})\cong H^2(T^2,\mathbb{R}^{2n})</math> as well as <math>\partial_su=0</math> and <math>\partial_tu=0</math> so the linearization of <math>\sigma:u\mapsto(u,\overline{\partial}_Ju)</math> is given by <div style="text-align: center;"> <math>\begin{align} D_u\overline{\partial}_J:\xi&\mapsto\left.\frac{d}{d\varepsilon}\overline{\partial}_J(u+\varepsilon\xi)\right|_{\varepsilon=0}\\&=\left.\frac{d}{d\varepsilon}(\partial_s(u+\varepsilon\xi)+J(u+\varepsilon\xi)\partial_t(u+\varepsilon\xi))\right|_{\varepsilon=0}\\&=\left.\frac{d}{d\varepsilon}((\varepsilon\partial_s\xi)+\varepsilon J(u+\varepsilon\xi)\partial_t\xi)\right|_{\varepsilon=0}\\&=\partial_s\xi+\left.\frac{d}{d\varepsilon}(\varepsilon J(u+\varepsilon\xi)\partial_t\xi)\right|_{\varepsilon=0}\\&=\partial_s\xi+\left.\left(J(u+\varepsilon\xi)\partial_t\xi+\varepsilon\frac{d}{d\varepsilon}(J(u+\varepsilon\xi)\partial_t\xi)\right)\right|_{\varepsilon=0}\\&=\partial_s\xi+J(u)\partial_t\xi=(\partial_s+J(u)\partial_t)\xi \end{align}</math> </div> which is to say that <math>D_u\overline{\partial}_J=\partial_s+J\partial_t</math>, where <math>J=J(u)</math> is constant. In other words, the linearization of <math>\sigma</math> at a constant map <math>u(s,t)\equiv p</math> is a Cauchy-Riemann operator. It then suffices to show that <math>D_u\overline{\partial}_J</math> is <math>sc</math>-Fredholm but such Cauchy-Riemann operators are Fredholm as maps <math>\mathcal{B}_0\to\left(\mathcal{E}_u\right)_0</math>, regularizing by elliptic regularity, and bounded at each level, all from the standard theory. Thus, by Lemma 3.6 of [//arxiv.org/abs/1209.4040?title= Fredholm notions in scale calculus and Hamiltonian Floer theory], by Katrin Wehrheim, <math>D_u\overline{\partial}_J</math> is <math>sc</math>-Fredholm.
</div></div>

== linear sc-Fredholm property on M-polyfold ==

Compute the linearization of <math>\sigma_Z</math> at points <math>u\in \mathcal{B}_Z</math> and <math>(u_-,u_+)\in \mathcal{B}_Z</math>, and show that both are linear sc-Fredholm operators. 


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==   nonlinear sc-Fredholm property ==

Show that <math>\sigma</math> 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 <math>\sigma_Z</math> in the M-polyfold and bundle charts arising from plus-minus-(pre)gluing. 

To simplify the technical setting in the following, restrict <math>\mathcal{B}_Z, \mathcal{E}_Z</math> to functions/sections in <math>H^k_{\rm av}(\ldots) =\{ f\in H^k \,|\, \textstyle\int_{t=0}^1f(s,t) {\rm d}t=0 \;\text{for all}\; s\}</math>. 

Then find a filled section for <math>\sigma_Z</math> by using the invertible operator <math>\partial_s + J(0) \partial_t : H^3_{\rm av}(\R\times S^1 , \R^{2n}) \to H^2_{\rm av}(\R\times S^1 , \R^{2n})</math>.

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 <math>\sigma:\mathcal{B}\to \mathcal{E}</math> of an M-polyfold bundle (modeled on scale-Hilbert spaces)
is transverse except for a compact subset <math>Z\subset \sigma^{-1}(0)</math>. Then, given any open neighbourhood <math>\mathcal{U}\subset\mathcal{B}</math> of <math>Z</math>, find a perturbation <math>\nu:\mathcal{B}\to \mathcal{E}</math> supported in <math>\mathcal{U}</math> such that <math>\sigma+\nu \pitchfork 0</math>, 
by working through the following steps.

(A) For any <math>z\in Z</math> find a ''stabilization'' <math>P_z: B_\epsilon(z) \times O_z \to \mathcal{E}|_{B_\epsilon(z)}</math> on a neighborhood <math>B_\epsilon(z) \subset \mathcal{B}</math> of <math>z</math> by a finite dimensional vector space <math>O_z</math>, so that <math>(\sigma + P_z)(b,o)=\sigma(b)+P_z(b,o)</math> is a transverse section over <math>B_\epsilon(z) \times O_z</math>.

(B) Use compactness of <math>Z</math> and the existence of smooth cutoff functions (on scale-Hilbert spaces) to construct a global section 
<math>\widetilde\sigma : \mathcal{B} \times O \to \mathcal{E}, \; (b,o) \mapsto \sigma(b) + \ldots </math> that is transverse after stabilization with a finite dimensional vector space <math>O</math> and agrees with <math>\sigma</math> outside of <math>\mathcal{U}\times O</math>.

(C) Explain why <math>\widetilde\sigma</math> is sc-Fredholm, then apply the polyfold implicit function theorem to it.

(D) Use the Sard theorem to find regular values of the projection <math>\widetilde\sigma^{-1}(0) \to O</math>, and understand why these yield transverse perturbations of <math>\sigma</math>. 
(Hint: This is analogous to the universal moduli space approach for finding regular almost complex structures or Hamiltonian perturbations in geometric regularization methods.)



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== Invariance  ==

Suppose that <math>\nu_0,\nu_1:\mathcal{B}\to \mathcal{E}</math> are two transverse perturbations for an sc-Fredholm section <math>\sigma:\mathcal{B}\to \mathcal{E}</math> of an M-polyfold bundle (modeled on scale-Hilbert spaces) without boundary or corners.
Explain how to obtain a (not yet necessarily compact) manifold with boundary <math>(\sigma+\nu_0)^{-1}(0) \sqcup (\sigma+\nu_1)^{-1}(0)</math>.



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== Compactness  ==

Suppose that an sc-Fredholm section <math>\sigma:\mathcal{B}\to \mathcal{E}</math> of an M-polyfold bundle has compact zero set. 
What can be said about the perturbed zero set <math>(\sigma+\nu)^{-1}(0)</math> for a scale-smooth perturbation <math>\nu:\mathcal{B}\to \mathcal{E}</math> if

* <math>\nu</math> is ''small'' in the sense that <math>\|\nu\|\leq \epsilon</math> for some norm <math>\| \cdot \|</math> on the fibers of <math>\mathcal{E}</math> ? 

* <math>\nu</math> is supported in a ''small'' neighborhood of <math>\sigma^{-1}(0)</math> ? 

Then understand how the <math>sc^+</math> property of a perturbation <math>\nu</math>, together with boundedness in an ''auxiliary norm'', and ''smallness'' in the second sense, can guarantee compactness of the perturbed zero set <math>(\sigma+\nu)^{-1}(0)</math>. 



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