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 $T^{2}=\{(s,t)\in \mathbb{R} ^{2}\}/(s,t)\sim (s+1,t)\sim (s,t+1)$

to $\mathbb{R} ^{{2n}}$ with a nonconstant almost complex structure $J:\mathbb{R} ^{{2n}}\to \mathbb{R} ^{{2n\times 2n}}$.

The advanced versions of the following problems study maps on cylinders

$Z_{R}=\{(s,t)\in \mathbb{R} ^{2}\,|\,-R\leq s\leq R\}/(s,t)\sim (s,t+1)$,

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

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

strong scale-smooth bundles

Understand in what sense ${\mathcal {E}}=\textstyle \bigcup _{{u\in {\mathcal {B}}}}H^{2}(T^{2},u^{*}{{\rm {T}}}\mathbb{R} ^{{2n}})$ forms a strong bundle over ${\mathcal {B}}=H^{3}(T^{2},\mathbb{R} ^{{2n}})$.

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strong M-polyfold bundles

Understand in what sense

$\textstyle {\mathcal {E}}_{Z}=\bigcup _{{u\in {\mathcal {B}}_{Z}}}H^{2}(Z_{R},u^{*}{{\rm {T}}}\mathbb{R} ^{{2n}})\;\cup \;\bigcup _{{(u^{-},u^{+})\in {\mathcal {B}}_{Z}}}H^{2}(Z^{-},u_{-}^{*}{{\rm {T}}}\mathbb{R} ^{{2n}})\times H^{2}(Z^{+},u_{+}^{*}{{\rm {T}}}\mathbb{R} ^{{2n}})$

forms a strong bundle over the M-polyfold $\textstyle {\mathcal {B}}_{Z}=\bigcup _{{R\geq 42}}H^{3}(Z_{R},\mathbb{R} ^{{2n}})\;\cup \;H^{3}(Z^{-},\mathbb{R} ^{{2n}})\times H^{3}(Z^{+},\mathbb{R} ^{{2n}})$

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regularizing and $sc^{+}$ sections

Verify that $\sigma :{\mathcal {B}}\to {\mathcal {E}},\;u\mapsto (u,\partial _{s}u+J(u)\partial _{t}u)$ is scale smooth and regularizing but not $sc^{+}$.

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

Are perturbations of the almost complex structure $sc^{+}$-perturbations of $\sigma$?

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sections of M-polyfold bundles

In what sense is the following section scale-smooth and regularizing?

$\sigma _{Z}:{\mathcal {B}}_{Z}\to {\mathcal {E}}_{Z},\;b\mapsto {\begin{cases}{\bigl (}u,\partial _{s}u+J(u)\partial _{t}u{\bigr )}&;b=u\\{\bigl (}(u_{-},u_{+}),{\bigl (}\partial _{s}u_{-}+J(u_{-})\partial _{t}u_{-},\partial _{s}u_{+}+J(u_{+})\partial _{t}u_{+}{\bigr )}{\bigr )}&;b=(u_{-},u_{+})\end{cases}}$

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linear sc-Fredholm property

Compute the linearization of $\sigma$ at a constant map $u(s,t)\equiv p\in \mathbb{R} ^{{2n}}$, 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 $\sigma _{Z}$ at points $u\in {\mathcal {B}}_{Z}$ and $(u_{-},u_{+})\in {\mathcal {B}}_{Z}$, and show that both are linear sc-Fredholm operators.

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nonlinear sc-Fredholm property

Show that $\sigma$ 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 $\sigma _{Z}$ in the M-polyfold and bundle charts arising from plus-minus-(pre)gluing.

To simplify the technical setting in the following, restrict ${\mathcal {B}}_{Z},{\mathcal {E}}_{Z}$ to functions/sections in $H_{{{\rm {av}}}}^{k}(\ldots )=\{f\in H^{k}\,|\,\textstyle \int _{{t=0}}^{1}f(s,t){{\rm {d}}}t=0\;{\text{for all}}\;s\}$.

Then find a filled section for $\sigma _{Z}$ by using the invertible operator $\partial _{s}+J(0)\partial _{t}:H_{{{\rm {av}}}}^{3}(\mathbb{R} \times S^{1},\mathbb{R} ^{{2n}})\to H_{{{\rm {av}}}}^{2}(\mathbb{R} \times S^{1},\mathbb{R} ^{{2n}})$.

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

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

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

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

(D) Use the Sard theorem to find regular values of the projection $\widetilde \sigma ^{{-1}}(0)\to O$, and understand why these yield transverse perturbations of $\sigma$. (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 $\nu _{0},\nu _{1}:{\mathcal {B}}\to {\mathcal {E}}$ are two transverse perturbations for an sc-Fredholm section $\sigma :{\mathcal {B}}\to {\mathcal {E}}$ 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 $(\sigma +\nu _{0})^{{-1}}(0)\sqcup (\sigma +\nu _{1})^{{-1}}(0)$.

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Compactness

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

• $\nu$ is small in the sense that $\|\nu \|\leq \epsilon$ for some norm $\|\cdot \|$ on the fibers of ${\mathcal {E}}$ ?

• $\nu$ is supported in a small neighborhood of $\sigma ^{{-1}}(0)$ ?

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

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