Scale calculus problems

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Scale calculus was developed to make reparametrization actions on function spaces smooth. All the key issues and ideas can already be seen at the example of S^{1}=\mathbb{R} /\mathbb{Z } acting on a space of nonconstant functions on S^{1}, for example the shift action

\sigma :S^{1}\times {\mathcal  {C}}_{{{\rm {id}}}}^{1}\;\to \;{\mathcal  {C}}_{{{\rm {id}}}}^{1},\quad (\tau ,u)\mapsto u(\tau +\cdot )

{\text{on}}\qquad \qquad \qquad \qquad {\mathcal  {C}}_{{{\rm {id}}}}^{1}:=\{u:S^{1}\to S^{1}\;|\;u\in {\mathcal  {C}}^{1}\;{\text{and homotopic to the identity}}\}.\qquad \qquad \qquad \qquad .

The purpose of the following exercises is to give the quotient of S^1-valued functions homotopic to the identity, modulo reparametrization by shifts, {\mathcal  {B}}:={\mathcal  {C}}_{{{\rm {id}}}}^{1}/S^{1} the structure of a scale-smooth manifold ... and to see that classical Banach space topologies fail at giving it a differentiable Banach manifold structure.

differentiability of shift map

Compute the directional derivatives of the shift map \sigma , first at (\tau ,u)=(0,0), then at (\tau ,u)\in S^{1}\times {\mathcal  {C}}_{{{\rm {id}}}}^{2}.

Next, understand how to identify T{\mathcal  {C}}_{{{\rm {id}}}}^{1}\simeq {\mathcal  {C}}^{1}(S^{1},\mathbb{R} ), and write down a conjectural formula for the differential d\sigma :\mathbb{R} \times {\mathcal  {C}}^{1}(S^{1},\mathbb{R} )\to {\mathcal  {C}}^{1}(S^{1},\mathbb{R} ).

Then recall the definition of differentiability in terms of uniform linear approximation and see whether it can be satisfied at any base point (\tau ,u).


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local charts

Verify that {\mathcal  {B}} has local Banach manifold charts modeled on E:=\{\xi \in {\mathcal  {C}}^{1}(S^{1},\mathbb{R} )\,|\,\xi (0)=0\}. To set up a chart near a given point, pick a representative [u_{0}]\in {\mathcal  {B}} with {\dot  u}_{0}(0)\neq 0 and consider the map

\phi _{{u_{0}}}:\{\xi \in E\,|\,\|\xi \|_{{{\mathcal  {C}}^{1}}}<\epsilon \}\to {\mathcal  {B}},\qquad \xi \mapsto [u_{0}+\xi ].


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transition maps

For [u_{0}]=[u_{1}]\in {\mathcal  {B}} with representatives u_{0}\neq u_{1} but {\dot  u}_{i}(0)\neq 0 determine the transition map \phi _{{u_{1}}}^{{-1}}\circ \phi _{{u_{0}}}.

Then find an example of a point and direction in which this transition map is not differentiable.


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alternative Banach norms

Explain why working with {\mathcal  {C}}^{k}, H\"older, or Sobolev spaces W^{{k,p}} does not resolve these differentiabliity issues. Possibly come up with other ideas for a Banach manifold structure on {\mathcal  {B}}.


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reparametrization on Riemann surfaces

Explore the same questions for a space of nonconstant functions u:\Sigma \to \mathbb{R} on a Riemann surface (\Sigma ,j) modulo the reparametrization action (\psi ,u)\mapsto u\circ \psi of a nontrivial automorphism group {{\rm {Aut}}}(\Sigma ,j)=\{\psi :\Sigma \to \Sigma \,|\,\psi ^{*}j=j\} (e.g. \Sigma =S^{2}{\text{or}}\;T^{2}).


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scale smoothness of the shift map

Show that the shift action \sigma is a scale-smooth map by the following steps:

a) Find a scale-Banach space {\mathbb  {E}}=(E_{\ell })_{{\ell \geq 0}} with E_{0}=C^{1}(S^{1},{\mathbb  R}^{2}).

b) Show that \widetilde \sigma :S^{1}\times {\mathbb  {E}}\to {\mathbb  {E}} is sc^{1} and calculate the linearization T\widetilde \sigma :(\tau ,u,t,\xi )\mapsto {\bigl (}\widetilde \sigma (\tau ,u),d_{{(\tau ,u)}}\widetilde \sigma (T,u){\bigr )}.

c) Iteratively show that \widetilde \sigma is sc^{k} for all k.

Note: The actual map \sigma above is a restriction of the map \widetilde \sigma to the subset {\mathcal  {C}}_{{{\rm {id}}}}^{1}\subset {\mathbb  {E}}, where we view S^{1}=\{|z|=1\}\subset \mathbb{R} ^{2}.


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scale-smooth structure on {\mathcal  {B}}

Now use the charts and transition maps discussed above to equip {\mathcal  {B}} with the structure of a scale-manifold (i.e. modeled on a scale-Banach space, with scale-smooth transition maps).


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