# Maps of loop spaces with infinity-bounded differential.

I am currently working with loop spaces of manifold and finite dimensional manifolds approximating these and the following comes up very naturally:

In the following piece-wise smooth means smooth on each set of a closed covering, implying continuity and boundedness of 1-sided derivatives.

For any closed Riemannian manifold $M$ define $\Lambda M$ as the space of piece-wise smooth maps from $S^1=I/\{0,1\}$ to M. define the energy of $\gamma \in \Lambda M$ by

$E(\gamma)=\int_{S^1} \mid\mid \gamma'(t)\mid\mid^2dt$

This follows Milnors book on Morse theory and the tangent space of $\Lambda M$ at $\gamma$ is defined to be piece-wise smooth tangent fields along $\gamma$ (WARNING: see comments by Andrew Stacey). Remark: with this definition $\gamma'$ may not be a tangent vector since it can be discontinuous. We may define the supremums norm on the tangent space by

$\mid\mid {\partial \gamma} \mid\mid_\infty = \sup_{t\in S^1} \mid\mid {\partial \gamma(s)}\mid\mid$

for any $\partial \gamma \in T\Lambda M$.

I define a bounded differentiable function $F\colon \Lambda M \to \Lambda N$ by the following criteria:

$E(F(\gamma)) \leq C_F E(\gamma)$ and $\mid\mid F_*(\partial \gamma)\mid\mid_\infty \leq C_F\mid \mid\partial \gamma\mid\mid_\infty$

for some $C_F>0$. Here $F_*$ is assumed to be well-defined using variations. Since I assume that $M$ and $N$ are closed this constant may depend on the Riemannian structures, but the notion does not. These arise e.g. as loops of differentiable maps $f\colon M\to N$, but I need them in their generality.

Question: Has anybody seen this notion of boundedness or maybe a similar local definition used anywhere?

Motivation: To begin with I felt this was an unnatural mix of $L^2$ and $L^\infty$, but working with these on the following spaces have made them feel much more natural: Define $\Lambda^\beta M$ as the space of loops with energy less than $\beta$, and define $\Lambda_r^\beta M$ as the space of piece-wise geodesics each piece parametrized by an interval of length $1/r$ with total energy less than $\beta$. If $\beta/r$ is small enough then this is a manifold given by the endpoints of the geodesics (see Milnors book on Morse theory). The above conditions are very suited for transfering arguments back and forth between $\Lambda^\beta M$ and $\Lambda_r^\beta M$ since the supremum norm is compatible with evaluations at points, but also the inclusions $\Lambda_r^\beta M \to \Lambda M$ for small $\beta/r$ is compatible. I need a lot of lemmas regarding these (e.g. existence of homotopy through such maps, when one have a continuous homotopy between two such which stayes constant outside a set of compact homotopy type) and if some one has already worked on this it can help me greatly.

One could ask why use the energy in the first place why not define $\Lambda_r^\beta M$ using length. The reasons is that the energy is more natural to use in the setting I am looking at, which is often the case since the energy is modelled on $L^2$ which is the nicest $L^p$ space.

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Minor technicality: $\Lambda M$ isn't really a manifold so you should be hesitant about speaking of "tangent spaces" and "tangent vectors" on it. This doesn't seem to be a problem in what you describe since either you are using approximations or aren't using the manifold structure at all, but it's important to keep in mind. (Incidentally, I haven't heard of the notion of "bounded differentiability" as you state it though it does remind me of the theory of inverse limit Hilbert manifolds) –  Loop Space May 11 '10 at 10:10
I know of these problems, but thought I would surpress it since it is not relevant to the question. I define my way out of this, and explicitly say that $F_*$ is defined by considering variations. –  Thomas Kragh May 11 '10 at 10:23
Fair enough, though someone reading your question may be fooled into thinking that there is such a manifold and I wish to guard against that trap. You could, though, take the space of all loops that have bounded energy. This would be some variation on C^1, I guess, or L^{2,1}. That space would be a bona-fide manifold, homotopy equivalent to smooth or continuous loops, so all your homotopy stuff should work. Incidentally, if you're working with things-up-to-homotopy then why not just work with smooth loops? –  Loop Space May 11 '10 at 10:46
The reason I am not using smooth loops is because I need to move back and forth between the finite-dimensional approximation manifolds and the infinte-dimensional loop spaces, and this is easiest in this case, e.g. because the inclusion $\Lambda_r^\beta M \to \Lambda M$ is easy to define. I need the finite-dimensionality because I am creating finite reductions of actions integrals in symplectic geometry, and the Morse index of the action integral is infinite on the infinite dimensional manifolds. –  Thomas Kragh May 11 '10 at 12:01
I feel as though there's more to this problem than you're telling us (which is completely fine), and I don't want to distract from the point of the question (especially as I don't have an answer), so I'm wary of pursuing this matter here as it is tangential (ha ha) to your actual question. But anything to do with loop spaces interests me so I'd like to learn more about what you're doing with them. So if you'd like to continue this discussion, I propose shifting it to the nForum: <math.ntnu.no/~stacey/Vanilla/nForum>; –  Loop Space May 11 '10 at 12:58