Maps of loop spaces with infinity-bounded differential.

2010-05-11T09:58:29Z

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.

Maps of loop spaces with infinity-bounded differential. - MathOverflow

Maps of loop spaces with infinity-bounded differential.