Maps of loop spaces with infinity-bounded differential. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:23:02Z http://mathoverflow.net/feeds/question/24229 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24229/maps-of-loop-spaces-with-infinity-bounded-differential Maps of loop spaces with infinity-bounded differential. Thomas Kragh 2010-05-11T09:58:29Z 2010-05-11T12:56:38Z <p>I am currently working with loop spaces of manifold and finite dimensional manifolds approximating these and the following comes up very naturally:</p> <p>In the following piece-wise smooth means smooth on each set of a closed covering, implying continuity and boundedness of 1-sided derivatives.</p> <p>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</p> <p>$E(\gamma)=\int_{S^1} \mid\mid \gamma'(t)\mid\mid^2dt$</p> <p>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</p> <p>$\mid\mid {\partial \gamma} \mid\mid_\infty = \sup_{t\in S^1} \mid\mid {\partial \gamma(s)}\mid\mid$</p> <p>for any $\partial \gamma \in T\Lambda M$.</p> <p>I define a <strong>bounded differentiable</strong> function $F\colon \Lambda M \to \Lambda N$ by the following criteria:</p> <p>$E(F(\gamma)) \leq C_F E(\gamma)$ and $\mid\mid F_*(\partial \gamma)\mid\mid_\infty \leq C_F\mid \mid\partial \gamma\mid\mid_\infty$</p> <p>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.</p> <p><strong>Question:</strong> Has anybody seen this notion of boundedness or maybe a similar local definition used anywhere?</p> <p><strong>Motivation:</strong> 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.</p> <p>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.</p>