Manifold-Valued Sobolev Spaces

Question

I have the following basic question about Sobolev-spaces which take their values in a Riemannian manifold $(M,g)$, i.e. functions $u:\Omega \to M$, $\Omega \subset \mathbb{R}^n$ bounded, such that for every chart $\varphi : M \to \mathbb{R}^d$, the composition $\varphi\circ u$ is in the usual $H^1$ space.

Question: 1. Is this notion well-defined (this is only easy to show for higher order smoothnes or $n =1$)? 2. Does the manifold-valued Sobolev space admit the structure of a Hilbert manifold? 3. Do there exist good references for manifold-valued Sobolev spaces (and their relations with Hilbert manifolds)? 4. Is there a better way to define manifold-valued Sobolev spaces?

Many thanks for your answers!

Answer 1

To my knowledge the usual approach to define manifold valued spaces is to embed the target manifold $M$ in some $R^N$, then consider the space of all functions $u:\Omega\to R^N$ which are in $H^1(\Omega,R^N)$ and in addition take their values in $M$: $u(x)\in M$ for a.e. $x\in \Omega$. For a basic example of this approach see the paper by Brezis and Mironescu (On some questions of topology for $S^1$ valued fractional Sobolev spaces) where functions with values in $S^1$ are considered. You will agree that in that case a very natural definition is to put a priori the constraint $|u|=1$ on the functions. Of course, you end up with considering bounded functions only, but this solves most of the problems you are facing with your definition.

Answer 2

Hi.

This isn't well-defined (even for higher order smoothness). Even the case $M=\mathbb{R}$ doesn't work, because arbitrary diffeomorphism can have arbitrarly bad growth towards infinity and therefore do not map H^1 map to H^1 maps.

Answer 3

Not sure if anybody is still interested in this, but one can indeed define Sobolev spaces (of high enough regularity, so that due to the Sobolev embedding theorem the maps your looking at are at least continuous) of mappings from a (compact) manifold with values in a manifold using a chart approach. While it is true (and has been mentioned above) that if you can check the $H^s$-property in some charts they will not hold in all charts (as I can always blow up things by composing with suitable diffeomorphisms). However, if one restricts to ''nice families of charts'' then the approach works and one obtains a Banach manifold (Hilbert if you are asking for derivatives in $L^2$). This has been worked out in the paper

Inci, Kappeler, Topalov: On the regularity of the composition ofdiffeomorphisms

Though this was well known for a long time, e.g. see Ebin,Marsden: Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, the new paper is the most complete account on all the technical details (also the point concerning the charts is at least not mentioned in the original Ebin/Marsden paper). Note however, that the results on the manifold structure require your $\Omega$ to be compact (so no chance to choose an open domain in $\mathbb{R}^N$. Otherwise, there is no way this could be a Hilbert or Banach space in any of the usual function space topologies.

I think that similar constructions, at least for the morphisms but not for the manifold structures, may be found in

J. Eichhorn: Global Analysis on Open Manifolds 2007

(where compactness of the source space is relaxed by requiring bounded geometries)

Answer 4

A (slightly) alternative approach to that suggested by Piero D'Ancona, and indeed quite similar to that proposed by Alexander Shamov, is to consider a partition of unity associated with a given atlas and define $H^1(M)$-functions locally wrt to said partition of unity. You can find a more detailed explanation of this approach in §1.7.3 in the first volume of the monograph by Lions-Magenes.