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

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.

Answer 2

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 3

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.