# Manifold of mappings between $M$ and $N$, with non-compact source $M$

EDIT: Let $M$ and $N$ are two smooth manifold and suppose $N$ is compact but $M$ is not necessarily compact. For my purpose, I just need to consider the case $M=\mathbb R \times S^1$ or $\mathbb R \times [0,1]$.

Thanks to David's comments that help me a lot, the problem I really concern is just the following.

Question:

1. Now that we know if $M$ is non-compact then $C^\infty(M,N)$ is modeled on spaces $C^\infty_c(M,N)$. Since we know $W^{k,p}_0(\Omega)$ is the closure of $C^\infty_c(\Omega)$ in $W^{k,p}(\Omega)$ for $\Omega\subset \mathbb R$, in analogy, can we say $W^{k,p}(M,N)$ is modeled on spaces $W^{k,p}_0(M,f^*TN)$? Any reference?
2. Can we consider $W^{k,p}_0(M,f^*TN)$ as a completion (in some sense?) of $C^\infty_c(M,f^*TN)$, the spaces of smooth sections with compact support of pullback bundles along $f:M\to N$?

Previous post:

(1) Can we define $W^{1,p}_0(M,N)$ in a similar manner that $W^{1,p}_0(\mathbb R)$ is defined by the completions of $C^\infty_0$ or $C^\infty_c$?

• For example, Floer in his paper (see Definition 2.1) actually discusses (roughly) $\mathcal W:=W_{loc}^{k,p}(\mathbb R \times [0,1], N)$ with a topology given by open sets as follows: $\mathcal O_{u,\rho,\epsilon} =\{ v\in \mathcal W \mid v= \exp_u \xi ~\text{on}~ [-\rho,\rho]\times[0,1], ~\text{and}~ ||\xi||_{W^{k,p}} <\epsilon \}$ On the other hand, Audin and Damian in their book (see Definition 8.2.2) consider Banach manifolds $\widetilde {\mathcal W} =W^{k,p}(\mathbb R \times [0,1], N)$ (actually $[0,1]$ should be replaced by $S^1$) in a quite different manner. Here open sets are the space of maps of the form $v=\exp_u \xi$ where $\xi \in W^{k,p}(\mathbb R \times [0,1], N)\equiv W_0^{k,p}(\mathbb R \times [0,1], N)$ and where $u$ is smooth and converges in some decay at the infinity.

Heuristically, $\mathcal W$ is like the completion of $C_c^\infty$ while $\widetilde {\mathcal W}$ is like that of $C_0^\infty$.

(2) Is $\mathcal W$ the same as $\widetilde{\mathcal W}$? Which one could be a better candidate for the definition of $W^{k,p}_0(M,N)$? Are they both Banach manifolds as in the case $M$ is compact?

• Recently I notice that when considering (infinite-dimensional) manifolds of mapping, say $C^\infty(M,N)$, we usually require $M$ to be compact. (See Section 4.2 of this paper: The inverse function theorem of Nash and Moser). Notice that as long as the domain $M$ is compact, the questions (1), (2) and (3) become trivial.

(3) In general, if $M$ is non-compact, then is $C^\infty(M,N)$ still a Frechet manifold as in the case $M$ is compact?

• $W^{1,p}(\mathbb{R})$ is not the completion of $C^\infty(\mathbb{R})$. It is the completion of $W^{1,p} \cap C^\infty$. Jun 11 '17 at 16:09
• I see, I have edited.
– Hang
Jun 11 '17 at 16:20
• If M is non-compact, $C^\infty(M,N)$ is locally convex, but not necessarily complete, as the model spaces are of compactly supported smooth sections. Jul 12 '17 at 5:22
• Jul 15 '17 at 5:19
• In answer to your question, without compact support one cannot define the relevant supremum seminorms on partial derivatives. Of course, the function spaces themselves contain all smooth functions. Jul 15 '17 at 6:20

Let $M$ and $N$ be Riemannian manifolds. In general, the space of Sobolev mappings $W^{k,p}(M,N)$ should not be defined as a completion of smooth mappings even if $k=1$ and manifolds are compact. The common definition (at least if $N$ is compact) is as follows. Take an isometric embedding of $N$ into a Euclidean space $\mathbb{R^\nu}$ and then define: $$W^{k,p}(M,N)=\{ f\in W^{m,p}(M,\mathbb{R}^\nu):\, f(x)\in N \text{ a.e.} \}.$$ This space is equipped with the metric inherited from the Sobolev norm and in general smooth mappings are not dense [1]. There are some problems when $N$ is not compact. You can find more papers about higher order Sobolev mappings between manifolds, including the case of non-compact target, at the homepage of Van Schaftingen.
Too long for a comment: If $M$ is a manifold, you have no volume form, and you cannot give a meaning to integrability of a function. To express integrability, you have to use densities, which are smooth sections of the fiber bundle of densities over the manifold $M$. Note that you don't have this difficulty with your example, since you have then the form $dx\wedge d\theta$ on $\mathbb R\times\mathbb R /2π\mathbb Z$.
• The issue is not with $M$, where you can simply restrict to functions that are period in the second input. The issue is how to define Sobolev spaces of maps into a manifold $N$. Jun 11 '17 at 17:40