8
$\begingroup$

Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there are several inequivalent definitions (completion of $C_{c}^{\infty}$ in Sobolev norm v.s. the classical definition using weak derivatives v.s. spectral theoretic definition). Furthermore, in the non-compact case - even if the manifold is complete - the Sobolev embedding theorem ($H^{k}\hookrightarrow C^{l}$ for $k>l+\frac{n}{2}$) does not necessarily hold.

Now, it is known that both these "problems" are solved for the special case when $(M,g)$ is of bounded geometry, which means that the injectivity radius of $(M,g)$ is non-zero and the Ricci curvature as well as all its covariant derivatives are uniformely bounded. A proof of this claim can for instance be found in the classical textbooks of Aubin [1] and Hebey [2].

On the other hand, similar problems also appear of couse in the more general case of sections $\Gamma(E)$ of some Hermitian (finite-rank) vector bundle $E$ over $M$ equipped with some compatible $\nabla$ connection. In the compact case, again it is not too hard to show that the Sobolev embedding theorem holds. Now, I have seen in several places that as in the scalar case, the Sobolev embedding theorem does hold for suitable vector bundles (vector bundles that are combatible with the bounded geometry structure, see e.g. the Appendix in Shubin [3] for details) over manifolds of bounded geometry. At least for differential forms, I remeber to have seen the claim of Sobolev embedding theorems somewhere, even though I dont remember the precise reference (the equivalence of the different definitions of Sobolev spaces is discussed in Dodziuk [4], but I do not remember the reference where the Sobolev embedding theorem was mentioned).

Question: Does anyone know a clean reference where the Sobolev embedding theorem is proven for general vector bundles (of bounded geometry) on Riemannian manifolds of bounded geometry?

Or maybe does it easily follow from the scalar case? Or is it just a standard folklore that is nowhere spelled out?

  1. T. Aubin: Some Nonlinear Problems in Riemannian Geometry, Springer, 1998.

  2. E. Hebey: Sobolev Spaces on Riemannian Manifolds, Springer, 1996.

  3. M. A. Shubin: Spectral theory of elliptic operators on non-compact manifolds, Méthodes semi-classiques Volume 1 - École d'Été (Nantes, juin 1991), Astérisque, no. 207 (1992), pp. 35-108.

  4. J. Dodziuk: Sobolev spaces of differential forms and de Rham-Hodge isomorphism, J. Differential Geom. 16(1): 63-73 (1981).

Improve this question
$\endgroup$
4
  • $\begingroup$ I have a couple of clarifications to ask: 1.) When you say that the Sobolev embedding theorem doesn't necessarily hold for a non-compact Riemannian manifold $(M,g)$, are you referring with respect to which definition of Sobolev spaces? It most certainly holds true with respect to $H^k_0$ (i.e. the completion of $C^\infty_c$ in the $H^k$ Sobolev norm w.r.t. $g$) since one can then argue locally using charts and a finite partition of unity. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Sep 14 at 17:30
  • $\begingroup$ 2.) The Sobolev embedding theorem holds for all closed balls of $(M,g)$ if the latter is complete (since then any closed ball of $(M,g)$ is a compact Riemannian manifold with Lipschitz boundary, by the Hopf-Rinow theorem). This means that $H^k$ functions are $C^l$ for $k>l+\frac{\dim(M)}{2}$ but not necessarily bounded w.r.t. the global $C^l$ norm w.r.t. $g$. In that regard, recall as well that by the Nomizu-Ozeki theorem any Riemannian metric is conformal to a complete one. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Sep 14 at 17:40
  • $\begingroup$ @PedroLauridsenRibeiro Thanks for the comments. (1) I see your points. To be honest, I didn't think about that. But in this case, the question concerns the "standard" definition using weak derivatives. (2) I was not aware of this, but it makes sense of course. In my understanding, when taking about the Sobolev embedding theorem, I want a continuous embedding, hence w.r.t. the global $C^{l}$-norm. $\endgroup$
    – G. Blaickner
    Commented Sep 14 at 17:43
  • $\begingroup$ $C^l$ functions are not necessarily bounded w.r.t. $C^l$ norm w.r.t. $g$, so you're actually talking about bounded $C^l$ functions w.r.t. $g$, thus the range space $C^l_b$ when you ask for a "continuous" embedding. The space $C^l$ is not a Banach space if $M$ is not compact - you have to take the sequence of $C^l$ seminorms on a compact exhaustion of $M$ (e.g. closed balls with the same center and all natural radii if $(M,g)$ is complete) to define a (Fréchet) topology on $C^l$, which btw doesn't depend on $g$. The Sobolev embedding is continuous w.r.t. the latter if $(M,g)$ is complete. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Sep 14 at 18:10

0

Reset to default

You must log in to answer this question.