# Sobolev imbedding on Riemannian manifolds

Let $(M, g)$ be a non-compact smooth Riemannian manifold of dimension $n \ge 2$, and $G$ a subgroup of the isometry group of $(M,g)$, say with $G$ contained in the component of the identy.

Let $W^{1,2}_{G}(M)=\{f \in W^{1,2}(M)| \quad f\circ \phi =\phi \quad \forall \phi \in G\}$.

Is there any known result concerning the compactness of the Sobolev imbedding $W^{1,2}_{G}(M) \hookrightarrow L^p(M)$ for some subgroup $G$?

• What do you mean by $\textrm{dim}\,G<\infty$? As I recall, the isometry group of a Riemannian manifold is finite dimensional. Commented May 3, 2011 at 21:09
• Sorry, it's a mistake. What I meant is: $G$ contained in the component of the identity Commented May 3, 2011 at 21:29
• You should not take the same letter for the metric and for elements of the group. Commented May 4, 2011 at 11:59
• There is an old note by Naceur Achtaich (circa 1988) when $M\subset\mathbb R^3$ has a rotational symmetry about the $z$-axis. Very localised, but at least an exmple. Commented May 4, 2011 at 12:02

Hi, You need additional geometric condition for the general case but considering the case of $\mathbb{R}^n$ with $G=SO(n)$, i.e. $H^1_ {radial}$, you have compact injection. You will find all the details in chapter 9 of the excellent book of Hebey Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities.
• I'm quite new in this field, but in the new book by Hebey [Sobolev spaces on Riemannian manifolds] he only considers the case when $M$ is compact. Anyhow, I have to the reference you've just mentioned. Commented May 4, 2011 at 9:54