# Rellich Embedding Theorem for the $2$-Sphere

I'm trying to understand the Rellich-Embedding Theorem in the non-flat case by looking at the $2$-sphere. To be precise, for $S$ the spinor bundle of $S^2$; $L^2(S^2)$ the space of square integrable spinors, and $H^1(S^2)$ the Sobolev space with respect to the Levi-Civita connection, the Rellich-Embedding Theorem states that the embedding $$H^1(S^2) \to L^2(S^2)$$ is a compact operator.

(i) Can anyone point me to a proof in this specific case.

(ii) The sphere is a (a) compact, (b) Kähler, and (c) projective space. Do any of this extra structures allow for a simplification of the statement or proof of the theorem?

I think $$X:=S^2$$ being a compact Riemannian manifold already gives you quite a lot, without any extra structure. The key seems to be the existence of geodesic normal coordinates, and in particular the Taylor expansion they give us for the metric tensor.

Leaving the details of how to do this to one side to begin with, let us suppose that we have constructed a collection of nice' local parameterisations $$(x_i:U_i\to X_i)_{i\in I}$$ such that the $$(X_i)_{i\in I}$$ is an open cover for $$X$$ and each $$H^1_0(X_i)$$ is isomorphic as a Hilbert space to the corresponding $$H^1_0(U_i)$$. Using compactness to pass to a finite subcollection if necessary, taking a partition of unity subordinate to $$(X_i)_{i\in I}$$ gives us a topological embedding

$$H^1(X) \to \prod_{i\in I}H_0^1(X_i)\cong \prod_{i\in I}H_0^1(U_i)$$

into a finite product of normal $$H^1$$ spaces on $$\mathbb{R}^2$$. At this point we can either use a diagonal sequence argument taking the compactness of the $$H^1(U_i)\to L^2(U_i)$$ for granted, or use the embedding to transfer local estimates back to $$H^1(X)$$ from each $$H^1_0(U_i)$$ (I think one $$\epsilon, C(\epsilon)$$-style interpolation inequality plus Freidrich's lemma is enough).

Of course, all this was based on the presumption of access to nice local parameterisations $$(x_i:U_i\to X_I)_{i\in I}$$. This is where I think geodesic normal coordinates come in. Pulling back the volume form on $$X$$ using a local parameterisation $$x:U\to X$$ always gives us a measure on $$U$$ and an identification between $$L^2(x(U))$$ and some weighted $$L^2$$ space $$L^2(U,w)$$ (where $$w$$ is the associated density w.r.t. Lebesgue measure). What I think geodesic normal coordinates (probably) let you do is make sure that this density is bounded away from zero, at which point $$L^2(U)\cong L^2(U,w)$$ and $$H^1(x(U)) \cong H^1(U)$$.

In case this isn't already clear, I'm a little sketchy on the details of this last bit: the MO post Volume of geodesic balls seems relevant here.

Actually: it looks like Hebey's book Nonlinear analysis on manifolds has all this and more. I've exhausted my google books preview allowance reading it this morning, but it looks like it covers everything I've said in spades. It also has the advantage of being much more citeable than an anonymous MO post.

• Re. relevance of extra structures: I don't think the Kahler structure really gives you any extra useful information.
– DCM
May 25, 2019 at 22:32
• This MO post seems relevant Re. the use of geodesic normal coordinates: mathoverflow.net/questions/327475/…
– DCM
May 26, 2019 at 9:44
• Ah! Proposition 2.4 of books.google.co.uk/… uses the kind of local charts needed here (the second sentence of the proof).
– DCM
May 26, 2019 at 10:02
• Technical geometric details aside, the crucial point here is that one can find parameterisations $U_i\to X_i\subset X$ such that pullback gives equivalence between the 'upstairs' norm $\Vert .\Vert_{H^1(X_i)}$ is equivalent to the downstairs' norm $\Vert .\Vert_{H^1(U_i)}$ for compactly supported smooth functions. Once you have this, I'm not sure there's any real difference between working on a compact manifold and working in a nice bounded domain in $\mathbb{R}^n$
– DCM
May 26, 2019 at 10:33
• Chapter X of "Seminar on the Atiyah-Singer index theorem" (Palais) seems like another useful reference for this sort of thing, especially if you're interested in Sobolev spaces associated with vector bundles. It's a very nice book, in any case.
– DCM
May 27, 2019 at 18:00