# What is the Sobolev space of maps between surfaces?

I am reading a book (Compact Riemann surfaces by Jost) that uses the concept of the Sobolev space of maps between surfaces. The exact words are

Suppose now that $$u:\Sigma_1\to\Sigma_2$$ is a map (between Riemann surfaces, with $$\Sigma_2$$ equipped with a conformal Riemannian metric $$\rho^2(u)dud\bar u$$) which carries the coordinate neighborhood $$V$$ of $$\Sigma_1$$ into the coordinate neighborhood $$U$$ of $$\Sigma_2$$. Then we can check whether the restriction lies in $$W^{1,2}$$ or not. If $$U$$ and $$V$$ are bounded, as we shall assume, then $$u\in V\to U$$ is bounded, hence in $$L^2$$ if it is measurable. To say that $$u\in W^{1,2}$$ is then equivalent to requiring that $$u$$ have weak derivatives $$D_zu,D_{\bar z}\bar u,D_{\bar z}u$$ and $$D_{\bar z}\bar u$$ on $$V$$, and the integral $$E(u,V)=\frac{1}{2}\int_V\rho^2(u(z))(D_zuD_{\bar z}\bar u+D_{\bar z}uD_z\bar u)idzd\bar z$$ be finite.

In particular, we can define the class $$C^0\cap W^{1,2}(\Sigma_1,\Sigma_2)$$, since small coordinate neighborhoods by continuous are mapped into coordinate neighborhoods by continuous maps. We can then also talk of the weak $$W^{1,2}$$-convergence of a sequence of continuous maps $$\Sigma_1\to\Sigma_2$$ of class $$W^{1,2}$$.

Questions and thoughts:

(1) What is a formal definition of the Sobolev space $$W^{1,2}(\Sigma_1,\Sigma_2)$$ of maps between Riemann surfaces? Actually, I am not even sure what $$L^2(\Sigma_1,\Sigma_2)$$ means. In the text above, $$u$$ is considered to be in $$L^2$$ because its representation in the local coordinates is bounded, clearly this is dependent of the choice of coordinate neighborhoods and does not seem to be well-defined.

(2) What are the definitions of these weak derivatives $$D_zu,D_{\bar z}\bar u,D_{\bar z}u$$ and $$D_{\bar z}\bar u$$? Also, in my opinion, weak derivatives should not be a local concept, but apparently, according to this text, it is.

(3) What is the "weak convergence mentioned" in the second paragraph above? I don't think there is a well defined inner product.

(4) Any other reference that has a systematic introduction of these concepts are welcome.

Update: Thanks to the answer below I was able to fashion a definition of $$C^0\cap W^{1,2}(\Sigma_1,\Sigma_2)$$ by myself, and I just want to know if this is what Jost means.

A function $$u\in C^0(\Sigma_1,\Sigma_2)$$ is said to be in $$W^{1,2}$$ if for every point $$p\in\Sigma_1$$, there is a coordinate neighborhood $$V$$ of $$p$$ and a coordinate neighborhood $$U$$ of $$u(p)$$, such that $$u(V)\subset U$$, and in every such pair of coordinate neighborhoods $$V,U$$, the local coordinate representation of $$u$$ is a $$W^{1,2}$$-function on $$V$$. And the weak derivatives are simply the weak derivatives in coordinate representations.

(1) Is this what his means by $$C^0\cap W^{1,2}$$? And we can verify this to be well-defined?

(2) Is the energy integral still defined using partition of unity, so that any integral formulae can be applied here too?

(3) Is the uniform convergence with respect to the metric given on $$\Sigma_2$$?

(4) Weak convergence needs to be defined with an inner product, is it supposed to be $$\langle u,v\rangle=\int_{\Sigma_1}\rho(u)\rho(v)D_zu\bar D_{\bar z}v(\frac{i}{2}dz\wedge d\bar z)$$ Well, this seems absurd to me, because things like $$u+v$$ doesn't really make sense. Yet this is the only thing I can come up with that has a well-defined integral (the uniform convergence is needed here to ensure that $$u,v$$ can lie in a common coordinate neighborhood, is this what you mean by "weak convergence makes sense as long as there is uniform convergence required?).

If you look in Jost's book closely, he does not attempt to define Sobolev spaces (and distributional derivatives, etc.) of maps between Riemann surfaces. Instead, he is content to define the intersection $$W^{1,2}\cap C^0$$. In this setting all what he says makes perfect sense although one needs to check that the notions are independent of choices of coordinate charts. Weak convergence also makes sense as long as uniform convergence is also required. In a footnote, Jost says that $$W^{2,1}$$ etc. can be defined without continuity assumptions. This is indeed true and can be also done in the context of (say, complete) Riemannian manifolds and even metric spaces. Some of these ways are discussed here. The most common solution is to use Nash's Isometric embedding theorem, which reduces everything to maps from manifolds to $${\mathbb R}^N$$. There are alternative intrinsic definitions which I find more natural.
• Thanks very much for this. Nonetheless, I want to know what Jost means when he writes $u\in C^0\cap W^{1,2}$ (even though he requires continuity, he does not specify what $C^0\cap W^{1,2}$ is), so I fashioned a definition that might work in this particular situation. I've updated it in my question. Can you take a look and see if it makes sense? Thanks again. Hopefully this is not too much to ask, given the 50 bounty and all :-) Sep 7, 2019 at 2:03
• What you wrote about the definition of $C^0\cap W^{1,2}$ is correct. One can verify that this does not depend on the choice of charts. (2) and (3) also work. I did not understand what you wrote in (4). One you work with charts, the weak convergence is the usual one. Sep 7, 2019 at 3:19
• The weak convergence that I know of is defined in a Hilbert space $H$, where an inner product induces a isomorphism between bounded linear functionals and elements in $H$. And a sequence in $H$ weakly converges if it converges as a sequence of linear functionals. But maps between surfaces don't constitue a vector space, let alone a Hilbert space. Hence my confusion. Sep 7, 2019 at 4:01
• Even if we work with charts, adding two maps could result in leaving the coordinate neighborhood on $\Sigma_2$ where everything is well-defined. Sep 7, 2019 at 4:07
• @trisct: It does not matter: You embed the space of maps $C(U,V)$ between two open subsets in $R^n$ in $C(U, R^n)$, which is a vector space. As for weak convergence, you test it for each pair of charts. Sep 8, 2019 at 3:56