Here's an awesome result that I saw yesterday:
Theorem of win (*): let \(\overline{M}\) be a connected compact Riemann surface with non-empty boundary, and \(K\) a smooth nonpositive function on \(\overline{M}\), then there exists a \(\mathbb{C}\)ompatible Riemannian metric \(g\) on \(\overline{M}\) which has \(K\) as its Gaussian curvature! Moreover, this metric \(g\) becomes unique (and still exists) if you designate a (smooth) metric on the boundary.
It's paraphrased slightly because I wanted to avoid asserting that every smooth Riemannian metric in a conformal class of metrics on a manifold can be written in the form: \[g=e^{2\varphi}g_0\] for some base metric \(g_0\) and any smooth function \(\varphi: \overline{M}\rightarrow\mathbb{R}\). In retrospect, maybe I should have just gone with that. Oh well, it's too late Liz Lemon, it's too late (Blast! Can't find a referencing Youtube clip for this 30 Rock quote).
On the plus side, I managed to introduce \(\mathbb{C}\)ompatible instead of saying compatible with its Riemann surface structure. Pretty Stokes'd with that...
Basically, this theorem tells us that as long as you don't deal with positive curvature, you can impose any kind of Gaussian curvature on your surface. This result isn't (to my understanding) all that hard to prove, although I still find some of the language a bit intimidating. But the fact that it's solving for a Dirichlet problem (i.e.: you specify the metric on the boundary) as opposed to something that let's you specify the geodesic curvature on the boundary, is unfortunate. Well, at least for me. Because I really want to prove stuff like, you can uniformize a bordered Riemann surface uniquely so that it's hyperbolic and has constant boundary geodesic curvature blah.
So I tried to prove this instead...and I think that I've done it. Almost all of the arguments are based on what Osgood, Phillips and Sarnak used in this little paper, all I did was that I:
- Came up with a family of functionals that link together the two that they study for surfaces with negative Euler characteristic, and found appropriate constraints on the domain of these functionals.
- Worked out a tweaked Poincare inequality that makes proves the boundedness of a sequence of functions tending to the infimum of one of our given functionals (step 3 of this).
- Emailed Ben Andrews at the ANU to see if elliptic regularity would hold for one of the steps that I needed. He was really quick about responding to my emails and politely nudged me towards a pretty sweet reference when I asked for one (+we both know that I know waaaay too little analysis, and need to do a bit more ground-work before I can make use of his suggestions/hints/glaringly obvious solution to anyone who's moderately okay with analysis).
Claim: given a Riemannian surface \((R,g)\), let \(W^{1,p}(R)\) denote its \((1,p)\)-th Sobolev space. Then for a constant \(C\) dependent only upon \(R\) and \(p\), \begin{align}\|\varphi\|_p=\left(\int_R|\varphi|^pdA\right)^{\frac{1}{p}}\leq C\left(\int_R|\nabla\varphi|^pdA\right)^{\frac{1}{p}}=\|\nabla\varphi\|_p\end{align} is true for functions \(\varphi\) in any subspace \(W^{1,p}(R)^{\tau}\subset W^{1,p}(R)\) constrained by: \begin{align}(1-\tau)\int_R\varphi dA+\tau\int_{\partial R}\varphi|_{\partial R} ds=0,\,0\leq\tau\leq1,\end{align}where \(\varphi|_{\partial R}\) denotes the trace of \(\varphi\) - the notion agrees with what you think it means when \(\varphi\) is continuously defined over all of \(R\).
Sigh...such a mouthful. Here comes the proof (**):
Let's prove this by contradiction. Assume that there is sequence of functions \(\{\varphi_k\}\) in \(W^{1,p}(R)^{\tau}\) so that \begin{align}\int_R|\varphi_k|^pdA>k\int_R|\nabla\varphi_k|^pdA.\end{align} This means that \(\varphi_k\) can't be \(0\), hence the norms \(\|\varphi_k\|_p\) are nonzero and we may assume without loss of generality that the \(\varphi\) were normalized to begin with. Note that the normalization of a function still lies in \(W^{1,p}(R)^\tau\) because it's a vector subspace of the Sobolev space \(W^{1,p}(R)\).
The normalization means that \(\|\varphi_k\|_p=1\) and \(\|\nabla\varphi_k\|_p<\frac{1}{k}\). The former fact, combined with the Rellich-Kondrachov theorem tells us that a subsequence of the \(\{\varphi_k\}\) must converge strongly (and hence a subsequence of that subsequence converges pointwise) to some \(\varphi\in L^p(R)\). Relabel \(\{\varphi_k\}\) as this subsequence, coz we can.
Okay, I lied slightly. It doesn't follow from the most broad versions of Rellich-Kondrachov theorem straight off. The idea is that for bounded domains and \(p\geq n\), we know that \(W^{1,p}(\Omega)\subset W^{1,n-\epsilon}(\Omega)\) and so if you bring \(\epsilon\) close enough to \(0\), you can get that \(W^{1,n-\epsilon}(\Omega)\) is compactly embedded in \(L^p(\Omega)\). Hence, \(W^{1,p}(\Omega)\) is compactly embedded in \(L^p(\Omega)\) for all \(p\in[1,\infty]\). But to be honest, I'm a bit worried about this argument, because it's based on my understanding of the \(n=p\) case argument presented in a remark at the end of section 5.7 in Evans (**). And maybe I've misunderstood him, because he invokes Morrey's inequality and Arzela-Ascoli to handle when \(p>n\). Worrisome.
Anyhoo, back to the proof.
Let's show that the \(\varphi\) we found actually lies in \(W^{1,p}(R)^{\tau}\). We're gonna do this by constructing a weak derivative in \(L^p(R)\). Take an arbitrary compactly supported smooth function \(\phi\in C^{\infty}_c(R)\) (i.e.: a test function), then we have that: \begin{align}\int_R \varphi\,\partial_{x_i}\phi dA=\lim_{k\to\infty}\int_R\varphi_k\,\partial_{x_i}\phi dA=-\lim_{k\to\infty}\int_R\partial_{x_i}\varphi_k\,\phi=0,\end{align}
where the last limit is because \(\|\partial_{x_i} \varphi_k\|_p\leq\|\nabla \varphi_k\|_p<\frac{1}{k}\). Therefore, we may take the weak derivative \(\partial_{x_i}\varphi\) to be the zero function, and so we see that any first order weak derivatives of \(\varphi\) are zero, which means that \(\varphi\in W^{1,p}(R)\). Now invoking either problem 11 of Evans (**) or corollary 2.1.9 of (***), we know that \(\varphi\) has to be constant. At the same time, we know that \(\varphi\) has to also satisfy the constraint that we initially placed upon \(W^{1,p}(R)\). Therefore, \(\varphi=0\), which contradicts the fact that \(\|\varphi\|_p=1\).
Hooray!
Or rather, maybe...hooray? I guess in the course of writing this up, I've realised that there are a few details here and there that I'm a tad worried about. For example, do I know that the constraint here in this last step extends to \(\varphi\)? I don't see why not, and it certainly does on the area integral, but I'm not so comfortable with what's happening on the boundary with the trace operator and jazz. (Update: yea, it works out because the trace operator is continuous.) Plus there's that whole business with showing that \(\varphi\) is in \(W^{1,p}(R)\). I'm not really sure why we had to play with those integrals. Is it because we cannot conclude from \(\|\nabla\varphi_k\|_p\) tending to \(0\) that \(\|\nabla\varphi\|_p\) is zero and we just had to go through that rigmarole? (Update: I now think that it's to show that zero is a valid candidate for the weak derivative.) I'm not comfortable enough with analysis to say for sure yet. But whatever the case, I've learned stuff man. I've learned stuff.
Oh and as per usual, I lament the fact that I still haven't finished writing up about Lie's line-sphere correspondence, and resolve to do it soon.
*: Prop 1.7 in Ch 14 of Taylor's "Partial Differential Equations Vol. 3". It's kind of a long chapter. In fact, chapter 14 is kind of the whole book minus some backgroundy stuff, I think?
**: the proof is adapted from Theorem 1 in section 5.7 of Evans's "Partial Differential Equations". It's a pretty thick book, but the writing is beautiful.
***: Ziemer's "Weakly Differentiable Functors".
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