Abstract Nonsense

Meromorphic Functions on the Riemann Sphere (Pt. II)

Point of Post: This is a continuation of this post.

$\text{ }$

October 7, 2012

Meromorphic Functions on the Riemann Sphere (Pt. I)

Point of Post: In this post we classify the meromorphic functions on the Riemann sphere $\mathbb{C}_\infty$.

$\text{ }$

Motivation

$\text{ }$

If a random kid off the street asked you “what are the continuous functions $(0,1)\to(0,1)$?” or “what are all the smooth  maps $S^4\to\mathbb{R}$?” you would probably replay with a definitive “Ehrm…well…they’re just…” This is because such a description (besides “they’re just the continuous functions!”)  is beyond comprehension in those cases! For example, it would take quite a bit of ingenuity to come up with something like the Blancmange function–in fact, such crazy continuous everywhere, differentiable nowhere functions are, in a sense dense in the space of continuous functions.

$\text{ }$

Thus, it should come as somewhat of a surprise that after reading this post you will be able to answer a kid asking “what are all the meromorphic functions on the Riemann sphere?” with a “Ha, that’s simple. They’re just…”. To be precise, we have already proven that $\mathbb{C}(z)\subseteq\mathcal{M}(\mathbb{C}_\infty)$, and we shall now show that the reverse inclusion is true!  Now, more generally, we shall be able to give a satisfactory (algebraic!) of the meromorphic functions of any compact Riemann surface. While this won’t be quite as impressive as the explicit, simple characterization of $\mathcal{M}(\mathbb{C}_\infty)$ but still a far cry from our situation with trying to characterize $C([0,1],[0,1])$, since we don’t really even have an algebraic (ring theoretic) description of this (in terms of more familiar objects). This should be, once again, another indication that the function theory of compact Riemann surfaces is very rigid–they admit meromorphic functions, but not so many that the computation (algebraically) of their meromorphic function field is untenable.

$\text{ }$

Ok, now that we have had some discussion about the philosophical implications of actually being able to describe $\mathcal{M}(\mathbb{C}_\infty)$ let’s discuss how we are actually going to prove $\mathcal{M}(\mathbb{C}_\infty)=\mathbb{C}(z)$. The basic idea comes from the fact that we can actually find a function $r$ which has prescribed zeros $\lambda_1,\cdots,\lambda_n$ and poles $p_1,\cdots,p_m$ such that $\text{ord}_{\lambda_i}(r)=e_i$ and $\text{ord}_{p_i}(r)=-g_i$ for any $e_i,g_i\in\mathbb{N}$–namely, the function $\displaystyle r(z)$ given by $(z-\lambda_1)^{e_1}\cdots(z-\lambda_n)^{e_n}(z-p_1)^{-g_1}\cdots(z-p_m)^{-g_m}$. Thus, if $f$ is a meromorphic function on $\mathbb{C}_\infty$ with zeroes and poles described as in the last sentence we see that $\displaystyle \frac{f}{r}$ is a meromorphic function on $\mathbb{C}_\infty$ and which has no zeros or poles on $\mathbb{C}$. In particular, $\displaystyle h=\frac{f}{r}$ is a function meromorphic on $\mathbb{C}_\infty$ but holomorphic on $\mathbb{C}$, and with no zeros. Now, it’s a common fact from complex analysis that the only entire function with a pole at infinity (recall that the somewhat confusing definition of pole at infinity now makes a lot more sense!) is a polynomial. Thus, $h$ is a polynomial, but since $h$ has no zeros on $\mathbb{C}$ we know from the fundamental theorem of algebra that $h$ is constant. Thus, $f$ is really just a constant multiple of $r(z)$! Note that the key to this proof is that ability specify poles and zeros of a given multiplicity, except perhaps specifying a pole at infinity, and that the point infinity is well-behaved (in the sense that things that have poles there are pretty tame).

$\text{ }$

October 7, 2012

Meromorphic Functions on Riemann Surfaces (Pt. III)

Point of Post: This is a continuation of this post.

$\text{ }$

October 7, 2012

Meromorphic Functions on Riemann Surfaces (Pt. II)

Point of Post: This is a continuation of this post.

$\text{ }$

October 7, 2012

Meromorphic Functions on Riemann Surfaces (Pt. I)

Point of Post: In this post we  define meromorphic functions on Riemann surfaces and discuss several properties of such maps, including the correspondence between holomorphic functions and holomorphic maps to the Riemann sphere.

$\text{ }$

Motivation

$\text{ }$

So, up until this we’ve point we’ve defined Riemann surfaces and the structure preserving maps between them (the holomorphic ones). We should feel accomplished, no? I mean, we at least have the bare basics to start trying to do complex analysis on surfaces/use complex analysis to analyze the surfaces, right? Well, anyone who has taken a serious course in complex analysis should be vehemently shaking their head (you are, right?). Indeed, looking only at holomorphic functions leaves out a huge part of complex analysis. In particular, we should be weary of any discussion of complex analytic concepts that ignore meromorphic functions. Indeed, without considering meromorphic functions we wouldn’t have things like Cauchy’s integral formula, the theorem that enriches all of our study of holomorphic functions.

$\text{ }$

Another very obvious reason that we would want to study meromorphic functions comes from our previous revelatio that compact Riemann surfaces admit no non-constant holomorphic functions! We saw, looking at the proof that, perhaps the issues with this is that for globally defined functions into $\mathbb{C}$ their is a boundedness is the issue. Thus, allowing functions with singularities (and consequently for functions to get unbounded near these singularities!) will perhaps allow us to create meaningful (non-constant) functions from compact Riemann surfaces into $\mathbb{C}$, which should be useful in the study of the compact Riemann surfaces themselves. Of course, such functions would be (if we were doing regular complex analysis!) meromorphic functions.

$\text{ }$

So, now that we have figured out why we would want to study meromorphic functions on Riemann surfaces, it remains to figure out precisely how to define them. Well, unsurprisingly, we define them like we define all local concepts on Riemann surfaces–via charts. Namely, just like in the case of holomorphic functions, we shall say that a function $f$ on a Riemann surface $X$ is meromorphic at $p$ if when we pretend that locally around $p$ and $f(p)$ we are dealing with subsets of $\mathbb{C}$, and when we pretend that $f$ is a mapping between these two subsets, that $f$ is meromorphic at $p$. Of course, this is rigorously done by checking the meromorphicity of $f$‘s coordinate map associated to a chart. We then have to verify that this meromorphicity of coordinate map is independent of chart transformation, but this is clear since if one conjugate a meromorphic map by biholomorphisms one gets a meromorphic map.

$\text{ }$

Ok, fine, so this is the obvious, not new definition of meromorphicity. But, as we have said before, not only does using abstract Riemann surfaces allow us to study such objects (you need to have Riemann surfaces and their pursuant concepts to study Riemann surfaces!) but it allows us to illuminate some old concepts. In particular, suppose that we wanted to try to extend a meromorphic function $f:\mathbb{C}\to\mathbb{C}$ to a holomorphic function? To put this on less confusing terms, we know that there is a discrete set $P$ of poles of $f$ such that $f:\mathbb{C}-P\to\mathbb{C}$ is holomorphic, and we want to extend this to a holomorphic map $\widetilde{f}:\mathbb{C}\to\mathbb{C}$. Well, naively, if this $\widetilde{f}$ is to be holomorphic, it better be continuous. Thus, we have no choice but to define $\displaystyle \widetilde{f}(p)=\lim_{z\to p}f(z)$ for $p\in P$. Now, there is an obvious issue with this since (by definition!) $\displaystyle \lim_{z\to p}|f(z)|=\infty$, and so (with this definition) $\widetilde{f}(p)$ is not a well-defined element of $\mathbb{C}$. Ah, but therein lies the rub. Namely, $\widetilde{f}(p)$ is not a well-defined element of the complex numbers. This doesn’t stop us from thinking about $\mathbb{C}$ as sitting inside a bigger Riemann surface $X$ and trying to extend $f:\mathbb{C}-P\to X$ (given by  the composition $\mathbb{C}-P\xrightarrow{f}\mathbb{C}\hookrightarrow X$) where, perhaps, $\displaystyle \lim_{z\to p}f(z)$ may exist!

$\text{ }$

Thus, we are left looking for a Riemann surface $X$ with an embedding $\mathbb{C}\hookrightarrow X$ and a notion of an “infinite complex number”. Hopefully you aren’t thinking too hard. Namely, the extended complex plane $\mathbb{C}_\infty$ is exactly such a Riemann surface! Not only does $\mathbb{C}$ sit comfily inside of $\mathbb{C}_\infty$, but the topology on $\mathbb{C}_\infty$ allows us to think of $\infty$ as an element for which defining $\displaystyle \widetilde{f}(p)$ makes sense! In particular, the knowledge that $|f(z)|$ gets arbitrarily large tells us that as $z\to p$ we have that $|f(z)|$ lies in discs $|z|>R$ for arbitrarily large values of $R$. Now, if one starts looking at what the discs $|z|>R$ on the Riemann sphere one sees that the only point in all such discs is $\infty$! Thus, it actually makes topological sense to define $\widetilde{f}(p)=\infty$. Thus, we see that meromorphic mappings $\mathbb{C}\to\mathbb{C}$ really correspond to maps $\mathbb{C}\to\mathbb{C}_\infty$, and if there is any justice in the world, these maps will be holomorphic. Luckily for us, this is in fact true (the just world part might be pushing it though), and we shall see that meromorphic mappings $X\to\mathbb{C}$ (where $X$ is any Riemann surface!) correspond exactly to holomorphic maps $X\to\mathbb{C}_\infty$. Well, almost, the only thing we need to watch out for is the condition that poles are discrete. But, since preimages of holomorphic non-constant maps are discrete we see the only trouble map is the map $X\to\{\infty\}\subseteq\mathbb{C}_\infty$, which we shall just ignore.

$\text{ }$

Well, once we have the notion of meromorphic functions, the most basic thing we’d want to do with them is inspect how “bad” of a singularity they have at a given poin, or  more specifically, how bad of a singularity either they or their inverse has at a given point. This leads us naturally to the notion of the order of a meromorphic function at a point. If we were doing plain old complex analysis on subsets of $\mathbb{C}$ this would be as simple as expanding our function as a Laurent series and finding the smallest non-zero coefficient (the integer attached to this coefficient would be the order). Simpleminded as we are, we would like to do the same thing here. Of course, the issue with this is the one that we have encountered at least fourteen times now–the fact that a Laurent series doesn’t make sense for a map $X\to\mathbb{C}$ when $X$ is an abstract Riemann surface. And, like the fourteen other times, we fix this by thinking locally. We will define the Laurent series of $f$ at $p$, relative some chart $(U,\varphi)$, to merely by the Laurent series of $f\circ\varphi^{-1}$ at $\varphi(p)$. Now, in a perfect world of sunshine and lollipops, the Laurent series of $f$ would be the same with respect to any chart. Sadly, this is not true, for if it were true then one would have that the Laurent series of a function $f:\mathbb{C}\to\mathbb{C}$ at $p$ is the same as the Laurent series of $f\circ\psi$ at $\psi^{-1}(p)$ for any biholomorphism $\psi$ (on a neighborhood of $p$)–you can find counterexamples to this. But, the key is that while the Laurent series is extremely chart dependent, the integer at which its lowest non-zero coefficient appears, does not. Thus, we get the well-defined notion of order that shall play such a pivotal role in our later studies.

$\text{ }$

October 7, 2012