## Meromorphic Functions on the Riemann Sphere (Pt. II)

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

## Meromorphic Functions on the Riemann Sphere (Pt. I)

**Point of Post: **In this post we classify the meromorphic functions on the Riemann sphere .

**Motivation**

If a random kid off the street asked you “what are the continuous functions ?” or “what are all the smooth maps ?” 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.

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 , 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 but still a far cry from our situation with trying to characterize , 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.

Ok, now that we have had some discussion about the philosophical implications of actually being able to describe let’s discuss how we are actually going to prove . The basic idea comes from the fact that we can actually find a function which has prescribed zeros and poles such that and for any –namely, the function given by . Thus, if is a meromorphic function on with zeroes and poles described as in the last sentence we see that is a meromorphic function on and which has no zeros or poles on . In particular, is a function meromorphic on but holomorphic on , 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, is a polynomial, but since has no zeros on we know from the fundamental theorem of algebra that is constant. Thus, is really just a constant multiple of ! 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).

## Meromorphic Functions on Riemann Surfaces (Pt. III)

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

## Meromorphic Functions on Riemann Surfaces (Pt. II)

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

## 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.

**Motivation**

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.

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 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 , 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.

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 on a Riemann surface is meromorphic at if when we pretend that locally around and we are dealing with subsets of , and when we pretend that is a mapping between these two subsets, that is meromorphic at . Of course, this is rigorously done by checking the meromorphicity of ‘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.

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 to a holomorphic function? To put this on less confusing terms, we know that there is a discrete set of poles of such that is holomorphic, and we want to extend this to a holomorphic map . Well, naively, if this is to be holomorphic, it better be continuous. Thus, we have no choice but to define for . Now, there is an obvious issue with this since (by definition!) , and so (with this definition) is not a well-defined element of . Ah, but therein lies the rub. Namely, is not a well-defined element of the *complex numbers*. This doesn’t stop us from thinking about as sitting inside a bigger Riemann surface and trying to extend (given by the composition ) where, perhaps, may exist!

Thus, we are left looking for a Riemann surface with an embedding and a notion of an “infinite complex number”. Hopefully you aren’t thinking too hard. Namely, the extended complex plane is exactly such a Riemann surface! Not only does sit comfily inside of , but the topology on allows us to think of as an element for which defining makes sense! In particular, the knowledge that gets arbitrarily large tells us that as we have that lies in discs for arbitrarily large values of . Now, if one starts looking at what the discs on the Riemann sphere one sees that the only point in all such discs is ! Thus, it actually makes topological sense to define . Thus, we see that meromorphic mappings really correspond to maps , 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 (where is any Riemann surface!) correspond exactly to holomorphic maps . 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 , which we shall just ignore.

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 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 when is an abstract Riemann surface. And, like the fourteen other times, we fix this by thinking locally. We will define the Laurent series of at , relative some chart , to merely by the Laurent series of at . Now, in a perfect world of sunshine and lollipops, the Laurent series of 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 at is the same as the Laurent series of at for any biholomorphism (on a neighborhood of )–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.

## Holomorphic Maps and Functions (Pt. I)

**Point of Post: **In this post we define holomorphic maps between surfaces and prove various properties that such maps possess. We then specialize this to looking at holomorphic functions.

**Motivation**

Now that we have defined Riemann surfaces and given ample examples we can define the arrows in our category–in other words, the structure preserving maps between Riemann surfaces. Of course, these should just be the maps between Riemann surfaces that are holomorphic locally, where this makes sense since locally Riemann surfaces are just open subsets of .

How do we make this rigorous? Precisely the way we did with smooth maps between smooth manifolds. Namely, if we have a map between two Riemann surfaces, and a point then should be holomorphic at if, when we pretend that is just an open subset of locally around , and that is just an open subset of around , and then pretend that is just a map between these open subsets of , that we get something holomorphic. Of course, we need to unravel this to make it slightly more rigorous. The first rigorization (new word?) we would like to perform is to make explicit what we mean by “pretending” things look, locally, like an open subset of . In particular, choosing charts and at and allows us to pretend that and are just the open subsets and of (we need to, for set theoretic technicalities, assume that –but this is just technical, and should [intuition wise!] just be ignored). Fine, but, how do we “pretend” that is a map ? Well, it’s fairly obvious that we would want the method of “pretending” to be consistent with the method of “pretend” we performed on the identifications and . In particular, we should pull back to a map on a subset of the same way we pulled back–using . Similarly, we should use to pull to a map on a subset of . Putting this all together we see that the map we should be considering is . Thus, *this* is the map we want to be holomorphic at (the “pretend” ).

Of course, just as in the case of smooth manifolds, the somewhat unsettling aspect of this definition is the idea that, a priori, this definition depends on our method of “pretend”. In particular, why can’t we pick different charts and get a function which is not holomorphic? Well, this is precisely why Riemann surfaces aren’t defined as just topological manifolds of one complex dimension. Namely, the requirement that our complex structure be internally (holomorphically) compatible is precisely so that this definition is independent of chart choice.

Ok, now that we have a notion of holomorphic mappings between Riemann surface we can start to ask how many of the theorems from complex analysis transfer over to this context? Does the Open Mapping Theorem hold? Does the Identity Theorem Hold? It turns out that the meta principle that (almost!) any mapping property that holds for true for holomorphic mappings between domains in holds true for mappings between Riemann surfaces. This should be intuitively true since holomorphic mapping properties are most often local theorems, and locally Riemann surfaces and the maps between them, are just holomorphic mappings between domains!

Of course, this is a meta principle, and as my parenthetical disclaimer indicates, things don’t always hold true for Riemann surfaces in general. What are some of the issues that would prevent us from carrying over a true statement about holomorphic maps between domains to a true statement about holomorphic mappings between Riemann surfaces? Probably the most egregious issue, and one of the simplest conceptually, is that perhaps there isn’t even an obvious analog for the theorem! For example, the Maximum Modulus Principle involves the notion of . Now, if we have a mapping for abstract Riemann surfaces and , what does even mean? That said, most of the issues are ones where we really only require to be not-so-abstract. Thus, we are naturally led (via trying to generalize theorems in complex analysis) to consider the special case of holomorphic mappings . Such holomorphic mappings shall be called (for historical reasons) holomorphic functions (i.e. the word function is reserved [instead of mapping] for when the codomain is ).

Now anyone familiar with smooth manifold theory will be saying to themselves “of course, holomorphic functions will undoubtedly be extremely important in our studies!” This intuition comes from the fact that in real manifold theory, this is very much true. Studying the smooth functions can tell us a surprisingly, fantastically huge amount of information about (pronounced Morse theory). This is, in fact, not entirely true. For example, we shall see that the *only* holomorphic functions on compact Riemann surfaces shall be constant maps (intuitively, this makes sense because any function shall have to assume it’s maximum from where [using our intuition given by the Maximum Modulus Principle on domains] we should guess our map is contant). This shall be the first indication of a very fundamental fact of Riemann surfaces. Namely, there are two types of Riemann surfaces: the compact and the non-compact. To be less cryptic, we shall see that the theorems/techniques used in the study of compact Riemann surfaces shall vary greatly from those in the study of non-compact Riemann surfaces. A good rule of thumb is that the study of compact Riemann surfaces feels algebraic/algebraic geometric (this has very precise, rigorous categorical statement) that the study of non-compact Riemann surfaces feels much more analysis/ geometric analysis like.

## Loci of Holomorphic Functions and the Inverse Function Theorem (Pt. III)

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