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.
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.
Holomorphic Functions and Examples
Now that we have the motivation we can unabashedly define the notion of holomorphic functions between Riemann surfaces. In particular, let and be Riemann surfaces and a map. Then, we say is holomorphic at if there exists charts at and at , such that and is holomorphic at .
Of course, as we stated, we have the somewhat comforting theorem:
Theorem: Let and be Riemann surfaces and a map. Then, is holomorphic at if and only if for every pair of charts at and at with one has that is holomorphic at .
Proof: We need only show that is holomorphic in a neighborhood of . So, by virtue of being holomorphic there exists a pair of charts at and such that and is holomorphic at . Now, is a neighborhood of (remember that is a homeomorphism!) and we have that
Now, the maps and are holomorphic because the compatibility of our complex structure, and the map is holomorphic at by assumption. Thus, we see that is holomorphic at as desired.
We define a map between Riemann surfaces to be holomorphic if is holomorphic at every point . Evidently, this is equivalent to the following:
Theorem: Let and be Riemann surfaces and . Then, is holomorphic if and only if there exists atlases and of and respectively such that and is holomorphic for all .
We denote the set of all holomorphic maps by .
A holomorphic map between Riemann surfaces and shall be called a biholomorphism if there exists a holomorphic map such that and . If there exists a biholomorphism we say that and are biholomorphic or conformally equivalent. It’s evident that the relation “is biholomorphic to” is an equivalence relation on Riemann surfaces. A biholomorphism shall be called an automorphism. The set of all automorphisms of shall be denoted .
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