# Abstract Nonsense

## Holomorphic Maps and Riemann Surfaces (Pt. IV)

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

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The last theorem we prove tells us that biholomorphisms are really just bijective holomorphic maps–in other words, if a holomorphic map has a set theoretic inverse, this inverse is necessarily holomorphic.

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Theorem(Holomorphic Inverse Theorem): Let $f:X\to Y$ be a bijective holomorphic map, then $f$ is a biholomorphism.

Proof: Let $y\in Y$ be arbitrary and let $x\in X$ be such that $f(x)=y$. Choose charts $(V,\psi)$ around $y$ and $(U,\varphi)$ around $x$ such that $f^{-1}(V)= U$. We need to check that $\varphi\circ f^{-1}\circ\psi^{-1}$ is holomorphic on $\psi(V)$. That said, by definition we know that $\psi\circ f\circ\varphi^{-1}$ is holomorphic $\varphi(U)\to\psi(V)$, and since since evidently $\varphi\circ f^{-1}\circ\psi^{-1}=(\psi\circ f\circ\varphi^{-1})^{-1}$ we know from the Holomorphic Inverse Theorem for domains in $\mathbb{C}$ that $\varphi\circ f^{-1}\circ\psi^{-1}$ is holomorphic. Since $y$ was arbitrary the conclusion follows. $\blacksquare$

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An interesting corollary of this is the following, very surprising, characterization of biholomorphisms between compact Riemann surfaces:

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Corollary: Let $X$ and $Y$ be compact Riemann surfaces. Then, a map $f:X\to Y$ is a biholomorphism if and only if it’s holomorphic and injective.

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Holomorphic Functions

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As promised, we now specialize the above discussion to mappings where the codomain is $\mathbb{C}$. In particular, we define a holomorphic function on a Riemann surface $X$ to be a holomorphic mapping $f:X\to \mathbb{C}$. We use the notation $\mathcal{O}(X)$ (or $\mathcal{O}_Z(X)$ if $X$ is a domain inside a Riemann surface $Z$ with the induced structure) for $\text{Hol}(X,\mathbb{C})$.

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As promised in the motivation we shall now show one of the indications that compact and non-compact Riemann surfaces are fundamentally different:

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Theorem: Let $X$ be a compact Riemann surface. Then, every holomorphic function on $X$ is constant.

Proof: Let $f\in\mathcal{O}(X)$, and suppose that $f$ is non-constant, then by previous theorem we see that this implies that $f$ is surjective and thus $\mathbb{C}$ compact, a contradiction. $\blacksquare$

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While this theorem is easy to prove, an observation based off of it actually can lead one to very deep intuition about the differences between compact and non-compact Riemann surfaces (if one is willing to take a little bit of knowledge on faith!). Namely, we shall prove later that every Riemann surface admits a non-constant meromorphic (to be defined, but should be clear–a holomorphic function with poles) function. Now, don’t try to prove this on your own. Don’t scoff at it either! This is one of the deepest, most powerful, and most difficult theorems to prove in all of (basic) Riemann surface theory (it will occupy a vast amount of our time!). Most of the theorems we shall prove generalize to higher dimensions, but this is truly a theorem of dimension one! Regardless, this existence allows us to notice something strange.

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Regardless of me telling you that the existence of non-constant meromorphic functions is amazing, it really does inspire a  ‘so what’? Why does this tell us anything important? Well, we shall prove in a soon-to-come post that, much like in the case of domains in $\mathbb{C}$, $\mathcal{O}(X)$ is a ring for any Riemann surface, but more specifically, is an integral domain (this is where the term ‘domain’ in integral domain came from, by the way–integral still seems to be a mystery[at least to me]). Moreover, when we define meromorphic functions on a Riemann surface $X$ we shall be able to prove that, much like in the case of domains in $\mathbb{C}$, one has that the set $\mathcal{M}(X)$ of mermorphic functions on $X$ is a field. Now, since $\mathcal{O}(X)$ is an integral domain, we can consider the field of fractions $\text{Frac}(\mathcal{O}(X))$. Well, then the above theorem tells us that if $X$ is compact then,

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$\text{Frac}(\mathcal{O}(X))=\mathcal{O}(X)=\{\text{constant functions}\}$

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Once again, so what? Well, combining these two ‘so what’ statements gives us the following fact: if $X$ is compact, then

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$\text{Frac}(\mathcal{O}(X))\subsetneq\mathcal{M}(X)$

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Well, depending on your point-of-view we may have just cashed in two ‘so what’ statements for another ‘so what’ statement. Ah, but here is where you are wrong! Remember that the statement $\text{Frac}(\mathcal{O}(D))=\mathcal{M}(D)$ is true for domains $D\subseteq\mathbb{C}$. The proof of this relied heavily on the Weierstrass product theorem, which allowed us to create entire functions with a specified (discrete!) zero set and specified multiplicities (we basically multiplied by an entire function whose zeros with multiplicity matched the poles with multiplicity, and then divided both sides by it). Thus, we see that $\text{Frac}(\mathcal{O}(X))\subsetneq\mathcal{M}(X)$ for compact $X$ is an indication that there isn’t a Weierstrass product theorem for compact Riemann surfaces.

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This tells us that we can’t just pick, willy-nilly, and set of zeros with specified multiplicties and find holomorphic functions on compact $X$ with them. This tells us that compact $X$ are less malleable, their structure more rigid. This intuition about compact Riemann surfaces is one that will hold true throughout all of our studies, and should be the first indication (before the big categorical justification of the intuition!) as to why compact Riemann surfaces are more algebraic–algebraic things are rigid. Conversely, we shall eventually prove that, spitting in the face of compacts, non-compact Riemann surfaces do have $\text{Frac}(\mathcal{O}(X))=\mathcal{M}(X)$, which is because non-compacts do have a Weierstrass theorem. This allows us to intuit that non-compacts are more malleable, more analytic.

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Ok, coming back from rants we can, as was alluded to in the motivation, actually find analogs in holomorphic maps on Riemann surfaces for every holomorphic mapping property between domains in $\mathbb{C}$. In particular, we can prove the big one that was left out of the discussion above: the maximum modulus principle:

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Theorem: Let $X$ be a Riemann surface, and $f\in\mathcal{O}(X)$. Then, if $|f|$ achieves its maximum on $X$ (i.e. $\displaystyle \sup_{x\in X}|f(x)|=f(x_0)$ for some $x_0\in X$) then $f$ is contant.

Proof: Suppose that $f$ achieves its maximum at $x_0\in X$. Choose a coordinate disc $(D,\varphi)$ centered at $x_0$. We see then that $f\circ\varphi^{-1}:\varphi(D)\to\mathbb{C}$ is a holomorphic mapping on $D$  which achieves its maximum modulus at $0$. Thus, by the Maximum Modulus Principle for domains this implies that $f\circ\varphi^{-1}$ is contant on $\varphi(D)$ and so $f$ is constant on $D$. Since $D$ is open it follows from the Identity Theorem that $f$ is actually constant on all of $X$.

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This allows us to give another proof that $\mathcal{O}(X)$ consists only of constants for compact $X$. Indeed, since $X$ is compact we know that $|f|$ achieves its maximum on $X$ for any $f\in\mathcal{O}(X)$, from where the Maximum Modulus Principle implies that $f$ is constant.

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The last theorem we currently prove is the adaptation of Riemann’s Removable Singularity theorem to holomorphic functions on Riemann surfaces:

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Theorem(Riemann’s Removable Singularity): Let $X$ be a Riemann surface and suppose that $f\in\mathcal{O}(X-\{p\})$ and $f$ is bounded in some punctured neighborhood of $p$. Then, then $f$ admits an extension $g\in\mathcal{O}(X)$.

Proof: Let $W-\{p\}$ be the punctured neighborhood for which $f$ is bounded on. Now, choose a coordinate disc $(D,\varphi)$ centered at $p$ such that $D\subseteq W$. Then, we know that $f\circ\varphi^{-1}$ is bounded on $\varphi(D)-\{0\}$ and holomorphic, and thus by Riemann’s Removable Singularity theorem, admits a holomorphic extension $h:\varphi(D)\to\mathbb{C}$. Define $g$ on $X$ by $g\mid_{X-\{p\}}=f$ and $g(p)=h(0)$. Clearly then $g$ is holomorphic at every point of $X-\{p\}$ and is holomorphic at $p$ since choosing the neighborhood $(D,\varphi)$ one sees that $h\circ\varphi^{-1}=g$ and thus is holomorphic. $\blacksquare$

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References:

[1] Varolin, Dror. Riemann Surfaces by Way of Complex Analytic Geometry. Providence, RI: American Mathematical Society, 2011. Print.

[2] Miranda, Rick. Algebraic Curves and Riemann Surfaces. Providence, RI: American Mathematical Society, 1995. Print.

[3] Forster, Otto. Lectures on Riemann Surfaces. New York: Springer-Verlag, 1981. Print.

[4] Conway, John B. Functions of One Complex Variable. New York: Springer-Verlag, 1978. Print.

[5] Gong, Sheng. Concise Complex Analysis. Singapore: World Scientific, 2001. Print

[6] Rudin, Walter. Real and Complex Analysis. New York: McGraw-Hill, 1966. Print.