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In this post I hope to start talking in a bit more depth about the global geometry of compact Kähler manifolds and their covers. Basic references for much of this post are the book Fundamental groups of compact Kähler manifolds by Amoros-Burger-Corlette-Kotschick-Toledo, and the paper Kähler hyperbolicity and L2 Hodge theory by Gromov. It turns out that there is a basic distinction in the world of compact Kähler manifolds between those that admit a holomorphic surjection with connected fibers to a compact Riemann surface of genus at least 2, and those that don’t. The existence or non-existence of such a fibration turns out to depend only on the fundamental group of the manifold, and in fact only on the algebraic structure of the cup product on $H^1$; thus one talks about fibered or nonfibered Kähler groups.

If X is a connected CW complex, by successively attaching cells of dimension 3 and higher to X we may obtain a CW complex Y for which the inclusion of X into Y induces an isomorphism on fundamental groups, while the universal cover of Y is contractible (i.e. Y is a $K(\pi,1)$ with $\pi$ the fundamental group of X). The (co)-homology of Y is (by definition) the group (co)-homology of the fundamental group of X. Since Y is obtained from X by attaching cells of dimension at least 3, the map induced by inclusion $H^*(Y) \to H^*(X)$ is an isomorphism in dimension 0 and 1, and an injection in dimension 2 (dually, the map $H_2(X) \to H_2(Y)$ is a surjection, whose kernel is the image of $\pi_2(X)$ under the Hurewicz map; so the cokernel of $H^2(Y) \to H^2(X)$ measures the pairing of the 2-dimensional cohomology of X with essential 2-spheres).

A surjective map f from a space X to a space S with connected fibers is surjective on fundamental groups. This basically follows from the long exact sequence in homotopy groups for a fibration; more prosaically, first note that 1-manifolds in S can be lifted locally to 1-manifolds in X, then distinct lifts of endpoints of small segments can be connected in their fibers in X. A surjection $f_*$ on fundamental groups induces an injection on $H^1$ in the other direction, and by naturality of cup product, if $V$ is a subspace of $H^1(S)$ on which the cup product vanishes identically — i.e. if it is isotropic — then $f^*V$ is also isotropic. If S is a closed oriented surface of genus g then cup product makes $H^1(S)$ into a symplectic vector space of (real) dimension 2g, and any Lagrangian subspace V is isotropic of dimension g. Thus: a surjective map with connected fibers from a space X to a closed Riemann surface S of genus at least 2 gives rise to an isotropic subspace of $H^1(X)$ of dimension at least 2.

So in a nutshell: the purpose of this blog post is to explain how the existence of isotropic subspaces in 1-dimensional cohomology of Kähler manifolds imposes very strong geometric constraints. This is true for “ordinary” cohomology on compact manifolds, and also for more exotic (i.e. $L_2$) cohomology on noncompact covers.

One of the nice things about living in Hyde Park is the proximity to the University of Chicago. Consequently, over the summer I came in to the department from time to time to work in my office, where I have all my math books, fast internet connection, etc. One day in early September (note: the Chicago quarter doesn’t start until October, so technically this was still “summer”) I happened to run in to Volodya Drinfeld in the hall, and he asked me what I knew about fundamental groups of (complex) projective varieties. I answered that I knew very little, but that what I did know (by hearsay) was that the most significant known restrictions on fundamental groups of projective varieties arise simply from the fact that such manifolds admit a Kähler structure, and that as far as anyone knows, the class of fundamental groups of projective varieties, and of Kähler manifolds, is the same.

Based on this brief interaction, Volodya asked me to give a talk on the subject in the Geometric Langlands seminar. On the face of it, this was a ridiculous request, in a department that contains Kevin Corlette and Madhav Nori, both of whom are world experts on the subject of fundamental groups of Kähler manifolds. But I agreed to the request, on the basis that I (at least!) would get a lot out of preparing for the talk, even if nobody else did.

Anyway, I ended up giving two talks for a total of about 5 hours in the seminar in successive weeks, and in the course of preparing for these talks I learned a lot about fundamental groups of Kähler manifolds. Most of the standard accounts of this material are aimed at people whose background is quite far from mine; so I thought it would be useful to describe, in a leisurely fashion, and in terms that I find more comfortable, some elements of this theory over the course of a few blog posts, starting with this one.

A couple of weeks ago, my student Yan Mary He presented a nice proof of Liouville’s theorem to me during our weekly meeting. The proof was the one from Benedetti-Petronio’s Lectures on Hyperbolic Geometry, which in my book gets lots of points for giving careful and complete details, and being self-contained and therefore accessible to beginning graduate students. Liouville’s Theorem is the fact that a conformal map between open subsets of Euclidean space of dimension at least 3 are Mobius transformations — i.e. they look locally like the restriction of a composition of Euclidean similarities and inversions on round spheres. This implies that the image of a piece of a plane or round sphere is a piece of a plane or round sphere, a highly rigid constraint. This sort of rigidity is in stark contrast to the case of conformal maps in dimension 2: any holomorphic (or antiholomorphic) map between open regions in the complex plane is a conformal map (and conversely). The proof given in Benedetti-Petronio is certainly clear and readable, and gives all the details; but Mary and I were a bit unsatisfied that it did not really provide any geometric insight into the meaning of the theorem. So the purpose of this blog post is to give a short sketch of a proof of Liouville’s theorem which is more geometric, and perhaps easier to remember.

The other day by chance I happened to look at Richard Kenyon’s web page, and was struck by a very beautiful animated image there. The image is of a region tiled by colored squares, which are slowly rotating. As the squares rotate, they change size in such a way that the new (skewed, resized) squares still tile the same region. I thought it might be fun to try to guess how the image was constructed, and to produce my own version of his image.

I am spending a few months in Göttingen as a Courant Distinguished Visiting Professor, and talking a bit to Laurent Bartholdi about rational functions — i.e. holomorphic maps from the Riemann sphere $\widehat{\mathbb C}$ to itself. A rational function is determined (up to multiplication by a constant) by its zeroes and poles, and can therefore generically be put in the form $f:z \to P(z)/Q(z)$ where P and Q are polynomials of degree $d$. If $d=1$ then $f$ is invertible, and is called a fractional linear transformation (or, sometimes, a Mobius transformation). The critical points are the zeroes of $P'Q-Q'P$; note that this is a polynomial of degree $\le 2d-2$ (not $2d-1$) and the images of these points under $f$ are the critical values. Again, generically, there will be $2d-2$ critical values; let’s call them $V$. Precomposing $f$ with a fractional linear transformation will not change the set of critical values.

The map $f$ cannot usually be recovered from $V$ (even up to precomposition with a fractional linear transformation); one needs to specify some extra global topological information. If we let $\overline{C}$ denote the preimage of $V$ under $f$, and let $C$ denote the subset consisting of critical points, then the restriction $f:\widehat{\mathbb C} - \overline{C} \to \widehat{\mathbb C} - V$ is a covering map of degree $d$, and to specify the rational map we must specify both $V$ and the topological data of this covering. Let’s assume for convenience that 0 is not a critical value. To specify the rational map is to give both $V$ and a representation $\rho:\pi_1(\widehat{\mathbb C} - V,0) \to S_d$ (here $S_d$ denotes the group of permutations of the set $\lbrace 1,2,\cdots,d\rbrace$) which describes how the branches of $f^{-1}$ are permuted by monodromy about $V$. Such a representation is not arbitrary, of course; first of all it must be irreducible (i.e. not conjugate into $S_e \times S_{d-e}$ for any $1\le e \le d-1$) so that the cover is connected. Second of all, the cover must be topologically a sphere. Let’s call the (branched) cover $\Sigma$ for the moment, before we know what it is. The Riemann-Hurwitz formula lets one compute the Euler characteristic of $\Sigma$ from the representation $\rho$. A nice presentation for $\pi_1(\widehat{\mathbb C}-V,0)$ has generators $e_i$ represented by small loops around the points $v_i \in V$, and the relation $\prod_{i=1}^{|V|} e_i = 1$. For each $e_i$ define $o_i$ to be the number of orbits of $\rho(e_i)$ on the set $\lbrace 1,2,\cdots,d\rbrace$. Then

$\chi(\Sigma) = d\chi(S^2) - \sum_i (d-o_i)$

If each $\rho(e_i)$ is a transposition (i.e. in the generic case), then $o_i=d-1$ and we recover the fact that $|V|=2d-2$.

This raises the following natural question:

Basic Question: Given a set of points $V$ in the Riemann sphere, and an irreducible representation $\rho:\pi_1(\widehat{\mathbb C} - V,0) \to S_d$ satisfying $\sum_i (d-o_i) = 2d-2$, what are the coefficients of the rational function $z \to P(z)/Q(z)$ that they determine (up to precomposition by a fractional linear transformation)?

The “header image” for this blog is an example of an interesting construction in 2-dimensional conformal geometry, due to Richard Kenyon, that I learned of some time ago; I thought it might be fun to try to explain where it comes from.

The example comes from the idea of a Riemann surface lamination. This is an object that geometrizes some ideas in 1-dimensional complex analysis. The basic idea is simple: given a noncompact infinite Riemannian $2$-manifold $\Sigma$, one gives it a new topology by declaring that two points on the surface are “close” in the new topology if there are balls of big radius in the surface centered at the two points which are “almost isometric”. Points that were close in the old topology are close in the new topology, but points that might have been far away in the old topology can become close in the new. For example, if $\Sigma$ is a covering space of some other Riemannian surface $S$, then points in the orbit of the deck group are “infinitely close” in the new topology. This means that the resulting topological space is not Hausdorff; one “Hausdorffifies” by identifying pairs of points that are not contained in disjoint open sets, and the quotient recovers the surface $S$ (assuming that the metric on $S$ is sufficiently generic; otherwise, it recovers $S$ modulo its group of isometries). Morally what one is doing is mapping $\Sigma$ into the space $\mathcal{M}$ of pointed locally compact metric spaces (which is itself a locally compact topological space), and giving it the subspace topology. In more detail, a point in $\mathcal{M}$ is a pair $(X,p)$ where $X$ is a locally compact metric space, and $p \in X$ is a point. A sequence $X_i,p_i$ converges to $X,p$ if there are metric balls $B_i$ around $p_i$ of diameter going to infinity, metric balls $D_i$ around $p$ also of diameter going to infinity, and isometric inclusions of $B_i,D_i$ into metric spaces $Z_i$ in such a way that the Hausdorff distance between the images of $B_i$ and $D_i$ in $Z_i$ goes to zero as $i \to \infty$. Any locally compact metric space $Y$ has a tautological map to $\mathcal{M}$, where each point $y \in Y$ is sent to the point $(Y,y) \in \mathcal{M}$. Gromov showed (see section 6 of this paper) that the space $\mathcal{M}$ itself is locally compact; in fact, this follows in an obvious way from the Arzela-Ascoli theorem.

If $\Sigma$ has bounded geometry — i.e. if the injectivity radius is uniformly bounded below, and the curvature is bounded above and below — then the image of $\Sigma$ in $\mathcal{M}$ is precompact, and its closure is a compact metric space $\mathcal{L}$. The path components of $\mathcal{L}$ are exactly the Riemann surfaces which are arbitrarily well approximated (in the metric sense) on every compact subset by compact subsets of $\Sigma$. If you were wandering around on such a component $\Sigma'$, and you wandered over a compact region, and were only able to measure the geometry up to some (arbitrarily fine) definite precision, you could never rule out the possibility that you were actually wandering around on $\Sigma$. Topologically, $\mathcal{L}$ is a Riemann surface lamination; i.e. a locally compact topological space covered by open charts of the form $U \times X$ where $U$ is an open two-dimensional disk, where $X$ is totally disconnected, and where the transition between charts preserves the decomposition into pieces $U \times \text{point}$, and is smooth (in fact, preserves the Riemann surface structure) on the $U \times \text{point}$ slices, in the overlaps. The unions of “surface” slices — i.e. the path components of $\mathcal{L}$ — piece together to make the leaves of the lamination, which are (complete) Riemann surfaces. In our case, the leaves have Riemannian metrics, which vary continuously in the direction transverse to the leaves. (Surface) laminations occur in other areas of mathematics, for example as inverse limits of sequences of finite covers of a fixed compact surface, or as objects obtained by inductively splitting open sheets in a branched surface (the latter can easily occur as attractors of certain kinds of partially hyperbolic dynamical systems). One well-known example is sometimes called the (punctured) solenoid; its Teichmüller theory is studied by Penner and Šarić  (question: does anyone know how to do a “\acute c” in wordpress? update 11/6: thanks Ian for the unicode hint).

A lamination is said to be minimal if every leaf is dense. In our context this means that for every compact region $K$ in $\Sigma$ and every $\epsilon>0$ there is a $T$ so that every ball in $\Sigma$ of radius $T$ contains a subset $K'$ which is $\epsilon$-close to $K$ in the Gromov-Hausdorff metric. In other words, every “local feature” of $\Sigma$ that appears somewhere, appears with definite density to within any desired degree of accuracy. Consequently, such features will “almost” appear, with the same definite density, in every other leaf $\Sigma'$ of $\mathcal{L}$, and therefore $\Sigma$ is in the closure of each $\Sigma'$. Since $\mathcal{L}$ is (in) the closure of $\Sigma$, this implies that every leaf is dense, as claimed.

In a Riemann surface lamination, the conformal type of every leaf is well-defined. If some leaf is elliptic, then necessarily that leaf is a sphere. So if the lamination is minimal, it is equal to a single closed surface. If every leaf is hyperbolic, then each leaf admits a unique hyperbolic metric in its conformal class (i.e. each leaf can be uniformized), and Candel showed that this family of hyperbolic metrics varies continuously in $\mathcal{L}$. Étienne Ghys asked whether there is an example of a minimal Riemann surface lamination in which some leaves are conformally parabolic, and others are conformally hyperbolic. It turns out that the answer to this question is yes; Richard Kenyon found an example, which I will now describe.

The lamination in question has exactly one hyperbolic leaf, which is topologically a $4$-times punctured sphere. Every other leaf is an infinite cylinder — i.e. it is conformally the punctured plane $\mathbb{C}^*$. Since the lamination is minimal, to describe the lamination, one just needs to describe one leaf. This leaf will be obtained as the boundary of a thickened neighborhood of an infinite planar graph, which is defined inductively, as follows.

Let $T_1$ be the planar “Greek cross” as in the following figure:

Inductively, if we have defined $T_n$, define $T_{n+1}$ by attaching four copies of $T_n$ to the extremities of $T_1$. The first few examples $T_1,\cdots,T_4$ are illustrated in the following figure:

The limit $T_\infty$ is a planar tree with exactly four ends; the boundary of a thickened tubular neighborhood is conformally equivalent to a sphere with four points removed, which is hyperbolic. Every unbounded sequence of points $p_i$ in $T_\infty$ has a subsequence which escapes out one of the ends. Hence every other leaf in the lamination $\mathcal{L}$ this defines has exactly two ends, and is conformally equivalent to a punctured plane, which is parabolic.

The header image is a very similar construction in $3$-dimensional space, where the initial seed has six legs along the coordinate axes instead of four; some (quite large) approximation was then rendered in povray.

When I was in graduate school, I was very interested in the (complex) geometry of Riemann surface laminations, and wanted to understand their deformation theory, perhaps with the aim of using structures like taut foliations and essential laminations to hyperbolize $3$-manifolds, as an intermediate step in an approach to the geometrization conjecture (now a theorem of Perelman). I know that at one point Sullivan was quite interested in such objects, as a tool in the study of Julia sets of rational functions. I have the impression that they are not studied so much these days, but I would be happy to be corrected.

Hermann Amandus Schwarz (1843-1921) was a student of Kummer and Weierstrass, and made many significant contributions to geometry, especially to the fields of minimal surfaces and complex analysis. His mathematical creations are both highly abstract and flexible, and at the same time intimately tied to explicit and practical calculation.

I learned about Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface as three quite separate mathematical objects, and I was very surprised to discover firstly that they had all been discovered by the same person, and secondly that they form parts of a consistent mathematical narrative, which I will try to explain in this post to the best of my ability. There is an instructive lesson in this example (for me), that we tend to mine the past for nuggets, examples, tricks, formulae etc. while forgetting the points of view and organizing principles that made their discovery possible. Another teachable example is that of Dehn’s “invention” of combinatorial (infinite) group theory, as a natural branch of geometry; several generations of followers went about the task of reformulating Dehn’s insights and ideas in the language of algebra, “generalizing” them and stripping them of their context, before geometric and topological methods were reintroduced by Milnor, Schwarz (a different one this time), Stallings, Thurston, Gromov and others to spectacular effect (note: I have the second-hand impression that the geometric point of view in group theory (and every other subject) was never abandoned in the Soviet Union).

Schwarz’s minimal surface (also called “Schwarz’s D surface”, and sometimes “Schwarz’s H surface”) is an extraordinarily beautiful triply-periodic minimal surface of infinite genus that is properly embedded in $\mathbb{R}^3$. According to Nitsche’s excellent book (p.240), this minimal surface closely resembles the separating wall between inorganic and organic materials in the skeleton of a starfish. The basic building block of the surface can be described as follows. If the vertices of a cube are $2$-colored, the black vertices are the vertices of a regular tetrahedron. Let $Q$ denote the quadrilateral formed by four edges of this tetrahedron; then a fundamental piece $S$ of Schwarz’s surface is a minimal disk spanning $Q$:

The surface may be “analytically continued” by rotating $Q$ through an angle $\pi$ around each boundary edge. Six copies of $Q$ fit smoothly around each vertex, and the resulting surface extends (triply) periodically throughout space.

The symmetries of $Q$ enable us to give it several descriptions as a Riemann surface. Firstly, we could think of $Q$ as a polygon in the hyperbolic plane with four edges of equal length, and angles $\pi/3$. Twelve copies of $Q$ can be assembled to make a hyperbolic surface $\Sigma$ of genus $3$. Thinking of a surface of genus $3$ as the boundary of a genus $3$ handlebody defines a homomorphism from $\pi_1(\Sigma)$ to $\mathbb{Z}^3$, thought of as $H_1(\text{handlebody})$; the cover $\widetilde{\Sigma}$ associated to the kernel is (conformally) the triply periodic Schwarz surface, and the deck group acts on $\mathbb{R}^3$ as a lattice (of index $2$ in the face-centered cubic lattice).

Another description is as follows. Since the deck group acts by translation, the Gauss map from $\widetilde{\Sigma}$ to $S^2$ factors through a map $\Sigma \to S^2$. The map is injective at each point in the interior or on an edge of a copy of $Q$, but has an order $2$ branch point at each vertex. Thus, the map $\Sigma \to S^2$ is a double-branched cover, with one branch point of order $2$ at each vertex of a regular inscribed cube. This leads one to think (like a late 19th century mathematician) of $\Sigma$ as the Riemann surface on which a certain multi-valued function on $S^2 = \mathbb{C} \cup \infty$ is single-valued. Under stereographic projection, the vertices of the cube map to the eight points $\lbrace \alpha,i\alpha,-\alpha,-i\alpha,1/\alpha,i/\alpha,-1/\alpha,-i/\alpha \rbrace$ where $\alpha = (\sqrt{3}-1)/\sqrt{2}$. These eight points are the roots of the polynomial $w^8 - 14w^4 + 1$, so we may think of $\Sigma$ as the hyperelliptic Riemann surface defined by the equation $v^2 = w^8 - 14w^4 + 1$; equivalently, as the surface on which the multi-valued (on $\mathbb{C} \cup \infty$) function $R(w):= 1/v=1/\sqrt{w^8 - 14w^4 + 1}$ is single-valued.

The function $R(w)$ is known as the Weierstrass function associated to $\Sigma$, and an explicit formula for the co-ordinates of the embedding $\widetilde{\Sigma} \to \mathbb{R}^3$ were found by Enneper and Weierstrass. After picking a basepoint (say $0$) on the sphere, the coordinates are given by integration:

$x = \text{Re} \int_0^{w_0} \frac{1}{2}(1-w^2)R(w)dw$

$y = \text{Re} \int_0^{w_0} \frac{i}{2}(1+w^2)R(w)dw$

$z = \text{Re} \int_0^{w_0} wR(w)dw$

The integral in each case depends on the path, and lifts to a single-valued function precisely on $\widetilde{\Sigma}$.

Geometrically, the three coordinate functions $x,y,z$ are harmonic functions on $\widetilde{\Sigma}$. This corresponds to the fact that minimal surfaces are precisely those with vanishing mean curvature, and the fact that the Laplacian of the coordinate functions (in terms of isothermal parameters on the underlying Riemann surface) can be expressed as a nonzero multiple of the mean curvature vector. A harmonic function on a Riemann surface is the real part of a holomorphic function, unique up to a constant; the holomorphic derivative of the (complexified) coordinate functions are therefore well-defined, and give holomorphic $1$-forms $\phi_1,\phi_2,\phi_3$ which descend to $\Sigma$ (since the deck group acts by translations). These $1$-forms satisfy the identity $\sum_i \phi_i^2 = 0$ (this identity expresses the fact that the embedding of $\widetilde{\Sigma}$ into $\mathbb{R}^3$ via these functions is conformal). The (composition of the) Gauss map (with stereographic projection) can be read off from the $\phi_i$, and as a meromorphic function on $\Sigma$, it is given by the formula $w = \phi_3/(\phi_1 - i\phi_2)$. Define a function $f$ on $\Sigma$ by the formula $fdw = \phi_1 - i\phi_2$. Then $1/f,w$ are the coordinates of a rational map from $\Sigma$ into $\mathbb{C}^2$ which extends to a map into $\mathbb{CP}^2$, by sending each zero of $f$ to $wf = \phi_3/dw$ in the $\mathbb{CP}^1$ at infinity. Symmetry allows us to identify the image with the hyperelliptic embedding from before, and we deduce that $f=R(w)$. Solving for $\phi_1,\phi_2$ we obtain the integrands in the formulae above.

In fact, any holomorphic function $R(w)$ on a domain in $\mathbb{C}$ defines a (typically immersed with branch points) minimal surface, by the integral formulae of Enneper-Weierstrass above. Suppose we want to use this fact to produce an explicit description of a minimal surface bounded by some explicit polygonal loop in $\mathbb{R}^3$. Any minimal surface so obtained can be continued across the boundary edges by rotation; if the angles at the vertices are all of the form $\pi/n$ the resulting surface closes up smoothly around the vertices, and one obtains a compact abstract Riemann surface $\Sigma$ tiled by copies of the fundamental region, together with a holonomy representation of $\pi_1(\Sigma)$ into $\text{Isom}^+(\mathbb{R}^3)$. Sometimes the image of this representation in the rotational part of $\text{Isom}^+(\mathbb{R}^3)$ is finite, and one obtains an infinitely periodic minimal surface as in the case of Schwarz’s surface. A fundamental tile in $\Sigma$ can be uniformized as a hyperbolic polygon; equivalently, as a region in the upper half-plane bounded by arcs of semicircles perpendicular to the real axis. Since the edges of the loop are straight lines, the image of this hyperbolic polygon under the Gauss map is a region in $\mathbb{R}^3$ also bounded by arcs of round circles; thus Schwarz’s study of minimal surfaces naturally led him to the problem of how to explicitly describe conformal maps between regions in the plane bounded by circular arcs. This problem is solved by the Schwarz-Christoffel transformation, and its generalizations, with help from the Schwarzian derivative.

Note that if $P$ and $Q$ are two such regions, then a conformal map from $P$ to $Q$ can be factored as the product of a map uniformizing $P$ as the upper half-plane, followed by the inverse of a map uniformizing $Q$ as the upper half-plane. So it suffices to find a conformal map when the domain is the upper half plane, decomposed into intervals and rays that are mapped to the edges of a circular polygon $Q$. Near each vertex, $Q$ can be moved by a fractional linear transformation $z \to (az+b)/(cz+d)$ to (part of) a wedge, consisting of complex numbers with argument between $0$ and $\alpha$, where $\alpha$ is the angle at $Q$. The function $f(z) = z^{\alpha/\pi}$ uniformizes the upper half-plane as such a wedge; however it is not clear how to combine the contributions from each vertex, because of the complicated interaction with the fractional linear transformation. The fundamental observation is that there are certain natural holomorphic differential operators which are insensitive to the composition of a holomorphic function with groups of fractional linear transformations, and the uniformizing map can be expressed much more simply in terms of such operators.

For example, two functions that differ by addition of a constant have the same derivative: $f' = (f+c)'$. Functions that differ by multiplication by a constant have the same logarithmic derivative: $(\log(f))' = (\log(cf))'$. Putting these two observations together suggest defining the nonlinearity of a function as the composition $N(f):= (\log(f'))' = f''/f'$. This has the property that $N(af+b) = N(f)$ for any constants $a,b$. Under inversion $z \to 1/z$ the nonlinearity transforms by $N(1/f) = N(f) - 2f'/f$. From this, and a simple calculation, one deduces that the operator $N' - N^2/2$ is invariant under inversion, and since it is also invariant under addition and multiplication by constants, it is invariant under the full group of fractional linear transformations. This combination is called the Schwarzian derivative; explicitly, it is given by the formula $S(f) = f'''/f' - 3/2(f''/f')^2$. Given the Schwarzian derivative $S(f)$, one may recover the nonlinearity $N(f)$ by solving the Ricatti equation $N' - N^2/2 - S = 0$. As explained in this post, solutions of the Ricatti equation preserve the projective structure on the line; in this case, it is a complex projective structure on the complex line. Equivalently, different solutions differ by an element of $\text{PSL}(2,\mathbb{C})$, acting by fractional linear transformations, as we have just deduced. Once we know the nonlinearity, we can solve for $f$ by $f = \int e^{\int N}$, the usual solution to a first order linear inhomogeneous ODE. The Schwarzian of the function $z^{\alpha/\pi}$ is $(1-\alpha^2/\pi^2)/2z^2$. The advantage of expressing things in these terms is that the Schwarzian of a uniformizing map for a circular polygon $Q$ with angles $\alpha_i$ at the vertices has the form of a rational function, with principal parts $a_i/(z-z_i)^2 + b_i/(z-z_i)$, where the $a_i = (1-\alpha_i^2/\pi^2)/2$ and the $b_i$ and $z_i$ depend (unfortunately in a very complicated way) on the edges of $Q$ (for the ugly truth, see Nehari, chapter 5). To see this, observe that the map has an order two pole near finitely many points $z_i$ (the preimages of the vertices of $Q$ under the uniformizing map) but is otherwise holomorphic. Moreover, it can be analytically continued into the lower half plane across the interval between successive $z_i$, by reflecting the image across each circular edge. After reflecting twice, the image of $Q$ is transformed by a fractional linear transformation, so $S(f)$ has an analytic continuation which is single valued on the entire Riemann sphere, with finitely many isolated poles, and is therefore a rational function! When the edges of the polygon are straight, a simpler formula involving the nonlinearity specializes to the “familiar” Schwarz-Christoffel formula.

(Update 10/22): In fact, I went to the library to refresh myself on the contents of Nehari, chapter 5. The first thing I noticed — which I had forgotten — was that if $f$ is the uniformizing map from the upper half-plane to a polygon $Q$ with spherical arcs, then $S(f)$ is real-valued on the real axis. Since it is a rational function, this implies that its nonsingular part is actually a constant; i.e.

$S(f) = \sum _i a_i/(z-z_i)^2 + b_i/(z-z_i) + c$

where $a_i$ is as above, and $z_i,b_i,c$ are real constants (which satisfy some further conditions — really see Nehari this time for more details).

The other thing that struck me was the first paragraph of the preface, which touches on some of the issues I alluded to above:

In the preface to the first edition of Courant-Hilbert’s “Methoden der mathematischen Physik”, R. Courant warned against a trend discernible in modern mathematics in which he saw a menace to the future development of mathematical analysis. He was referring to the tendency of many workers in this field to lose sight of the roots of mathematical analysis in physical and geometric intuition and to concentrate their efforts on the refinement and the extreme generalization of existing concepts.

Instead of using a word like “menace”, I would rather take this as a lesson about the value of returning to the points of view that led to the creation of the mathematical objects we study every day; which was (to some approximation) the point I was trying to illustrate in this post.