Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface

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 . 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 -colored, the black vertices are the vertices of a regular tetrahedron. Let denote the quadrilateral formed by four edges of this tetrahedron; then a fundamental piece of Schwarz’s surface is a minimal disk spanning :

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

The symmetries of enable us to give it several descriptions as a Riemann surface. Firstly, we could think of as a polygon in the hyperbolic plane with four edges of equal length, and angles . Twelve copies of can be assembled to make a hyperbolic surface of genus . Thinking of a surface of genus as the boundary of a genus handlebody defines a homomorphism from to , thought of as ; the cover associated to the kernel is (conformally) the triply periodic Schwarz surface, and the deck group acts on as a lattice (of index in the face-centered cubic lattice).

Another description is as follows. Since the deck group acts by translation, the Gauss map from to factors through a map . The map is injective at each point in the interior or on an edge of a copy of , but has an order branch point at each vertex. Thus, the map is a double-branched cover, with one branch point of order at each vertex of a regular inscribed cube. This leads one to think (like a late 19th century mathematician) of as the Riemann surface on which a certain multi-valued function on is single-valued. Under stereographic projection, the vertices of the cube map to the eight points where . These eight points are the roots of the polynomial , so we may think of as the hyperelliptic Riemann surface defined by the equation ; equivalently, as the surface on which the multi-valued (on ) function is single-valued.

The function is known as the Weierstrass function associated to , and an explicit formula for the co-ordinates of the embedding were found by Enneper and Weierstrass. After picking a basepoint (say ) on the sphere, the coordinates are given by integration:

The integral in each case depends on the path, and lifts to a single-valued function precisely on .

Geometrically, the three coordinate functions are harmonic functions on . 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 -forms which descend to (since the deck group acts by translations). These -forms satisfy the identity (this identity expresses the fact that the embedding of into via these functions is conformal). The (composition of the) Gauss map (with stereographic projection) can be read off from the , and as a meromorphic function on , it is given by the formula . Define a function on by the formula . Then are the coordinates of a rational map from into which extends to a map into , by sending each zero of to in the at infinity. Symmetry allows us to identify the image with the hyperelliptic embedding from before, and we deduce that . Solving for we obtain the integrands in the formulae above.

In fact, any holomorphic function on a domain in 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 . 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 the resulting surface closes up smoothly around the vertices, and one obtains a compact abstract Riemann surface tiled by copies of the fundamental region, together with a holonomy representation of into . Sometimes the image of this representation in the rotational part of is finite, and one obtains an infinitely periodic minimal surface as in the case of Schwarz’s surface. A fundamental tile in 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 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 and are two such regions, then a conformal map from to can be factored as the product of a map uniformizing as the upper half-plane, followed by the inverse of a map uniformizing 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 . Near each vertex, can be moved by a fractional linear transformation to (part of) a wedge, consisting of complex numbers with argument between and , where is the angle at . The function 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: . Functions that differ by multiplication by a constant have the same logarithmic derivative: . Putting these two observations together suggest defining the nonlinearity of a function as the composition . This has the property that for any constants . Under inversion the nonlinearity transforms by . From this, and a simple calculation, one deduces that the operator 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 . Given the Schwarzian derivative , one may recover the nonlinearity by solving the Ricatti equation . 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 , acting by fractional linear transformations, as we have just deduced. Once we know the nonlinearity, we can solve for by , the usual solution to a first order linear inhomogeneous ODE. The Schwarzian of the function is . The advantage of expressing things in these terms is that the Schwarzian of a uniformizing map for a circular polygon with angles at the vertices has the form of a rational function, with principal parts , where the and the and depend (unfortunately in a very complicated way) on the edges of (for the ugly truth, see Nehari, chapter 5). To see this, observe that the map has an order two pole near finitely many points (the preimages of the vertices of under the uniformizing map) but is otherwise holomorphic. Moreover, it can be analytically continued into the lower half plane across the interval between successive , by reflecting the image across each circular edge. After reflecting twice, the image of is transformed by a fractional linear transformation, so 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 is the uniformizing map from the upper half-plane to a polygon with spherical arcs, then 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.

where is as above, and 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.