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 
The surface may be “analytically continued” by rotating 
The symmetries of 
Another description is as follows. Since the deck group acts by translation, the Gauss map from 
The function 
The integral in each case depends on the path, and lifts to a single-valued function precisely on
Geometrically, the three coordinate functions 
In fact, any holomorphic function 
Note that if 
For example, two functions that differ by addition of a constant have the same derivative: 
(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
where 
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.
, a quasimorphism is a function 
for which there is some least real number 
(called the defect) such that for all pairs of elements 
there is an inequality 
. Bounded functions are quasimorphisms, although in an uninteresting way, so one is usually only interested in quasimorphisms up to the equivalence relation that 
if the difference 
is bounded. It turns out that each equivalence class of quasimorphism contains a unique representative which has the extra property that 
for all 
and 
. Such quasimorphisms are said to be homogeneous. Any quasimorphism may be homogenized by defining 
(see e.g. 
nevertheless admit rich families of homogeneous quasimorphisms. For example, groups that act weakly properly discontinuously on word-hyperbolic spaces admit infinite dimensional families of homogeneous quasimorphisms; see e.g. 
be a knot invariant. Then one can extend 
strands by 
where 
, and the “hat” denotes plat closure. It is an interesting question to ask: under what conditions on 
in the 
in ![S^3 \times [0,1]](https://s0.wp.com/latex.php?latex=S%5E3+%5Ctimes+%5B0%2C1%5D&bg=ffffff&fg=333333&s=0 "S^3 \times [0,1]")
with 
and 
. Concordance is an equivalence relation, and the equivalence classes form a group, with connect sum as the group operation, and orientation-reversed mirror image as inverse. The only subtle aspect of this is the existence of inverses, which we briefly explain. Let 
be an arbitrary knot, and let 
denote the mirror image of 
in space so that they are symmetric with respect to reflection in a dividing plane. There is an immersed annulus 
which connects each point on 
. A ribbon surface (in particular, a ribbon disk) can be pushed into a (smoothly) embedded surface in a 
-ball bounding 
to the unknot, as claimed.
of knots. Since connect sum is commutative, this group is abelian. Notice as above that a slice knot — i.e. a knot bounding a locally flat disk in the 
. If 
are loops in 
to be the linking number of 
, which is obtained from 
by pushing it to the positive side of 
, and after choosing generators, it can be expressed in terms of a matrix 
(called the Seifert matrix of 
, is the signature (in the usual sense) of the symmetric matrix 
. Changing the orientation of a knot does not affect the signature, whereas taking mirror image multiplies it by 
. Moreover, if 
are Seifert surfaces for 
for which there is some sphere 
that intersects 
, and therefore the Seifert matrix of 
; it follows that 
. In fact it turns out that 
is a homomorphism from 
; equivalently (by the arguments above), it is zero on knots which are topologically slice. To see this, suppose 
in the 
is an embedded bicollared surface in the 
whose interior may be taken to be disjoint from 
induces a map 
whose kernel is Lagrangian (with respect to the usual symplectic pairing on 
of an oriented surface). Geometrically, this means we can find a basis for the homology of 
(which is equal to the homology of 
be obtained by pushing off 
block is identically zero. It follows that 
also has a zero 
lower right block, and therefore its signature is zero.
; equivalently, to the structure of the maximal metabelian quotient of 
. More sophisticated “twisted” and 
signatures can be obtained by studying further derived subgroups of 
gives rise to a quasimorphism on braid groups if there is a constant 
so that ![|I([K])| \le C\cdot\|K\|_g](https://s0.wp.com/latex.php?latex=%7CI%28%5BK%5D%29%7C+%5Cle+C%5Ccdot%5C%7CK%5C%7C_g&bg=ffffff&fg=333333&s=0 "|I([K])| \le C\cdot\|K\|_g")
, where 
denotes 
, 
and 
. It is shown that the connect sum 
bounds a Seifert surface whose genus may be universally bounded in terms of the number of strands in the braid group. Pushing this Seifert surface into the 
. Properties of 
one can (usually) form a hyperbolic 
which fibers over the circle, with fiber 
. The 
-invariant of 
where 
is a 
with a product metric near the boundary, and 
is the first Pontriagin form on 
(eg on a subgroup consisting entirely of pseudo-Anosov elements)?
preserves a probability measure on 
, one can take a sequence of almost invariant probability measures, supported on the subset ![[-n,n] \cdot p](https://s0.wp.com/latex.php?latex=%5B-n%2Cn%5D+%5Ccdot+p&bg=ffffff&fg=333333&s=0 "[-n,n] \cdot p")
(where 
is arbitrary), and any weak limit will be invariant. For a general amenable group, in place of the subsets ![[-n,n] \subset \mathbb{Z}](https://s0.wp.com/latex.php?latex=%5B-n%2Cn%5D+%5Csubset+%5Cmathbb%7BZ%7D&bg=ffffff&fg=333333&s=0 "[-n,n] \subset \mathbb{Z}")
, one works with a sequence of Folner sets; i.e. subsets with the property that the ratio of their size to the size of their boundary goes to zero (so to speak).
, then 
). Let 
denote the space of probability measures on 
defined by the formula
stationary under 
under a randomly chosen 
that 
is absolutely continuous with respect to 
; in fact, the Radon-Nikodym derivative satisfies 
. Substituting 
, we have 
, where 
denotes word length with respect to the given generating set.
. In one dimension, a measure (at least one of full support without atoms) can be “integrated” to a path metric. Consequently, any finitely generated group of homeomorphisms of the circle is conjugate to a group of bilipschitz homeomorphisms (if the harmonic measure associated to the original action does not have full support, or has atoms, one can “throw in” another random generator to the group; the resulting action can be assumed to have a harmonic measure of full support without atoms, which can be integrated to give a structure with respect to which the group action is bilipschitz). In fact, 
acts in a non-bilipschitz way on the interval — see 
for some manifold 
. The action of 
determines a foliated circle bundle 
; i.e. a circle bundle, together with a codimension one foliation transverse to the circle fibers. To see this, first form the product 
with its product foliation by leaves 
, where 
denotes the universal cover of 
. The foliation makes 
. The foliation allows us to associate to each path 
in 
to the fiber over 
; integrability (or flatness) implies that this homeomorphism only depends on the relative homotopy class of 
of a manifold 
, by finding a fixed point of the leafwise heat flow on the space of probability measures on 
in its harmonic measure. Let 
everywhere. The integrability condition for a foliation is expressed in terms of the 
, and we can write 
where 
. More intrinsically, 
is the diffeomorphism of 
, then
-structure on the bundle 
, and we have a formula
under the identification of 
; i.e. the absolute value of the curvature of the connection is equal to 
times the area enclosed by 
by a discrete group of isometries. A positive harmonic function on 
; identifying positive harmonic functions on hyperbolic space with distributions on the sphere at infinity, one sees that the “worst case” is the harmonic extension of an atomic measure concentrated at a single point at infinity, since every other positive harmonic function is the weighted average of such examples. As one moves towards or away from a blob at infinity concentrated near this point, the radius of the blob expands like 
; since the sphere at infinity has dimension 
, the conclusion follows. But this means that the speed of 
. A circle of length 
, so the curvature of the connection has absolute value pointwise bounded by 
.
, where 
is the Euler number of the bundle (a topological invariant). For, the surface 
is an immersion which is injective on 
. There is a cover 
of 
on the circle at infinity of 
. Integrating as above over the image of 
is pointwise bounded by 
. If 
.
for 
, and better estimates for higher 
be a polynomial in two variables; i.e. 
where each 
is non-negative, and the coefficients 
are complex numbers which are nonzero for only finitely many pairs 
determines a smooth complex curve 
, consider the finite set consisting of the points with integer coordinates 
for which the coefficient 
denote the partial derivative of 
is a holomorphic 
for some polynomial 
of degree 
and it has 
interior lattice points. Geometrically one can “see” the surface by projecting to the 
has two distinct square roots, so the map is 2 to 1. However, at the 
. The Newton polygon in this case is the triangle with vertices 
and it has 
interior lattice points. One way to “see” the surface in this case is to first imagine 
points of intersection. After deforming the curve, these points of intersection are resolved into tubes, so one obtains 
tubes are needed to tube the lines together into a (multiply)-punctured plane, and the remaining 
; i.e. remove the intersection with the coordinate axes. For generic 
. The image is a subset of 
or 
and with vertices at integer lattice points (by 
and take the limit as 
corresponding to a linear equation 
whose solution set in 
) is a line minus two points — i.e. a pair of pants.
be the graph dual to the triangulation, with edges dual to boundary edges of the triangulation extended indefinitely. The surface 
. This can be subdivided into twelve triangles of area 
The dual spine is then the following:
-structure on a manifold 
. If we fix an “initial” value 
, then we are guaranteed short time existence and uniqueness of a solution (providing the function 
defines a vector field on 
. The group in this case is 
.
. The group in this case is 
, acting on 
is preserved; this corresponds to the fact that 
is always a solution of a homogeneous linear ODE). The Lie algebra is 
is the general solution. In fact, in the previous example, the Lie algebra of the group of translations is also identified with 
. The group in this case is the affine group 
where the first factor acts by dilations and the second by translation. The affine group is not abelian, so one cannot “integrate” a vector field directly, but it is solvable: there is a short exact sequence 
. The image in the Lie algebra of the group of dilations is the term 
, which can be integrated as before to give an integrating factor 
. Setting 
gives 
which is an indefinite integral, and can be solved by a further integration. In other words, we do one integration to change the structure group from 
where each 
surjects onto an abelian group, with kernel 
, and after finitely many steps, the last kernel is trivial.
. In this case, it is well-known that the equation can blow up in finite time, and one does not obtain a group of transformations of 
; another point of view says that one obtains a pseudogroup of transformations of subsets of 
, acting by projective linear transformations. Let 
be a 
, with 
. Matrices act on 
for 
. Differentiating 
at 
one obtains 
which is the general form of the Ricatti equation. Since the group 
, one can find all other solutions as follows. Do a change of co-ordinates, by sending the solution 
“to infinity”; i.e. define 
. Then as a function of 
, the Ricatti equation reduces to a linear inhomogeneous ODE. In other words, the structure group reduces to the subgroup of 
.
can be transformed into a system of two first order linear equations in the variables 
. A system of first order ODEs in 
. A system of linear equations corresponds to the structure group 
, hence in the case of a 
system, to 
. The determinant map is a homomorphism from 
with kernel 
; hence, after multiplication by a suitable integrating factor, one can reduce to a system which is (equivalent to) the Ricatti equation.
is homeomorphic to 
is topologically a foliation by planes. One important special case is when any two leaves of 
acts on the leaf space (which is 
for which 
is properly contained in 
denote the two endpoints of the interval 
by defining 
to be the supremum of the set of values 
over all 
. The non-nesting property, and the fact that every leaf of 
is a strictly increasing (i.e. fixed-point free) homeomorphism of 
of homeomorphisms that commute with integer translation. One says in this case that the manifold 
be a Jordan curve in the Riemann sphere. The image of the curve decomposes the sphere into two regions homeomorphic to disks. Each open disk region can be uniformized by a holomorphic map from the open unit disk, which extends continuously to the boundary circle. These uniformizing maps are well-defined up to composition with an element of the Möbius group 
called the welding homeomorphism. Conversely, given a homeomorphism of the circle, one can ask when it arises from a Jordan curve in the Riemann sphere as above, and if it does, whether the curve is unique (up to conformal self-maps of the Riemann sphere). Neither existence nor uniqueness hold in great generality. For example, if the image 
has positive (Hausdorff) measure, any quasiconformal deformation of the complex structure on the Riemann sphere supported on the image of the curve will deform the curve but not the welding homeomorphism. One significant special case in which existence and uniqueness is assured is the case that 
are contained in the quasicircle, and the spherical distance between the two points is 
, then at least one arc of the quasicircle joining 
to 
has spherical diameter at most 
. In other words, there are no bottlenecks where two points on the quasicircle come very close in the sphere without being close in the curve. Welding maps corresponding to quasicircles are precisely the quasisymmetric homeomorphisms. A homeomorphism is quasisymmetric if for every sufficiently small interval in the circle, the image of the midpoint of the interval under the homeomorphism is not too far from being the midpoint of the image of the interval; i.e. it divides the image of the interval into two pieces whose lengths have a ratio which is bounded below and above by some fixed constant. Other classes of geometric structures can be detected by welding: smooth Jordan circles correspond to smooth welding maps, real analytic circles correspond to real analytic welding maps, round circles correspond to welding maps in 
, there is a constant 
so that
is the product of the distinct primes appearing in the product 
. Informally, this conjecture says that for triples 
, the inequality 
has only finitely many solutions in coprime integers 
to within 
. In fact, the rationals appearing in the continued fraction approximation to 
for each coprime pair 
This circle packing nests down on the 
. Some “space” appears between neighboring circles, and they no longer pack tightly (the following picture shows 
).
The total cross-sectional width of these circles, restricted to pairs 
, can be estimated as follows. Each 
which converges for positive 
.
for some integer 
acts on the upper half-plane, which we think of now as the complex numbers with non-negative imaginary part, by fractional linear transformations 
for some constant 
out the cusp (this is a kind of equidistribution fact – the thickness of the cusp goes to zero like 
, so if one chooses a sequence of points in a hyperbolic surface at random with respect to the uniform (area) measure, it takes about 
for any 
.
. The matrix 
has 
and 
as eigenvectors (here 
denotes the “conjugate” 
), and thus preserves a “wedge” in 
integer matrix whose eigenvalues are all real, positive, irrational and distinct. A collection of 
subgroup of 
consisting of matrices with the same set of eigenvectors (this is a consequence of 
, where 
denote the set of homotopy classes of closed oriented curves on 
denote the free abelian group with generating set ![[\alpha,\beta] = \sum_{p \in \alpha \cap \beta} \epsilon(p; \alpha,\beta) |\alpha_p\beta_p| \in \mathbb{Z}\hat{\pi}](https://s0.wp.com/latex.php?latex=%5B%5Calpha%2C%5Cbeta%5D+%3D+%5Csum_%7Bp+%5Cin+%5Calpha+%5Ccap+%5Cbeta%7D+%5Cepsilon%28p%3B+%5Calpha%2C%5Cbeta%29+%7C%5Calpha_p%5Cbeta_p%7C+%5Cin+%5Cmathbb%7BZ%7D%5Chat%7B%5Cpi%7D&bg=ffffff&fg=333333&s=0 "[\alpha,\beta] = \sum_{p \in \alpha \cap \beta} \epsilon(p; \alpha,\beta) |\alpha_p\beta_p| \in \mathbb{Z}\hat{\pi}")
are 
represents their product in 
, and 
represents the resulting conjugacy class in 
is the oriented intersection number of ![[\cdot,\cdot]:\mathbb{Z}\hat{\pi} \times \mathbb{Z}\hat{\pi} \to \mathbb{Z}\hat{\pi}](https://s0.wp.com/latex.php?latex=%5B%5Ccdot%2C%5Ccdot%5D%3A%5Cmathbb%7BZ%7D%5Chat%7B%5Cpi%7D+%5Ctimes+%5Cmathbb%7BZ%7D%5Chat%7B%5Cpi%7D+%5Cto+%5Cmathbb%7BZ%7D%5Chat%7B%5Cpi%7D&bg=ffffff&fg=333333&s=0 "[\cdot,\cdot]:\mathbb{Z}\hat{\pi} \times \mathbb{Z}\hat{\pi} \to \mathbb{Z}\hat{\pi}")
. Goldman shows that this bracket makes 
be a (smooth) class function (i.e. a function which is constant on conjugacy classes) on a Lie group 
by the formula
is some (fixed) 
-invariant orthogonal structure on the Lie algebra 
(for example, if 
). The tangent space to the character variety 
determine a pairing
determines a function 
by sending a (conjugacy class of) representation ![[\phi]](https://s0.wp.com/latex.php?latex=%5B%5Cphi%5D&bg=ffffff&fg=333333&s=0 "[\phi]")
to 
. Note that 
only depends on the conjugacy class of 
by 
when 
if 
, so that one component of 
, and then 
where 
is angle and 
is length, and the sum is taken over a maximal system of disjoint essential simple curves ![[\alpha,\alpha^3]](https://s0.wp.com/latex.php?latex=%5B%5Calpha%2C%5Calpha%5E3%5D&bg=ffffff&fg=333333&s=0 "[\alpha,\alpha^3]")
is zero. Nevertheless, the full geometric meaning of the Goldman bracket remains mysterious, and a topic worthy of investigation.
