The genus g Teichmüller space Teich(g) is the space of all marked hyperbolic metrics on a genus g surface , modulo homotopy, or equivalently the space of all conformal structures, modulo biholomorphisms isotopic to the identity.
Recall it can be given a topology in a number of ways. Could we also give the space a geometry?
In line with the general philosophy of deformation theory, we may attempt to get a handle on the local geometry of this space—if indeed it exists—by describing infinitesimal deformations of hyperbolic metrics or conformal structures.
How could we describe deformations? One way is to by using a pair of transverse measured foliations on our surface—roughly speaking, “nice” partitions of the surface (minus finitely many singular points) into 1-dimensional manifolds—which record the directions of maximal and minimal stretch of an deformation on the surface. These can be encoded, complex analytically, using holomorphic quadratic differentials.
Alternatively, and in some sense more directly, we may describe infinitesimal deformations using quasiconformal maps—continuous maps whose infinitesimal distortion is uniformly bounded, from both above and below—which, by the measurable Riemann mapping theorem (which despite its name is not directly related to the Riemann mapping theorem) may be encoded, complex analytically, using Beltrami differentials.
Now recall that tangent vectors are, in general, derivations, i.e. infinitesimal deformations, and cotangent vectors are what tangent vectors—which are canonically identifiable with cocotangent vectors—eat (for variety’s sake, they also eat Koko Krunch for breakfast.) With this in mind, we assert (without quite proving, although one could) that QD(S), the complex vector space of holomorphic quadratic differentials on S, is (or, rather, is isomorphic to, or in other words can be used to describe) the cotangent space to Teich(g) at [(S, h)], and BD(S), the complex vector space of Beltrami differentials modulo infinitesimally-trivial ones the tangent space.
Distances and geodesics
This characterization of the tangent and cotangent spaces allows us to define a metric, the Teichmüller metric, by , where the infimum is taken over all quasiconformal homeomorphisms , and is the dilatation of f.
By construction (and the properties of quasiconformal maps), this is symmetric and satisfies the triangle inequality; proving positivity requires Teichmüller’s existence and uniqueness theorems, which assert that there exists a unique (up to homotopy) minimal-dilatation quasiconformal map between any two points in Teichmüller space.
The Teichmüller metric is Finsler, not Riemannian: it is a -type norm; more precisely, we may show (after some relatively straightforward, but non-trivial work) that the norm it induces on the tangent spaces is given by , where the supremum is taken over all holomorphic quadratic differentials q with .
(In particular, the norm is dual to a -type norm on QD(S), is hence strictly convex, and in particular not something induced by a bilinear form—in other words, not Riemannian.)
With a metric in hand we can ask what the geodesics in this space look like, i.e. what are the “straight lines” (which are the geodesics in our familiar Euclidean space) in this space?
Well, Teichmüller geodesics (sometimes called Teichmüller lines) are given by fixing a holomorphic quadratic differential—or, from a geometric point of view, fixing a pair of transverse measured foliations on the surface—and then scaling that differential, i.e. continuously stretching and contracting along these foliations. What this does to the hyperbolic metric / conformal structure is fairly straightforward to visualize on a single coordinate patch, but globally, on the surface as a whole, the effect can be rather more complicated.
[One day I’ll have a visualization here. Maybe. Hopefully.]
This characterization of Teichmüller geodesics can also be used to show that the Teichmüller metric is (geodesically, and hence metrically by Hopf-Rinow) complete.
The Teichmüller metric is not negatively-curved. Since we are not dealing with a Riemannian metric here, we need to specify what we mean here, and what we will sketch below is that the metric is not negatively-curved in the sense of Busemann: distinct geodesic rays originating from the same point do not diverge exponentially.
Masur constructed more-or-less explicit examples of distinct geodesic rays (in his thesis, which then became an Annals paper) which stay a bounded distance apart. To build these, he starts with a pair of Strebel differentials—holomorphic quadratic differentials whose horizontal foliations consist of closed leaves, which may be grouped into finitely many annuli, which together with a finite number of singular leaves form a partition of the surface—with the same annuli but different relative lengths between the annuli. (Ralph Strebel proved that such differentials always exist, with prescribed relative lengths, hence their name.)
Now do affine stretches along each annuli to obtain the geodesic rays, which are called Strebel rays. The construction of the Strebel differentials, and more precisely the structure of their horizontal foliations, ensures that corresponding points on the resulting rays are always bounded quasiconformal dilatation, i.e. bounded Teichmüller distance, apart.
We may also ask: what are the isometries of this space? Or, slightly less precisely but perhaps more descriptively, what symmetries does this space have?
To answer this, we define the mapping class group Mod(g) (sometimes called the Teichmüller modular group) as the group of all orientation-preserving homeomorphisms , modulo those which are homotopic to the identity.
It is clear that any element of the mapping class group induces an isometry of Teichmüller space: pre-composing a Teichmüller map with any representative of mapping class still yields a Teichmüller map.
The key lemma involved, which Royden proves by mucking around with the space of holomorphic quadratic differentials and its analytic / algebraic underpinnings, is this: any complex linear isometry between the (co)tangent spaces induces a conformal map between the underlying points in Teichmüller space—which, recall, may be thought of as Riemann surfaces.
(In fact Royden’s lemma—really a theorem—characterizes these linear isometries very precisely: any such complex linear isometry is given by , where has modulus 1 and is a conformal map between the surfaces, although we will not need the full strength of it here.)
Now any isometry induces complex linear isometries between cotangent spaces and between tangent spaces, and hence conformal maps—in particular orientation-preserving self-homeomorphisms, which are well-defined mod homotopy, or in other words, mapping classes—between the underlying points.