Articles

We previously defined a Finsler metric on Teichmüller space, the Teichmüller metric; recall this was a $L^\infty$-type norm defined using extremal quasiconformal maps; it was complete, but not negatively-curved.

There is also a Riemannian metric on Teichmüller space, which is negatively-curved (although not in a terribly nice way) but not complete (although its metric completion has reasonably nice properties.) This is the Weil-Petersson metric, which we shall explore presently.

To define this metric, recall that we identified the tangent spaces to Teichmüller space as spaces of Beltrami differentials, or more precisely bounded (-1,1)-differentials (modulo infinitesimally-trivial ones.) Now we define the metric on the tangent space $T_{[(S,h)]} \mathcal{T}_g$ by $\langle \varphi, \psi \rangle_{WP} := \int \varphi\bar{\psi} \,ds$, where ds denotes the hyperbolic metric on S.

Alternatively (and equivalently), we may define it first as a cometric, on the cotangent spaces $T^*_{[(S,h)]} \mathcal{T}_g$ (which, recall, are identified with holomorphic quadratic differentials, i.e. (2,0)-differentials) by $\langle \varphi, \psi \rangle_{WP} := \int \varphi\bar{\psi} \,(ds)^{-1}$. Either of these expressions checks out formally, but why else on Earth would we consider an expression like that?

Perhaps the most natural way to arrive at such an expression (that I can think of, anyhow) is to start with the pairing $T_{[(S,h)]} \mathcal{T}_g \times T^*_{[(S,h)]} \mathcal{T}_g \to \mathbb{C}$ given by $(\varphi, \psi) \mapsto \int_S \varphi \psi$: Given a Beltrami differential—which, recall, represents an infinitesimal deformation—we can feed it (pair it with) a holomorphic quadratic differential—which encodes infinitesimal changes under deformation—and average the resulting (infinitesimal) distortion across the surface. (This is also the [admittedly not altogether precise] sense in which the Weil-Petersson metric records a $L^2$-smoothed distortion, rather than the $L^\infty$-distortion that the Teichmüller metric records.)

But now any quadratic differential may be associated to a Beltrami differential by $\psi \mapsto \psi (ds)^{-1}$: again, this checks out formally in terms of degrees, and the correspondence doesn’t seem so unreasonable if we remember how both of these objects encode infinitesimal change in some way. Going through this correspondence and applying what comes out to our pairing results in the Weil-Petersson metric / cometric expressions above.

cf. the Petersson inner product on entire modular forms … apparently Weil first defined this metric taking inspiration from the Petersson inner product.

This post describes some salient geometric properties of this metric; a subsequent post will describe some of the reformulations / novel constructions of this metric, as well as applications of the Weil-Petersson geometry on Teichmüller space.

### Kählerity and Fenchel-Nielsen coordinates

(full disclosure: this part somewhat shamelessly stolen off this blogpost of Carlos Matheus.)

The real part $g_{WP} = \mathrm{Re} \langle \cdot, \cdot \rangle_{WP}$ induces a real inner product (also inducing the Weil-Petersson metric), while the imaginary part $g_{WP} = \mathrm{Im} \langle \cdot, \cdot \rangle_{WP}$ induces an anti-symmetric bilinear form, i.e., a symplectic form$\omega_{WP}$.

By definition, if we let J denote the complex structure on Teich(S), we have $g_{WP}(q_1, q_2) = \omega_{WP}(q_1, Jq_2)$. Moreover, as firstly discovered by Weil by means of a “simple-minded calculation” (“calcul idiot”) and later confirmed by Ahlfors and others (including McMullen, who produces an explicit Kähler potential), the Weil-Petersson symplectic form $\omega_{WP}$ is closed, and so the Weil-Petersson metric is Kähler.

Using these properties, Wolpert (see also Section 7.8 in Hubbard’s book) showed that $\omega_{WP} = \frac 12 \sum_{\alpha \in P} d\ell_\alpha \wedge d\tau_\alpha$, where P is any pants decomposition on our surface, and $\ell$ and $\tau$ denote the corresponding length and twist parameters. In this sense, Fenchel-Nielsen coordinates are canonical (even if any particular manifestation of them involves an arbitrary choice of pants decomposition).

The proofs actually involve quite explicit considerations of twist deformations, and how the length parameters vary along these deformations. In particular, we have that the infinitesimal generator $\partial / \partial\tau_\alpha$ of the twist about $\alpha$ is the symplectic gradient for the Hamiltonian function $\frac 12 \ell_\alpha$, that is $\frac 12 d\ell_\alpha = \omega_{WP}(-, \partial/\partial\tau_\alpha)$.

These considerations are also the starting point for Wolpert’s expansion formulas for the Weil-Petersson metric, which appear and are heavily useful below.

### Geodesics

It is rather more difficult to describe Weil-Petersson geodesics than Teichmüller geodesics. (One reason, apparently, for a comment of Curt McMullen’s to the effect that the metric is “useless”.) Nevertheless we do know several large-scale properties:

The Weil-Petersson metric is uniquely geodesic.

Length functions, as well as their square roots, are convex along Weil-Petersson geodesics.

It is a result of Burns-Masur-Wilkinson that the Weil-Petersson geodesic flow is ergodic, so generically Weil-Petersson geodesics are equidistributed.

### Incompleteness and metric completion

The Weil-Petersson metric is not complete: using Wolpert’s formula above to perform a first-order expansion of $\omega_{WP}$ near a cusp, we may conclude that it is possible to degenerate a given curve $\alpha$ to zero length within finite Weil-Petersson distance $\sim\ell_\alpha^{1/2}$

The metric completion of Teichmüller space with the Weil-Petersson metric is augmented Teichmüller space $\overline{\mathcal{T}_g}$—Teichmüller space with strata glued in consisting of points corresponding to noded surfaces, where some finite set $\sigma$ of simple closed curves on our surface has degenerated to zero length; the set $\sigma$ exactly determines the stratum $\mathcal{T}_\sigma$.

At a point $X \in \overline{\mathcal{T}_g}$ near a stratum $\mathcal{T}_\sigma$, we have an adapted length basis, consisting of a set of curves $(\sigma, \chi)$, where  $\sigma$ is precisely the set of degenerated curves for the stratum (“short curves”, identifying the nearby stratum), and $\chi$ is a collection of simple closed curves disjoint from those in $\sigma$, such that the tangent vectors $\{d \ell_\alpha^{1/2}(X), J d\ell_\alpha^{1/2}(X), d\ell_\beta(X) \}_{\alpha \in \sigma, \beta \in \chi}$, where J is the linear endomorphism on the $T_X \mathcal{T}_g$ inducing the natural conformal structure on X, is a basis of $T_X\mathcal{T}_g$.

We can extend such a basis to a relative basis at $X_\sigma \in \mathcal{T}_\sigma$, now in the glued stratum, if the length parameters $\{\ell_\beta\}_{\beta\in\chi}$ give a local system of coordinates for the stratum $\mathcal{T}_\sigma$ near $X_\sigma$. In such a relative basis, we have Wolpert’s first-order expansion $\langle \cdot, \cdot \rangle_{WP} \sim \sum_{\alpha \in \sigma} \left( (d\ell_\alpha^{1/2})^2 + (d\ell_\alpha^{1/2} \circ J)^2\right) + \sum_{\beta \in \chi} (d\ell_\beta)^2$, and the implied constant is uniform in a neighborhood $U \subset \overline{\mathcal{T}_g}$ of $X_\sigma$.

### Curvature

Wolpert’s first-order expansion/s and “second-order Masur-type expansions” lead to the following estimates for any adapted length basis $(\sigma, \chi)$ and any $\alpha, \alpha' \in \sigma$, $\beta, \beta' \in \chi$, with uniform constants on suitable Bers regions $\Omega(\sigma,\chi,c)$ (regions where all the “short curves” in $\sigma$ have length $\geq \frac 1c$, and all curves in $\sigma \cup \chi$ have length $\leq c$.)

• $\langle d\ell_\alpha^{1/2}, d\ell_{\alpha'}^{1/2} \rangle = \frac 1{2\pi} \delta_{\alpha\alpha'} + O((\ell_\alpha\ell_{\alpha'})^{3/2})$;
• $\langle d\ell_\alpha^{1/2}, J d\ell_{\alpha'}^{1/2} \rangle =\langle J d\ell_\alpha^{1/2}, d\ell_\beta \rangle = 0$;
• $\langle d\ell_\beta, d\ell_{\beta'} \rangle \sim 1$, and this inner product extends continuously to the boundary stratum $\mathcal{T}_\sigma$;
• $\langle d\ell_\alpha^{1/2}, d\ell\beta \rangle = O(\ell_\alpha^{3/2})$.

From these we obtain

• $d(X, \mathcal{T}_\sigma) = \sqrt{2\pi \sigma_\alpha \ell_\alpha} + O(\sum_{\alpha\in\sigma} \ell_\alpha^{5/2})$;
• estimates of covariant derivatives;

and then the following estimates for sectional curvatures:

• for any complex line $\mathbb{R} d\ell_\alpha^{1/2} + \mathbb{R} J d\ell_\alpha^{1/2}$, $\langle R(d\ell_\alpha^{1/2}, Jd\ell_\alpha^{1/2} ) Jd\ell_\alpha^{1/2}, d\ell_\alpha^{1/2} \rangle = \frac 3{16\pi^2\ell_\alpha} + O(\ell_\alpha)$;
• for any quadruple of vectors in the adapted length basis, not a curvature-preserving permutation of the previous quadruple, $R(v_1, v_2)v_3, v_4 \rangle = O(1)$, and  each $d\ell_\alpha^{1/2}$ or $Jd\ell_\alpha^{1/2}$ in the quadruple introduces a multiplicative factor $O(\ell_\alpha)$ in the estimate.

This yields (after non-trivial computation) that Teichmüller space with the Weil-Petersson metric is negatively-curved, although with sectional curvatures not bounded away from 0 or from $-\infty$. As a consequence the Weil-Petersson geometry is not coarsely (Gromov-)hyperbolic except for topologically simple surfaces (with $3g - 3 + n < 3$)

Thus augmented Teichmüller space, being the metric completion of a uniquely geodesic negatively-curved space, has nonpositive curvature in the sense of the CAT(0) condition: triangles in this space are no fatter than they are in Euclidean space.

### Curve complexes and pants graphs

The combinatorial structure of the strata in augmented Teichmüller space is determined by the set of simple closed curves which degenerate to zero length, and hence is described by the curve complex. (Recall the curve complex is the flag simplicial complex on the 1-skeleton with vertices corresponding to homotopy classes of simple closed curves on the surface, and edges between two vertices if the corresponding homotopy classes have disjoint representatives.)

Moreover, Brock and Margalit established that augmented Teichmüller space is quasi-isometric to the pants graph, which is a graph with vertices corresponding to pants decompositions (i.e. to maximal simplices in the curve complex), and edges between two vertices if the corresponding pants decompositions differ by replacing a curve $\alpha$ with another curve $\beta$ with minimal intersection number with $\alpha$; in other words, the Weil-Petersson metric, when considered between (maximally degenerate) strata, can be seen as coarsely encoding distances between the corresponding pants decompositions in the pants graph. The (highly non-unique) quasi-isometries in questions are given by sending a point $X \in \mathcal{T}_g$ to a pants decomposition describing a Bers region X lies in, and conversely sending a pants decomposition to a point in Teichmüller space lying in the corresponding Bers region.

### Symmetries

We may use the combinatorial structure described by the curve complex to describe the isometry group of the Weil-Petersson geometry as the extended mapping class group $\mathrm{Mod}^\pm_g$ : Weil-Petersson isometries extend to the metric completion $\overline{\mathcal{T}_g}$ and preserve the combinatorial structure of the strata, and hence preserve the combinatorial structure of curve complex. By a result of Ivanov, order-preserving bijections of the curve complex are induced by elements of the $\mathrm{Mod}_g$; hence there is a mapping class which yields our isometry on the maximally degenerate structure in $\overline{\mathcal{T}_g}$, and hence on the closed convex hull of these maximally degenerate structures, i.e. all of $\overline{\mathcal{T}_g}$.

The Nielsen-Thurston classification of elements of the $\mathrm{Mod}_g$, together with the general theory of isometries of CAT(0) spaces, then tell us a good deal about the geometry of these isometries: they either have fixed points in $\overline{\mathcal{T}_g}$, or else have positive translation length realized on a cloesd convex set isometric to a metric space product $\mathbb{R} \times Y$, on which our isometry acts as $\mathrm{translation} \times \mathrm{id}_Y$.

### Alexandrov cones

There is a well-defined notion of angles between two geodesics emanating from a common initial point in a CAT(0) space, which allows to define tangent spaces in terms of sets of constant speed geodesics, modulo those at zero angle and having the same speed, with some natural topology on them.

In the interior of augmented Teichmüller space, the CAT(0) notion of angle coincides with the angles given by the Riemannian Weil-Petersson metric, and we get the usual tangent spaces; on the boundary stratum,  the CAT(0) Alexandrov angles yield Alexandrov tangent cones, which are isometric to Euclidean orthants (corresponding to the degenerate curvve “directions”) cross tangent spaces to strata.

These tangent-space-like structures have a number of applications here: e.g. they allow us to classify flat subspaces, yield a first variation formula for distance and non-refraction of geodesics (length-minimizing paths may change strata only at endpoints), allow us to construct combinatorial harmonic maps in certain cases, etc. (for slightly more detail, see Section 8 of Wolpert’s survey on Weil-Petersson metric geometry.)