## Slightly elevated Teichmuller theory

Last week at my invitation, David Dumas spoke in the U Chicago geometry seminar and gave a wonderful introductory talk on the theory of convex real projective structures on surfaces. This is the first step on the road to what is colloquially known as “Higher Teichmüller theory”, and the talk made such an impression on me that I felt compelled to summarize it in a blog post, just to organize and clarify the material in my own mind.

Fix a surface $S$ (for convenience closed, oriented of genus at least 2). We are interested in the space $C(S)$ of convex real projective structures on $S$. This has at least 3 incarnations:

1. it is a connected component of the $\mathrm{SL}(3,\mathbf{R})$ character variety $X(S)$, the space of homomorphisms from $\pi_1(S)$ into $\mathrm{SL}(3,\mathbf{R})$ up to conjugacy (note: since all representations in this component are irreducible, one can really take the naive quotient by the conjugation action rather than the usual quotient in the sense of geometric invariant theory);
2. it is topologically a cell, homeomorphic to $\mathbf{R}^{16g-16}$, and can be given explicit coordinates (analogous to Fenchel-Nielsen coordinates for hyperbolic structures) in such a way that a coordinate describes an explicit method to build such a structure from simple pieces by gluing; and
3. it has the natural structure of a complex variety; explicitly it is a bundle over the Teichmuller space of $S$ whose fiber is isomorphic to the vector space of cubic differentials on $S$.

This last identification is quite remarkable and subtle, since $\mathrm{SL}(3,\mathbf{R})$ is not a complex Lie group, and its action on the projective plane does not leave invariant a complex structure. Here one should think of $\mathrm{Teich}(S)$ as the space of (marked) conformal structures on $S$, rather than as the space of (marked) hyperbolic structures on $S$.

Following Dumas, we explain these three incarnations in turn. Some references for this material are Goldman, Benoist, and Loftin.

1. Real projective structures

Let’s start with the definition of a real projective structure. This is an example of what is called a $(G,X)$ structure in the sense of Ehresmann; i.e. an atlas of charts modeled on some real analytic manifold $X$ with transition functions in some (pseudo)group of real analytic transformations $G$. Here $X$ is the real projective plane $\mathbf{RP}^2$, which can be thought of as the ordinary plane together with a circle at infinity, or as the space of lines through the origin in ordinary 3-space; and $G$ is the group $\mathrm{SL}(3,\mathbf{R})$, acting linearly on 3-space and thereby projectively on the (projective) plane.

Associated to such a structure is a developing map $D:\tilde{S} \to \mathbf{RP}^2$ defined as follows. Pick a basepoint and a chart around that point, and use the chart to identify the chart with a subset of the projective plane. Extend the map along each path based at the basepoint by analytic continuation, using the transition functions to move from chart to chart. The result is well-defined on homotopy classes of paths rel. endpoints and determines a map from the universal cover — this is the developing map. It is independent of choices, up to composition with a projective automorphism. In particular, the deck group of the covering acts on the projective plane in a unique manner which makes $D$ equivariant. Thus a projective structure determines a holonomy representation $\rho:\pi_1(S) \to \mathrm{SL}(3,\mathbf{R})$.

It follows from a general theorem of Ehresmann-Thurston (valid for any $(G,X)$ structure) that projective structures on $S$ near any given structure are parameterized (locally) by the conjugacy class of the representation associated to the developing map; technically, the map from the space of $(G,X)$ structures to the space of representations up to conjugacy is a local homeomorphism. There are two parts to this claim: first, that any deformation of the representation is associated to a deformation of the structure; and second, that nearby $(G,X)$ structures with the same holonomy are isomorphic.

The first claim can be proved as follows. Think of a representation from $\pi_1(S) \to G$ as an $X$ bundle $E$ over $S$ with a flat $G$ structure giving a foliation $\mathcal{F}$ transverse to the fibers. In this language a $(G,X)$ structure is determined by a section $\sigma$ of the bundle transverse to $\mathcal{F}$; charts are given locally by the composition of this section with projection along leaves of $\mathcal{F}$ to a fiber. The key point is that as we deform the flat bundle structure and the foliation $\mathcal{F}$ by deforming the representation, the section $\sigma$ stays transverse so there is an accompanying deformation of the $(G,X)$ structure.

The second claim can be seen by covering $S$ by small open charts $U_i$ and choosing subcharts $V_i$ with $\overline{V}_i \subset U_i$, and then noting that if $\phi_i,\phi_i':U_i \to X$ are sufficiently close, the image $\phi_i'(V_i)$ is contained in $\phi_i(U_i)$, and we obtain an isomorphism of $(G,X)$ structures by patching together local isomorphisms $\phi_i^{-1} \phi_i'|_{V_i}$.

Note that the point stabilizers of $\mathrm{SL}(3,\mathbf{R})$ acting on the projective plane are noncompact, and there is therefore no canonical metric on a real projective surface. On the other hand, projective transformations permute the set of straight lines in the plane, so that projective surfaces have canonical families of lines through every point in every tangent direction. One refers to these lines as geodesics, even in the absence of a natural metric.

2. Convex structures

A real projective structure on a surface is convex if the developing map $D$ is a homeomorphism onto a proper convex (open) subset $\Omega$ of the projective plane. Thus all such structures arise from a projective action of $\pi_1(S)$ that stabilizes some $\Omega$ and acts freely, properly discontinuously and cocompactly there.

Example. Let $T$ be the open triangle in the projective plane with vertices at the (projective) points $(1:0:0), (0:1:0), (0:0:1)$. Let $\alpha,\beta$ be the diagonal matrices with entries $(a,a^{-1},1)$ and $(1,b,b^{-1})$ for some $a,b$. Then the projective action of $\langle \alpha,\beta\rangle$ stabilizes $T$ with quotient a torus. The figure below shows $T$ together with a tiling by fundamental domains.

Example. Not every real projective structure is convex. Here is the image under the developing map of another real projective torus; a fundamental domain is the immersed annulus between the green and red curves. Observe that the holonomy representation is not faithful (as it must be for a convex projective structure):

I love how a picture like this lets you “see” a surface immersed in 3-space in terms of the projective impression it leaves on your retina.

Notice that the core of the immersed annulus is not homotopic in the projective torus to a “geodesic” representative. On the other hand, every essential loop in a surface has a geodesic representative in any convex structure. On a nonconvex surface, some loops have geodesic representatives, and some don’t. A fundamental theorem of Choi says that there is always a canonical collection of disjoint simple geodesics which decompose the surface into convex pieces:

Theorem (Choi): Every real projective surface with negative Euler characteristic has a unique collection of disjoint simple closed geodesics whose complementary pieces are either annuli covered by an affine half-space, or the interior of a compact convex real projective manifold of negative Euler characteristic.

Building on this result, Choi-Goldman obtained a complete classification of real projective structures on a surface $S$ into combinatorial data (associated to the decomposing curves) and moduli (associated to the convex pieces):

Theorem (Choi-Goldman): The space of real projective structures on a surface of genus $g>1$ is a countable disjoint union of open cells of dimension $16g-16$. The space of convex structures can be identified with a connected component of the moduli space of representations of the fundamental group.

3. Hilbert metric

Although an arbitrary real projective surface does not carry a canonical metric, the convex ones do. Equivalently, a convex, compact domain $C$ carries a canonical metric invariant under projective automorphisms, namely the Hilbert metric.

Let’s start with the simplest case, that of an interval in the projective line. For concreteness, think of this interval as the projectivization of the positive orthant in the plane, so that the endpoints have projective coordinates $(1:0)$ and $(0:1)$, and a typical point has coordinates $(x:y)$ with $x$ and $y$ both non-negative, and at least one positive. The group of projective automorphisms of this interval (preserving orientation) is just $\mathbf{R}$, acting by $\lambda\cdot (x:y) = (e^\lambda x:y)$. Thus we can use this action to define a distance, by $d_H((x_1:y_1),(x_2:y_2)) = \log(x_1y_2/x_2y_1)$. If we parameterize this interval instead as $[0,1]$ then the relationship to the projective coordinates is $(x:y) \to y/(x+y)$ with inverse $t \to (1-t:t)$ and we obtain the formula $d_H(s,t) = \log((1-s)t/(1-t)s)$. More generally, if 4 points $p,q,r,s$ lie (in order) on a straight line in projective space, the interval $[p,q]$ carries a Hilbert metric in which

$d_H(q,r) = \log((s-q)(r-p)/(s-r)(q-p))$

i.e. the logarithm of the cross-ratio of the four points.

If $C$ is an arbitrary bounded convex domain, then we can define the Hilbert metric on $C$ as follows: for each pair $p,q$ of points in the interior of $C$, let $\ell$ with endpoints $\ell(0),\ell(1)$ be the maximal straight line in $C$ containing $p,q$ in the interior. The (Hilbert) distance from $p$ to $q$ is the logarithm of the cross ratio of $\ell(0),p,q,\ell(1)$. This function is monotone in the sense that if $C \subset C'$ is an inclusion of convex domains, then for any $p,q$ in $C$ there is an inequality $d_C(p,q) \ge d_{C'}(p,q)$ with equality if and only if the maximal straight segments through $p,q$ in $C$ and in $C'$ are equal. Note further that when $C$ is the region bounded by a conic, the Hilbert metric becomes the hyperbolic metric in the Klein model. From this and monotonicity the triangle inequality follows (showing that this is an honest metric): if $p,q$ are arbitrary and contained in a maximal segment $\ell \subset C$ we can projectively embed $C$ in the interior of a region $C'$ bounded by a conic in such a way that $\ell$ is still properly embedded in $C'$. The Hilbert metrics for $C$ and $C'$ agree on $\ell$, and the triangle inequality is satisfied in $C'$ (because it is satisfied for the usual hyperbolic metric) so for any $r$ we have

$d_C(p,q) = d_{C'}(p,q) \le d_{C'}(p,r) + d_{C'}(r,q) \le d_C(p,r) + d_C(r,q)$

Because of this monotonicity, a (geodesic) triangle $\Delta$ in any domain $C$ is thinner than the same triangle in any $C'$ with $\Delta \subset C' \subset C$. By comparison with suitable quadrics, Benoist showed that the Hilbert metric is $\delta$-hyperbolic if and only if the boundary is “quasisymmetrically convex”; this is a slightly technical condition, which can be expressed prosaically as saying that the limits of projective rescalings near a point on the boundary are strictly convex. It implies, in particular, that the boundary is $C^{1+\epsilon}$ for some $\epsilon$. Note that this part of the story is dimension-independent (and even makes sense in infinite dimensional projective spaces).

If $X$ is a real projective manifold which is not necessarily convex, it still carries a canonical Hilbert pseudo-metric defined as follows: for any points $p,q$ define $d_H(p,q)$ to be the infimum of sums $\sum_i d_{\ell_i}(p_i,p_{i+1})$ over all finite sequences $p=:p_0,p_1,\cdots,p_n:=q$ such that each successive pair $p_i,p_{i+1}$ is contained in a straight segment $\ell_i$, and $d_{\ell_i}(p_i,p_{i+1})$ means the distance from $p_i$ to $p_{i+1}$ in the Hilbert metric on $\ell_i$. This construction is the analog of the construction of the Kobayashi metric on a complex manifold, and the monotonicity of the Hilbert metric plays the role of the Schwarz Lemma. If $X$ is convex, this recovers the ordinary Hilbert metric, but otherwise it is necessarily degenerate (the degeneracy, when $X$ is compact, is equivalent to the existence of an entire straight line in $X$; i.e. a real projective immersion of $\mathbf{R}$; this is the analog of Brody’s Lemma in the projective context). I believe that for a surface $S$ this metric should be degenerate precisely on the decomposing annuli in Choi’s theorem, but I have not checked this carefully (note: I am not saying this should give a new proof of Choi’s theorem (although maybe it does?), but that a posteriori one could use the Hilbert pseudo-metric to understand the canonical decomposition).

4. Construction of examples

Now let’s explicitly construct some examples of convex projective structures on surfaces of positive genus. The simplest examples are simply the hyperbolic structures: the region enclosed by a quadric is stabilized by a conjugate of $\mathrm{SO}(2,1)$ in $\mathrm{SL}(3,\mathbf{R})$, and can be thought of as the ordinary hyperbolic plane in the Klein model. Such domains are symmetric, since the group of projective symmetries acts transitively on the interior.

Some genuinely new examples can be obtained from this one by bending, much as one obtains quasifuchsian deformations of fuchsian groups. Let’s start with a hyperbolic structure on a surface $S$ (which is a special case of a real projective structure) and pick an essential closed curve $\gamma$ which divides $S$ into two subsurfaces $A,B$. Thus $\pi_1(S) = \pi_1(A) *_{\langle \gamma \rangle} \pi_1(B)$. Choose some $\beta$ which is in the centralizer of $\gamma$ in $\mathrm{SL}(3,\mathbf{R})$. Then we can deform the representation by conjugating $\pi_1(B)$ by $\beta$. Appealing to Ehresmann-Thurston, this deformation of representations is accompanied by a deformation of projective structures.

A hyperbolic element of $\mathrm{SO}(2,1)$ has three real eigenvectors; two correspond to the fixed points $p,q$ on the quadric at infinity, and one corresponding to the point which is the intersection of the tangents to the quadric at $p$ and $q$. Thus we may always conjugate $\gamma$ to a diagonal matrix with entries $(a,a^{-1},1)$. The centralizer of $\gamma$ is thus isomorphic to the diagonal matrices; these are spanned by shears $(\lambda,\lambda^{-1},1)$ (these fix the given quadric, and just deform the hyperbolic structure) and bends $(b,b^{-2},b)$.

Geometrically, choose coordinates in which the quadric looks like a round circle in the plane, and the fixed points $p,q$ are the top and bottom points (i.e. the intersections with the $y$ axis). The centralizer of $\gamma$ preserves the eigenvectors, which is to say it preserves the two horizontal tangencies to the circle. Thus the image of the “right hand side” of the circle under conjugation is a new convex curve which fits together with the “left hand side” of the circle to make a $C^1$ convex curve (in general it will no longer be $C^2$ at $p,q$). For example, in these specific coordinates, conjugating the right hand side by a “bend” as above turns the half-circle into half an ellipse, sliced along one of its axes. Propagating this bending to the other images of the axis of $\gamma$, we obtain the new limit set as a limit of a sequence of uniformly convex, $C^1$ domains (since the deformations are uniform on all scales, the limit is automatically Hölder, which is to say $C^{1+\epsilon}$, as Benoist says it must be). This new domain is (by construction) invariant under a proper cocompact group of projective transformations (namely $\pi_1(S)$) but generically, by no other symmetries; one says the domain is divisible.

The figure below shows the “before” and “after” picture for a hyperbolic structure on a once-punctured torus bent along the edges of an ideal square fundamental domain (yes I know a once-punctured torus is not closed, and I am bending along proper geodesics rather than closed ones, but this is easier to draw and gives the essential idea).

Although it seems hard to believe, the existence of a divisible but non-symmetric convex bounded projective domain was (apparently) unknown until Kac-Vinberg constructed examples in 1967.

5. Goldman’s coordinates

Suppose that $S$ is a closed surface with a convex projective structure. A maximal collection of $3g-3$ essential non-parallel simple closed curves can be realized by a family of disjoint geodesics, which decompose $S$ into $2g-2$ pairs of pants $P_i$. Each cuff of a pair of pants has three real eigenvectors, and it is determined up to conjugacy by two numbers: its trace, and the trace of the inverse.

The centralizer of a cuff is 2-dimensional (as explained above), so there are an additional two parameters for each geodesic explaining how adjacent pants are glued along each cuff. Finally, Goldman showed that there are two additional real parameters describing the geometry of each pair of pants (once the cuff parameters have been prescribed). Thus, after choosing a pair of pants decomposition, one determines a system of $16g-16$ real numbers which describe the structure up to isomorphism. In other words, the space of convex projective structures is homeomorphic to $\mathbf{R}^{16g-16}$.

Notice that the dimension of the space of convex projective structures on a pair of pants is easily seen to be 8, since this is just the dimension of the character variety: a pair of pants has fundamental group which is free on two generators, so the space of representations has twice the dimension of $\mathrm{SL}(3,\mathbf{R})$, i.e. 16, while the conjugation action cuts down this dimension by 8.

How to describe the parameters for a pair of pants geometrically? Thurston showed how to understand hyperbolic structures on surfaces with geodesic boundary by decomposing them into ideal triangles which can be “spun” around the boundary components (thus finessing the issue of where the ideal vertices should land). A similar construction makes sense for convex projective structures on surfaces with boundary. Goldman obtains his coordinates by understanding the way in which two projective triangles can be glued along their edges in pairs in such a way that the resulting (incomplete) structure on a pair of pants is convex. There does not seem to be a straightforward way to see that these conditions cut out a (topological) cell, fibering naturally over the space of cuff lengths.

6. Complex structure

Now let $X$ be a strictly convex domain in the projective plane (we have in mind that this is the image of the universal cover of our convex projective surface $S$ under the developing map). Put it in $\mathbf{R}^3$ as a convex subset of the horizontal plane $z=1$. Each point $x \in X$ determines a ray $r_x$ through the origin and passing through $x$, and the union of these rays sweeps out a (strictly convex) cone. We would like to construct, in a “natural” (i.e. projectively invariant) way, a surface $\Sigma$ intersecting each ray $r_x$ at a point $u^{-1}(x)\cdot x$ (so that we can think of $u$ as a function on the domain $X$ in the projective plane going to zero at the boundary). The surface $\Sigma$ will be strictly convex exactly when the hessian $\mathrm{Hess}(u)$ (i.e. the matrix of 2nd partial derivatives) is positive definite. Such a positive definite form determines a Riemannian metric, and thereby an area form on $\Sigma$, and we would like equal area regions to subtend equal volume cones to the origin. Since volume is preserved by $\mathrm{SL}(3,\mathbf{R})$, this is a projectively invariant notion. As a formula, this says that $u$ solves the following Monge-Ampère equation in $X$:

$\mathrm{det}(\mathrm{Hess}(u))=u^{-4}$

The existence and uniqueness of a (smooth) solution $u$ when $X$ is strictly convex was established by Cheng-Yau.

Let’s consider the special case where $X$ is the unit disk $x^2+y^2<1$. In this case we expect $\Sigma$ to be the hyperboloid $x^2+y^2-z^2=-1$ and the area form on $\Sigma$ should be the hyperbolic area. In this case we have an explicit formula $u(x,y) = (1-x^2-y^2)^{1/2}$. Thus $\partial u/\partial x = -x/u$ and $\partial u/\partial y = -y/u$, and we see that $u$ solves the Monge-Ampère equation. A similar calculation shows that $1/u\cdot \mathrm{Hess}(u)$ gives the hyperbolic metric on the unit disk (in the Klein model).

The surface $\Sigma$ with its Riemannian metric is invariant under projective symmetries, and gives rise to a canonical Riemannian metric on $S$ associated to the projective structure. The conformal class of this metric thus determines a map from the space of convex projective structures to the Teichmüller space of $S$.

The surface $\Sigma$ carries two natural connections — a flat affine connection $\nabla$ coming from the projection to $X$, whose straight lines are the intersection of $\Sigma$ with planes through the origin, and a Levi-Civita connection $\hat{\nabla}$ coming from the Riemannian metric defined as above. The difference of these two connections defines a cubic form on $\Sigma$, by the formula

$C(u,v,w) = \langle u,\nabla_v w - \hat{\nabla}_v w\rangle$

and it turns out that this cubic form is symmetric, and holomorphic with respect to the conformal structure associated to the metric on $\Sigma$ (for a longer discussion of cubic forms see this post). Thus, the space of convex projective structures on $S$ is isomorphic to the total space of the bundle of holomorphic cubic differentials over Teichmuller space!

As a sanity check, let’s verify that dimensions work out. Teichmuller space is a complex manifold of complex dimension $3g-3$. The Riemann-Roch formula says for any line bundle $L$ there is a formula

$h^0(L) - h^0(L^{-1}\otimes K) = \mathrm{deg}(L) + 1 - g$

where $K$ is the bundle of holomorphic 1-forms (which is the cotangent bundle on a Riemann surface). Now, $\mathrm{deg}(K)=-\chi(S)=2g-2$ so taking $L=K^3$ we get $h^0(K^3) = 5g-5$. Thus the space of convex projective structures has complex dimension $8g-8$, and real dimension $16g-16$.

Monge-Ampère equations arise in minimal surface theory, and one may think of this instance in a similar way. A convex real projective structure determines a holonomy representation of the fundamental group into $\mathrm{SL}(3,\mathbf{R})$, and one may look for a harmonic equivariant map from the universal cover to the symmetric space. A harmonic map is a minimal surface if it is conformal; thus an equivariant minimal surface in the symmetric space picks out a conformal class. Associated to such a minimal surface one obtains a holomorphic cubic differential, much as a suitable triple of holomorphic 1-forms determine a minimal surface in Euclidean 3-space by the Weierstrass parameterization.

This construction is due (independently) to Loftin and Labourie.

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