Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds
The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We provide a variation formula for the renormalized volume in terms of the extremal length of , and an upper bound on .
We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poincaré-Einstein metrics, the other (specifically for conformally flat structures) of the second fundamental form of a hypersurface in a “constant curvature” space with a degenerate metric, interpreted as the space of horospheres in hyperbolic space. This clarifies a relation between linear Weingarten surfaces in hyperbolic manifolds and Monge-Ampère equations.
- 1.1 The measured foliation at infinity
- 1.2 A variational formula for the renormalized volume
- 1.3 From the boundary of the convex core to the boundary at infinity
- 1.4 Surfaces associated to metrics at infinity
- 1.5 Geometric structures on a hyperbolic end
- 1.6 Main relations
- 1.7 Linear Weingarten surfaces
- 1.8 The Thurston metric at infinity
- 1.9 Content
- 2 Background material
- 3 The measured foliation at infinity
- 4 The second fundamental form at infinity and the space of horospheres
- 5 Poincaré-Einstein ends
- 6 Linear Weingarten surfaces and Monge-Ampère equations
- A Proof of Lemma 3.1
1.1. The measured foliation at infinity
Consider a quasifuchsian manifold homeomorphic to , where is a closed oriented surface of genus at least . We call the Teichmüller space of , the space of measured laminations on , and the space of holomorphic quadratic differential on , which can be considered as a bundle over with fibre over . We denote by the space of complex projective structures on , which can through the Schwarzian derivative be considered as an affine bundle over with fiber over (see §2.1).
We also denote by , etc, the corresponding notions but on rather than on . If is a quasifuchsian manifold homeomorphic to then is the disjoint union of two copies of , which we denote by and , one with the opposite orientation.
Recall that the boundary at infinity of , , can be identified with the quotient by the action of of the domain of discontinuity of :
Here is the holonomy representation of , and is its limit set.
Since acts on by complex projective transformations, is endowed with a -structure . Denote by the underlying complex structure, and by the complex projective structure obtained by applying to the Uniformization Theorem. The Schwarzian derivative of the holomorphic map isotopic to the identity between and is a holomorphic quadratic differential (see §2.4).
We will consider a naturally defined measured foliation at infinity on . In the point of view developed here, is an analog at infinity of the measured bending lamination on the boundary of the convex core of .
The foliation at infinity of , denoted by , is the horizontal foliation of the holomorphic quadratic differential of .
1.2. A variational formula for the renormalized volume
We consider here the renormalized volume of quasifuchsian hyperbolic manifolds, see §3.4. There is a simple variational formula for the renormalized volume, in terms of and of the variation of the conformal structure at infinity, Equation (16) below. Here we write this variational formula in another way, involving the measured foliation at infinity.
In a first-order variation of , we have
Here is the extremal length of , considered as a function over the Teichmüller space of the boundary . The right-hand side is the differential of this function, evaluated on the first-order variation of the complex structure on the boundary.
Equation (1) is remarkably similar to the dual Bonahon-Schläfli formula. The dual volume of the convex core of is defined as
where and are the induced metric and measured bending lamination on the boundary of the convex core of . The dual Bonahon-Schläfli formula is then:
1.3. From the boundary of the convex core to the boundary at infinity
Theorem 1.2, and its analogy to the dual Bonahon-Schläfli formula, suggests an analogy between the properties of quasifuchsian manifolds considered from the boundary of the convex core and from the boundary at infinity. For instance, on the boundary of the convex core, we have the following upper bound on the length of the bending lamination, see [6, Theorem 2.16].
Theorem 1.3 (Bridgeman, Brock, Bromberg).
Similarly, on the boundary at infinity, we have the following result, proved in §3.7.
|On the convex core||At infinity|
|Induced metric||Conformal structure at infinity|
|Thurston’s conjecture on prescribing||Bers’ Simultaneous Uniformization Theorem|
|Measured bending lamination||measured foliation|
|Hyperbolic length of for||Extremal length of for|
|Volume of the convex core||Renormalized volume|
|Dual Bonahon-Schläfli formula||Theorem 1.2|
|Bound on [8, 6]||Theorem 1.4|
|Brock’s upper bound on ||Upper bound on |
This analogy, briefly described in Table 1, suggests a number of questions (see §3.8) since it could be expected that, at least up to some point, phenomena known to hold on the boundary of the convex core might hold also on the boundary at infinity, and conversely.
Another series of questions arises from comparing the data on the boundary of the convex core to the corresponding data on the boundary at infinity. For instance, it is well known that is uniformly quasi-conformal to (see [16, 17]), and one can ask whether similar statements hold for other quantities. We do not expand on those questions here.
1.4. Surfaces associated to metrics at infinity
We now consider another point of view on the Schwarzian derivative at infinity.
Let be an open domain, and let be a Riemannian metric on compatible with the conformal structure of . We can associate to two distinct but related surfaces, each immersed in a 3-dimensional manifold.
One can associate to a smooth surface in a the space of horospheres of , see . This surface is dual (see 2.8) to the Epstein surface . The space of horospheres, denoted by below, has a degenerate metric but a rich geometric structure, and is equipped with an induced metric, , and a “second fundamental form”, . They satisfy the Codazzi equation, , and a modified form of the Gauss equation, . There is a natural embedding of in with image . The pull-back is equal to , while is a bilinear symmetric tensor field on naturally associated to .
The second geometric data, given by and , is perhaps less obvious than the first. However it is also quite natural and, as we will see below, it is an efficient tool in relating (1) to (3), (4) and (4’) below.
1.5. Geometric structures on a hyperbolic end
Consider now a hyperbolic end , for instance an end of a quasifuchsian or convex co-compact hyperbolic 3-manifold (the notion of hyperbolic end is recalled in 2.5). We are interested here in three geometric structures that occur quite naturally on the boundary at infinity of . They are related to (1) and (2) above when is the universal cover of the boundary at infinity and is invariant under the action of on .
Extending to hyperbolic ends the construction made in §1.1, is equiped with a complex structure , with a complex projective structure , and with a holomorphic quadratic differential , defined as the Schwarzian derivative of the holomorphic map isotopic to the identity between and , where is the Fuchsian complex projective structure associated to .
Given any metric in the conformal class at infinity of , there is a section if the bundle of bilinear symmetric forms on . In , and are defined in terms of equidistant foliations of a neighborhood of infinity in : given , there is a unique foliation such that the hyperbolic metric can be written, in a neighborhood of infinity, as
and are called the induced metric and second fundamental form at infinity of , since they satisfy the Codazzi equation, , and a modified version of the Gauss equation for surfaces in 3-dimensional space-forms: . and completely characterize .
The hyperbolic metric on can be written as
where is a one-parameter family of metrics on . Moreover can be written as
The metric is always in the conformal class on determined by the complex structure . Conversely, any such metric is obtained in a unique way. The bilinear form depends on and in a simple way (see 2.6), so the geometry of is encoded solely in and .
There are some well-known relations between the geometric structures above. First, (4) and (4’) are related in a particularly simple way. Given , it defines a unique equidistant foliation near infinity such that (3) and (4) hold. If both and are determined by the same equidistant foliation, they are related by:
The geometric quantities (1)–(4’) defined above extend, to various extents, in higher dimension. In particular:
(1) and (2) extend to higher dimensions, with , for .
(3) extends (in a way) to the situation where is replaced by any conformally flat metric, for instance a Riemannian metric in the conformal class at infinity of a hyperbolic end in dimension . The Schwarzian derivative is then replaced by the Schwarzian tensor, defined in 2.2.
(4) extends to hyperbolic ends in higher dimension.
(4’) extends to the setting where is replaced by an end of a Poincaré-Einstein manifold, as recalled in 2.6.
1.6. Main relations
We will show that the geometric structures (1)-(4’) above are strongly related, in particular when is hyperbolic. We also intend to clarify the notions of “induced metric” and “second fundamental forms” at infinity, denoted by and here, and the corresponding notions for surfaces in , denoted by and here. (The index is not present in  but is introduced here to limit ambiguities.) Those relations lead to a simple conformal transformation rule for , Theorem 1.9 and Corollary 1.10 below, which in turn provides a potentially useful relation between special surfaces in and Monge-Ampère equations on surfaces.
We now consider a hyperbolic end , along with a metric in the conformal class at infinity. This metric determines an equidistant foliation of near infinity by surfaces , which in turns determines a metric and a field of bilinear symmetric forms on .
The following can be found e.g. in [28, Lemma 8.3], but we will provide here a much simpler proof.
Suppose that is the hyperbolic metric in the conformal class on . Then the traceless part of is equal to .
The proof can be found in 4.2.
Together with Proposition 1.5, we obtain the following direct consequence.
Suppose again that is the hyperbolic metric at infinity of . Then , while .
The relation between the data at infinity and the description by the induced metric and second fundamental form of the dual surface in is quite simple.
, while .
The proof is in 4.1.
A key tool in the paper is a simple variational formula for under a conformal deformation of . We state here directly in the setting of Poincaré-Einstein manifolds. Here is the Schwarzian tensor of Osgood and Stowe , a generalization of the Schwarzian derivative recalled in 2.2.
Let be a -dimensional Poincaré-Einstein manifold, , and let and be two metrics in the conformal class at infinity on . Let and be the one-parameter families of metrics on determined by (see above) and let and be the second terms in the asymptotic developments of and . Then:
As a consequence, the traceless part of and are related by:
The proof can be found in §5. For it is a direct consequence of an explicit relation between and , while for it uses the relation with surfaces in the space of horospheres.
As a consequence, we can describe or when is any metric in the conformal class at infinity of . We will see some interesting examples below. To simplify notations, we use the following notation. If is a Riemannian metric on a surface and is a smooth function, then
Note that is a kind of non trace-free version of the Schwarzian tensor, and is the traceless part of .
Suppose that , where is a metric in the conformal class at infinity of . Then , while
Those relations extend without change to conformally flat metrics in higher dimension, we do not elaborate on this point here.
1.7. Linear Weingarten surfaces
We consider linear Weingarten surfaces in , or in hyperbolic 3-manifolds, defined as a smooth surface satisfying an equation of the form
where are constants. Here is the mean curvature of , where is its shape operator, while is its extrinsic curvature, related to the Gauss curvature by the Gauss equation, .
We are particularly interested in some well-behaved surfaces that play a particular role in some situations, in particular when studying quasifuchsian 3-manifolds. We will say that a smooth hypersurface is horospherically tame, or h-tame for short, if its principal curvatures are everywhere in . Note that the hyperbolic Gauss map of a complete h-tame surface in is injective, so that it defines a data at infinity on an open domain , see 2.7.
If an oriented hypersurface is h-tame then the corresponding second fundamental form at infinity is positive definite. Any admissible pair in an open subset with positive definite determines a smooth h-tame surface.
The definition of an “admissible pair” is given in §2.5.
Let be a h-tame surface in , and let be the corresponding data at infinity. Suppose that . Then satisfies (7) if and only if the data at infinity satisfies the relation
Here is the “shape operator at infinity”, the unique bundle morphism self-adjoint for such that , and is the identity.
Note that the case when (or , after changing the orientation) corresponds to the case treated e.g. in .
Together with (5) this leads to the following characterization of linear Weingarten surfaces in terms of solutions of Monge-Ampère equations.
Let be an admissible pair defined on an open domain , and let . The surface defined by the metric at infinity is h-tame and satisfies (7) if and only if:
is positive definite,
satisfies the Monge-Ampère equation
Equation (9) can be written as
where is the gradient of for and is the Hessian of considered as a 1-form with values in the tangent space of .
The behavior of those linear Weingarten surfaces will likely be simpler if the following two conditions are satisfied:
, since (10) is then of elliptic type,
We now outline three interesting special cases.
Here we can take (this corresponds to changing the orientation of the surface). Then , so Proposition 1.13 cannot directly be used. However following the same computations as in 6.2 shows that (9) becomes simply
Convex constant Gauss curvature surfaces
This is the case of surfaces of constant curvature . We can then take . Then , , and .
1.8. The Thurston metric at infinity
We outline here another special case of the relations described above, that was also a motivation for writing those notes. It is based on the work of Dumas .
Let be a hyperbolic end and let be the convex pleated surface which is the non-ideal boundary of . The data at infinity corresponding to is quite interesting: is the “Thurston metric” on associated to the pleated surface , while has rank at most at each point, and determines a measured lamination on . It is zero on the subset of points projecting to the totally geodesic part of the boundary of the convex core, and, on regions projecting to the support of the measured bending lamination, it is zero in directions of the lamination.
We can also consider on the hyperbolic metric , and the corresponding second fundamental form at infinity . Then
where is the solution of the equation
where is the curvature of the Thurston metric , which takes values in . Moreover,
Taking the trace-free part of this relation leads precisely to [12, Theorem 7.1].
Section 2 contains background material used in the rest of the paper. Section 3 then contains details on the measured foliation at infinity and the proof of Theorems 1.2 and 1.4. Section 4 focuses on hypersurfaces in the space of horospheres and contains the proofs of Theorems 1.6 and 1.8. Finally, Section 5 presents the proof of Theorem 1.9, and Section 6 gives some details on the application to linear Weingarten surfaces.
I am grateful to Sergiu Moroianu for helpful remarks.
2. Background material
2.1. The Schwarzian derivative
Let , and let be holomorphic. The Schwarzian derivative of is a meromorphic quadratic differential defined as
It has two remarkable properties.
if and only if is a Möbius transformation,
As a consequence of those two properties, the Schwarzian derivative is defined for any holomorphic map from a surface equiped with a complex projective structure to another (see next section). It is a meromorphic quadratic differential on the domain, holomorphic if everywhere.
2.2. The Schwarzian tensor
Osgood and Stowe  generalized the Schwarzian derivative to the notion of Schwarzian tensor, associated to a conformal map between two Riemannian manifolds of the same dimension.
Let be a Riemannian -dimensional manifold, and let . The Schwarzian tensor associated to the metrics and on is defined as
where the index denotes the traceless part with respect to .
The Schwarzian tensor is a natural generalization of the Schwarzian derivative in the sense that if and is a holomorphic map, then
The Schwarzian tensor also shares some key properties of the Schwarzian derivative.
It behaves well under compositions of conformal maps: if is a Riemannian manifold and are smooth functions, then
It behaves well under diffeomorphism: if is a diffeomorphism and and are two conformal metrics on , then
If is the hyperbolic metric on the ball given by the Poincaré model, and is the Euclidean metric on , then .
Similarly, if is the spherical metric on given as the push-forward of the spherical metric by the stereographic projection, then .
Here we give two simple interpretations of the Schwarzian tensor:
as the difference between the second terms in the asymptotic development of metrics on Poincaré-Einstein manifolds, when one conformally varies the first term,
as the variation in second fundamental forms of certain hypersurface in a -dimensional space associated to conformal (and conformally flat) metrics.
The second interpreation is related to the interpretation in , but more in the setting of isometric embeddings in the space of horospheres as developed in . The hypersurfaces that appear are dual, in a sense that will be made precise below, to the Epstein hypersurfaces of the metrics.
2.3. Complex projective structures
We will need to consider complex projective structures on closed surfaces. Recall that a complex projective structure (also called -structure) is a -structure (see [39, 21]), where and . In other terms, they are defined by atlases with values in , with change of coordinates in . We denote by the space of -structures on .
Given a complex structure on , there is by the Riemann Uniformization Theorem a unique hyperbolic metric on compatible with . This hyperbolic metric determines a complex projective structure on , because the hyperbolic plane can be identified with a disk in , on which hyperbolic isometries act by elements of fixing the boundary circle. We denote this complex projective structure by , and call it the Fuchsian complex projective structure of .
Let , and let be the underlying complex structure. There is a unique map holomorphic for the underlying complex structure. Let be its Schwarzian derivative. This construction defines a map , sending to , with the holomorphic quadratic differential considered as a cotangent vector to at . The map is known to be a homeomorphism .
2.4. The holomorphic quadratic differential at infinity of quasifuchsian manifolds
We now consider a quasifuchsian manifold homeomorphic to , where is a closed surface of genus at least . We note its boundary at infinity by , so that is the disjoint union of two surfaces , each homeomorphic to .
The boundary at infinity is the quotient of the complement of the limit set of by the action of on . Since hyperbolic isometries act on by complex projective transformations, is equiped with a complex projective structure, that we denote by . We denote by the complex structure underlying .
We denote by the “holomorphic quadratic differential at infinity” of .
Therefore, can be considered as minus the “difference” between the quasifuchsian complex projective structure on and the Fuchsian complex projective structure obtained by applying the Riemann uniformization theorem to the complex structure .
2.5. Hyperbolic ends
A hyperbolic end as considered here is non-complete hyperbolic manifold, diffeomorphic to , where is a closed surface of genus at least , which is complete on the side corresponding to but has a metric completion obtained by adding a concave pleated surface on the boundary corresponding to . A typical example is a connected component of the complement of the convex core in a convex co-compact hyperbolic manifold.
Given a hyperbolic end , we denote by its ideal boundary, corresponding to in the identification of with , and by the concave pleated surface which is the boundary of its metric completion corresponding to . We will denote by the space of hyperbolic ends homeorphic to , considered up to isotopy.
Let be a hyperbolic end. Its ideal boundary is equipped with a -structure . This is clear in the simpler case when the developing map of is injective, since in this case is the quotient of a domain in (identified with ) by an action of by elements of . In the general case this picture has to be slightly generalized, and is a quotient of a simply connected surface which has a (not necessarily injective) projection to , see [38, 39].
In fact, this map from to is one-to-one, and a hyperbolic end can be constructed for any -structure on , see .
Given a metric in the conformal class at infinity of , it defines an equidistant foliation of a neighborhood of infinity in in the following way. Given a real number , consider the Epstein surface of . Then for large enough, is smooth and embedded, and even locally convex. Moreover, if are large enough, then and are at fixed distance . The hyperbolic metric then has the asymptotic expansion (2), and the term does not change if one replaces with .
In addition, and satisfy two relations (see [28, 5]): is Codazzi for , that is, , where is the Levi-Civita connection of , and , where is the curvature of . We will say that is an admissible pair if it satisfies those two equations.
2.6. Poincaré-Einstein manifolds
A Poincaré-Einstein manifold is a complete Riemannian manifold such that the Riemannian metric is Einstein and can be written as
where is a smooth metric on a compact manifold with boundary , is the interior of , and on , see .
Poincaré-Einstein manifolds have a well-defined boundary at infinity , identified with , endowed with a conformal class of metrics, defined as the conformal class of the restriction of to . In the neighborhood of each connected component of the boundary at infinity, one can write , where is a one-parameter family of metrics on .
When is even, has the asymptotic expansion
where are one-parameter families of tensors on depending smoothly on . The tensor has a Taylor expansion at given by
where are formally determined by if and formally determined by the pair for ; for , the tensors have a Taylor expansion at formally determined by and .
When is odd, has the simpler asymptotic expansion
where . All terms for are formally determined by , while the traceless part of is “free”. All other terms in the asymptotic development are determined by and by the traceless part of .
Let be an oriented hypersurface. We will denote by its induced metric (classically called its “first fundamental form”).
Let be the oriented unit normal vector field to . The shape operator of is the bundle morphism defined as follows:
where is the Levi-Civita connection on . Then is self-adjoint with respect to .
The second fundamental form of is defined as
and its third fundamental form as
The hyperbolic Gauss map of is the map sending a point to the endpoint of the geodesic ray starting from in the direction of the oriented normal .
We say that is horospherically tame, or h-tame, if its principal curvatures are everywhere in .
If is complete and h-tame, than its hyperbolic Gauss map is a diffeomorphism between and a connected component of .
Suppose that the hyperbolic Gauss map of is injective. The data at infinity associated to is the pair defined on by
The shape operator at infinity is the bundle morphism which is self-adjoint for and such that
One can then prove (see ) that is an admissible pair, as defined above.
2.8. The space of horospheres
We denote by the space , with the degenerate metric , where . The notation comes from the fact that it can be identified with the future light cone of a point in the 4-dimensional de Sitter space. Equivalently, it can be identified with the space of horospheres in the 3-dimensional hyperbolic space with the natural metric defined in terms of intersection angles, see [33, §2],
This space has a number of features that are strongly reminiscent of a 3-dimensional space of constant curvature.
acts by isometries on . The simplest way to see this is by considering as the space of horospheres in .
There is a notion of “totally geodesic planes”, which are the 2-dimensional spheres with induced metric isometric to the round metric on . Those planes can be identified with the set of horospheres going through a given point in , and the action of on those totally geodesic planes is transitive.
There is a 2-dimensional space of totally geodesic planes going through each point in .
In addition, there is at each point a distinguished “vertical” direction, corresponding to the kernel of the metric. Moreover the integral lines of those vertical directions have a canonical affine structure.
Although the metric is degenerate, it is possible to define an analog of the Levi-Civita connection at a point . However it depends on the choice of a non-degenerate plane , and is defined for vector fields tangent to (see the last paragraph of [33, §2]).
Using this connection, one can define a notion of second fundamental form of a surface which is nowhere vertical (see [33, §5]). Using the induced metric , one can then define the shape operator as the self-adjoint operator such that . It satisfies two equations (see [33, §6]):
the Codazzi equation , where is the Levi-Civita connection of on ,
a modified form of the Gauss equation: the curvature of is equal to .
3. The measured foliation at infinity
3.1. The Fischer-Tromba metric
Let be a hyperbolic metric on . The tangent space can be identified with the space of symmetric 2-tensors on that are traceless and satisfy the Codazzi equation for . (In other terms, the real parts of holomorphic quadratic differentials in .) We call the space of those traceless Codazzi symmetric 2-tensors for .
Let be two such tensors and let be the corresponding vectors in . Then the Weil-Petersson metric between and can be expressed as
The right-hand side of this equation is sometimes called the Fischer-Tromba metric on .
We can also relate the scalar product on symmetric 2-tensors to the natural bracket between holomorphic quadratic differentials and Beltrami differentials as follows.
Let be a closed Riemann surface, and let be the hyperbolic metric compatible with its complex structure. Let be a first-order deformation of , and let be the corresponding Beltrami differential. Then for any holomorphic quadratic differential on ,
The proof is in Appendix A.
3.2. The energy of harmonic maps and the Gardiner formula
Let , and let be its dual real tree. For each , there is a unique equivariant harmonic map from to . Let be its energy, and let be its Hopf differential. Then
Here is considered as a Beltrami differential, and is the duality product between Beltrami differentials and holomorphic quadratic differentials. (See e.g. [43, Theorem 1.2].)
We use below the same notations, but with replaced by .
3.3. Extremal lengths of measured foliations
Let be a measured foliation on and, for given , let be the holomorphic quadratic differential on with horizontal foliation .
The extremal length of at is the integral over of ,
A more classical definition can be given in terms of modulus of immersed annuli, see .
Theorem 3.3 ().
3.4. The renormalized volume of quasifuchsian manifolds
The renormalized volume of quasifuchsian manifolds is closely related to the Liouville functional in complex analysis, see [36, 35, 37, 27]. However it can also be considered as a special case, in dimension 3, of the renormalized volume of conformally compact Einstein manifolds as seen in §5, see [26, 23, 22].
A definition of the renormalized volume of quasifuchsian manifolds can be found in [28, Def 8.1]. It satisfies a simple variational formula, which can be written as
where is the traceless part of the “second fundamental form at infinity” which, together with the metric at infinity , completely characterizes a hyperbolic end.
3.5. The measured foliation at infinity
We now introduce a measured foliation at infinity, which can be thought of as an analog at infinity of the measured bending lamination on the boundary of the convex core.
The measured foliation at infinity is the horizontal measured foliation of . We denote it by .
It follows from Theorem 3.3 that .
Let , and let . Then is the horizontal measured foliation of the quasifuchsian hyperbolic metric determined by if and only if the function defined as
is critical at .
Suppose first that is the horizontal measured foliation of , the holomorphic quadratic differential at infinity of the quasifuchsian manifold .
But we have seen that , and it follows that .
Conversely, if , the same argument as above shows that , so that is the horizontal measured foliation of . ∎