Harmonic maps and wild Teichmüller spaces
We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has non-compact ends with boundary cusps. This extends Wolf’s parametrization of the Teichmüller space of a closed surface using holomorphic quadratic differentials. Our proof involves showing the existence of a harmonic map from a punctured Riemann surface to a crowned hyperbolic surface, with prescribed principal parts of its Hopf differential which determine the geometry of the map near the punctures.
The Teichmüller space of a closed surface of genus is the space of marked conformal or hyperbolic structures on , and admits parametrizations that reflect various aspects (metric, complex-analytic, symplectic) of its geometry. Many of these naturally extend to the case when the surface has boundaries and punctures, in which case the uniformizing hyperbolic metrics are such that the boundaries are totally geodesic, and the punctures are cusps.
This paper deals with the case when each boundary component has an additional “decoration”, namely, finitely many distinguished points. These arise naturally in various contexts involving the arc complex (see for example [KP06]), and such a surface can be uniformized to a non-compact hyperbolic surface with crown ends (Definition 2.7) where the distinguished boundary points become “boundary cusps” (see [Che07]).
The associated Teichmüller spaces, and their symplectic and algebraic structures have been studied before (see [Che07], [Huab], [Pen04]).
In this paper, we provide a parametrization of the Teichmüller space of such crowned hyperbolic surfaces using meromorphic quadratic differentials with higher order poles.
Holomorphic quadratic differentials and their geometry play a crucial role in Teichmüller theory, and our result provides a link with this analytical aspect of the subject.
In the case of a closed surface, the work of M. Wolf ([Wol89]) parametrized Teichmüller space by the vector space of holomorphic quadratic differentials on any fixed Riemann surface .
Namely, Wolf proved that we have a homeomorphism
that assigns to a hyperbolic surface the Hopf differential of the unique harmonic diffeomorphism that preserves the marking.
This turns out to be equivalent to Hitchin’s parametrization of in [Hit87] using self-dual connections on (see [Nag91]); both these approaches rely on a fundamental existence results for certain non-linear PDE (see §2.3 and Theorem 2.19) that crucially depend on the compactness of the surfaces.
In this paper, we shall establish what can be described as a “wild” analogue of the above correspondence - see [Sab13] for the far more general context of what is called the “wild Kobayashi-Hitchin correspondence”.
That is, shall be a Riemann surface with punctures, and the target for the harmonic maps would be the non-compact crowned hyperbolic surfaces introduced earlier. It turns out that the Hopf differential of such a map has poles of higher orders (greater than two) at the punctures; if the order of the pole is , the number of boundary cusps of the corresponding crown end is . Further, the analytic residue of the differential at the pole determines the “metric residue” of the crown (see Definition 2.9). This residue is part of the data of a principal part that is a meromorphic -form comprising the negative powers of in a Laurent expansion with respect to a choice of coordinate chart around the pole. (See §2 for details.)
Our key theorem is an existence result for such harmonic maps:
Theorem 1.1 (Existence).
Let be a closed Riemann surface with a non-empty set of marked points with fixed coordinate disks around each. For let , and let be a crowned hyperbolic surface having crowns with vertices, together with a homeomorphism (or a marking) .
Then there exists a harmonic diffeomorphism
that is homotopic to , taking a neighborhood of each to .
Moreover, such a map is unique if one prescribes, in addition, the principal parts at the poles, having residues compatible with the metric residues of the corresponding crowns.
(See Definition 2.30 for the notion of compatible residues.)
As a consequence, we establish the following extension of the Wolf-Hitchin parametrization:
Theorem 1.2 (Parametrization).
Let be a closed Riemann surface with a set of marked points with coordinate disks around them as above. For a collection of principal parts having poles of orders for , let
be the space of meromorphic quadratic differentials on with principal part at , and
be the space of marked crowned hyperbolic surfaces homeomorphic to , with crowns, each having boundary cusps, with metric residues compatible with the residues of the principal parts .
Then we have a homeomorphism
that assigns, to any marked crowned hyperbolic surface , the Hopf differential of the unique harmonic map from to with prescribed principal parts.
Remark. Throughout this article, we shall assume that the punctured surface has negative Euler characteristic, though this is not strictly necessary. Indeed, we shall discuss the case of ideal polygons (genus zero and one puncture) in §3.3.
We briefly recount some previous related work:
In the case of meromorphic quadratic differentials with poles of order one, the appropriate Teichmüller space is that of cusped hyperbolic surfaces; the analogue of Wolf’s parametrization was established in [Loh91]; see also §8.1 of the recent work [BGPR].
When the hyperbolic surface in the target has geodesic boundary, the analogue of the existence result (Theorem 1.1) is implicit in the work of Wolf in [Wol91b]. There, he proves the existence of harmonic maps from a noded Riemann surface to hyperbolic surfaces obtained by “opening” the node; the Hopf differential of such a map has a pole of order two at either side of the node.
The existence of harmonic maps from a punctured Riemann surface to ideal polygonal regions in the Poincaré disk was recently shown by A. Huang in [Huaa]. This was the first major advance since the work [AW05], [ATW02], [HTTW95], [TW94] which dealt with the case when the domain surface was the complex plane , and the Hopf differentials were polynomial quadratic differentials (see Theorem 2.26). We point out a description of the subspace of polynomial differentials that correspond to a fixed ideal polygonal image - see Proposition 3.12.
Furthermore, the work in [GWa] established a generalization of the Hubbard-Masur Theorem by proving the existence of harmonic maps to leaf-spaces of measured foliations with pole singularities; as in Theorem 1.1, such a harmonic map is unique if one further prescribes a principal part that is “compatible” with the foliation. In analogy with the case of a compact surface in the work of Wolf, we expect that that such foliations form the analogue of the Thurston compactification for the “wild” Teichmüller space that we describe in this paper, and our main result in [GWa] is a “limiting” case of the results in this paper. We hope to pursue this in later work.
Another key inspiration for the present work is the seminal work of Sabbah ([Sab99]) and Biquard-Boalch ([BB04]) who prove the existence of the corresponding harmonic metrics for self-dual -connections with irregular singularities, that lead to a “wild” Kobayashi-Hitchin correspondence for that complex Lie-group.
The work in this paper pertains, instead, to the “wild” theory for the real Lie group ; implicit in this paper is a geometric understanding of the associated equivariant harmonic maps to the symmetric space, which in this case is the Poincaré disk.
It should be possible to generalize the methods in this paper to the case of other real Lie groups of higher rank (for the case of tame singularities see the recent work of [BGPR]); and in particular explore the structure of the spaces of “wild” geometric structures whose monodromies would lie in the corresponding wild character variety. Our hope is that establishing an analytic parametrization of such spaces will shed light on compactifications of the usual “tame” spaces.
Strategy of the proof. Existence results for harmonic maps to non-compact targets is, in general, difficult as one needs to prevent the escape of energy out an end. Moreover, the traditional methods of existence results include heat-flow or sub-convergence of a minimizing sequence which crucially use finite energy (see, for example, [LT91]). The key difficulty in proving the existence result in Theorem 1.1 is the fact that the desired harmonic map has infinite energy as its Hopf differential has higher order poles.
We start by proving an existence result for harmonic maps from a punctured disk to hyperbolic crowns (see Theorem 3.2) having Hopf differentials with prescribed principle parts. This is crucial to the subsequent argument as these maps serve as models of the asymptotic behavior of the desired map on the punctured Riemann surface. Our proof of this Asymptotic Models Theorem relies heavily on the technical results for the work concerning harmonic maps from to ideal polygons that we mentioned above; in particular, on estimates on the geometry of the harmonic map that is inspired by the work of Wolf ([Wol91b]) and Minsky ([Min92]).
The strategy of the proof of Theorem 1.1 then is to consider an energy minimizing sequence defined on a compact exhaustion of the punctured surface , restricting to a desired model map around the punctures. An important step to ensure convergence is to provide a uniform energy estimate. Our previous work in [GWa], required a “symmetry” of the model map, that was a crucial to obtain a spectral decay of harmonic functions along a cylinder. Here, a new argument using ideas of A. Huang (see the Doubling Lemma in §4.2), dispenses with the need for this symmetry.
Moreover, our previous work involved equivariant harmonic maps to certain augmented -trees, here the target is two-dimensional; this necessitates various modifications.
Acknowledgments. The idea of the project arose from a conversation with Yi Huang at a conference at the Chern Institute of Mathematics at Tianjin, and I thank him for his interest, and the conference organizers for their invitation. It is a pleasure to thank Jérémy Toulisse for his comments on a previous version of this article, Andrew Huang for conversations, and Michael Wolf, as some of these ideas emerged from a collaboration with him on a related project. I also acknowledge the support by the center of excellence grant ’Center for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95) - this work was completed during a stay at QGM at Aarhus, and I am grateful for their hospitality. The visit was supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Union Framework Programme (FP7/2007-2013) under grant agreement no. 612534, project MODULI - Indo European Collaboration on Moduli Spaces.
2.1. Principal part
We begin by defining the relevant space of quadratic differentials on the punctured disk that we shall use later:
Let and let be the space of meromorphic quadratic differentials on . Any such differential has a representative of the form:
where the coefficients are complex numbers.
The fundamental complex invariant one can associate with such a differential is:
Definition 2.2 (Analytic Residue).
The residue of a meromorphic differential (-form) at a pole is the integral
where is a loop around . (Note that this is independent of the choice of such a loop). We shall ignore the ambiguity of sign.
For a meromorphic quadratic differential , the (quadratic) residue shall be the residue of a choice of square root , that can be defined locally around the pole .
However in this paper we shall need the finer information of a principal part, that we shall soon define.
Definition 2.3 (Principal differential).
A principal (quadratic) differential on of order is defined to be a holomorphic quadratic differential of the form
on where and if is even and if is odd. The space of such principal differentials shall be
via the map to the coefficients .
Given a meromorphic quadratic differential on with a pole of order at the origin, there is a unique principal differential such that is a holomorphic quadratic differential on .
Suppose the quadratic differential is of the form (1) up to an addition of a holomorphic quadratic differential on .
Then on completing the square in the expression (2) and comparing coefficients, we have: , , , and in general
for each . Inductively, determines , determines , and determines . Thus the map
is well-defined and injective, and thus determines such that the lemma holds. ∎
Remark. For a as in the above Lemma, we shall sometimes say:
although this only makes sense formally (as it depends on a suitable choice of a branch of the square root).
Using this lemma, we can finally define:
Definition 2.5 (Principal part).
Given a meromorphic quadratic differential on with a pole of order at , and a choice of coordinate disk around the point , we define its principal part (relative to the choice of ) to be the unique element of such that is a holomorphic quadratic differential on .
Recalling the definition of the space of quadratic differentials (Definition 2.1), we have:
Let and fix a principal differential . Then the subspace of the quadratic differentials in with principal part is homeomorphic to if is odd, and if is even.
As a consequence of the computations in the proof of Lemma 2.4, we have is uniquely determined by the coefficients where . The remaining coefficients are a set of complex numbers, which contributes real parameters if is even, and real parameters if is odd. ∎
2.2. Hyperbolic crowns
Throughout this paper, will be the Poincaré disk, or equivalently, the hyperbolic plane with the hyperbolic metric . The hyperbolic distance function will be denoted by .
Definition 2.7 (Crown, polygonal end).
A crown with “boundary cusps” is an incomplete hyperbolic surface bounded by a closed geodesic boundary , and a crown end comprising bi-infinite geodesics arranged in a cyclic order, such that the right half-line of the geodesic is asymptotic to the left half-line of geodesic , where . A crown comes equipped with a labelling of the boundary cusps inbetween adjacent geodesic lines; the labels are in a cyclic order.
A polygonal end of a crown is the -invariant bi-infinite chain of geodesic lines in obtained by lifting the cyclically ordered collection of geodesics in to its universal cover, where is the group generated by the hyperbolic translation corresponding to the geodesic boundary . A post-composition by an isometry ensures that the lift of is the bi-infinite geodesic with endpoints at , and that the cusp labelled “1” is at ; this normalization specified the polygonal end associated with the crown uniquely.
Remark. The ideal endpoints of the chain of geodesics of the polygonal end limit to two points ; the axis of the lift of is between these two limit points. Thus, a polygonal end together with the data of the hyperbolic translation that it is invariant under, determines the crown by taking a quotient.
See Figure 1.
The following associated notions will be useful later:
Definition 2.8 (Truncation).
A truncation of a crown with boundary cusps is obtained by removing a choice of disjoint neighborhoods at each ideal vertex of a , to obtain a subset that is convex in the hyperbolic metric. The height of the truncation of a cusp is the distance of the boundary of the deleted neighborhood, from the boundary of a maximal neighborhood of the cusp.
Remark. The convexity of the truncated crown would be useful later (see the remark following Theorem 2.19). This means we need to truncate along geodesic arcs across the cusps, rather than horocycles.
Definition 2.9 (Metric residue).
The metric residue of the hyperbolic crown with boundary cusps is defined to be when is odd, and equal to the alternating sum of lengths of geodesic sides of a truncation, when is even; like the analytic residue (see Den. 2.2), we ignore the ambiguity of sign.
It is easy to see that:
For even the metric residue is well-defined, that is, it is independent of the choice of truncation.
Truncating at a different height changes the length of the geodesic sides associated to that truncated sum by the same amount. However, in the alternating sum, the lengths of these sides appear with opposite signs, and hence the metric residue remains unchanged. ∎
The final notion we need for our crowns is:
Definition 2.11 (Boundary twist).
The hyperbolic crowns we shall consider shall come with an additional real parameter, the boundary twist, that we associate with the geodesic boundary. In the corresponding polygonal end in the universal cover, this can be thought of as the choice of a marked point on the bi-infinite line that is the lift of the geodesic boundary; the parameter is then the signed distance of this point, from the foot of the orthogonal arc from the cusp labelled “1” to .
We can now define the relevant spaces of geometric objects:
Definition 2.12 (Spaces of crowns).
For , the space of hyperbolic crowns with labelled boundary cusps and a boundary twist is denoted by . For any , the subspace will comprise crowns with the specified metric residue .
Lemma 2.13 (Parametrizing the spaces).
For , the space of polygonal ends , and the space of those having metric residue equal to , is homeomorphic to .
Consider a crown with boundary cusps. As described earlier, its universal cover can be thought of as the region in bounded between a geodesic line and a chain of geodesics invariant under the hyperbolic translation . One can normalize such that is along an axis with end points , and one of the lifts of a boundary cusp has an ideal vertex at . A fundamental domain of the -action is then uniquely determined by the points on the ideal boundary that are the endpoints of the remaining boundary cusps. These real parameters thus determine the hyperbolic crown with labelled cusps. The final real parameter is the choice of the basepoint which is anywhere along the axis , which is the boundary twist.
For a fixed residue , in the universal cover, a fundamental domain is in fact determined by the positions of ideal vertices (where one is already fixed at by the usual normalization).This is because the remaining ideal vertex is then uniquely determined from the metric residue, by the following hyperbolic geometry fact: If you fix two disjoint horodisks and , and let , then there is a unique choice of an ideal point such that any horodisk based at satisfies . We leave the verification of this to the reader. ∎
2.3. Crowned hyperbolic surface
Throughout this paper, let be a compact orientable surface of genus and boundary components.
A crowned hyperbolic surface is obtained by attaching crowns to a compact hyperbolic surface with geodesic boundaries by isometries along their closed boundaries. This results in an incomplete hyperbolic metric of finite area on the surface. Topologically, the underlying surface is , where is a set of finitely many points on each boundary component.
Remark. On doubling a crowned hyperbolic surface, one obtains a cusped hyperbolic surface with an involutive symmetry. A compact Riemann surface with finitely many punctures admits a unique uniformizing metric that is hyperbolic and has cusps at the punctures. Thus a crowned hyperbolic surface can be thought of as the unique uniformizing metric for a compact Riemann surface with boundary, having distinguished points on each boundary component.
Definition 2.15 (Teichmüller space of crowned hyperbolic surfaces).
such that the -th crown has boundary cusps, for each .
Here, the marking is a homeomorphism that takes each boundary component to the crown ends. Two marked surfaces are equivalent, that is, if there is an isometry that is homotopic to via a homotopy that keeps each boundary component fixed, and is a homeomorphism that does not Dehn-twist around any crown end.
We then have the following parametrization:
Let and fix integers . The Teichmüller space of crowned hyperbolic surfaces is homeomorphic to where .
It is well-known that a hyperbolic surface with geodesic boundary components is determined by real parameters. This can be seen, for example, by considering Fenchel-Nielsen parameters on the bordered surface.
By Definition 2.14, a crowned hyperbolic surface is obtained by an isometric identification of a bordered hyperbolic surface and a collection of crowns along their geodesic boundaries. (See Figure 2.)
By Lemma 2.13 each hyperbolic crown is determined by real parameters. The additional “boundary twist” parameter determines the “twist” in the identification of the crown boundary with that of the bordered hyperbolic surface. However, the length of the geodesic boundary must match with that of the bordered hyperbolic surface to achieve the isometric identification.
Thus, each crown adds real parameters to the crowned hyperbolic surface, and we obtain a total of parameters, as claimed. ∎
Remark. Our Definition 2.15 differs from that in Y.Huang [Huab] or Chekhov-Mazzocco [CM] in that the parameter spaces they define are “decorated”, and in particular also record the data of a choice of truncation for each boundary cusp; this results in an additional real parameters in the notation above. In any case, the above parameterization can also be derived from their work.
Finally, we note the following parametrization of marked crowned hyperbolic surfaces when we fix the metric residues for crowns with an even number of boundary cusps. The proof follows from that of Lemma 2.13, and we leave the details to the reader:
Let be the subset of indices such that is even for . Fix an ordered tuple of real numbers . Let be the subspace of comprising marked hyperbolic surfaces with the crown end having metric residue , for each . Then where .
2.4. Harmonic maps between surfaces
In this section we shall recall some basic facts about harmonic maps with negatively curved targets.
Let and be Riemann surfaces with a conformal metrics.
Throughout this paper, shall be a hyperbolic metric, that is, has constant negative curvature .
Definition 2.18 (Harmonic map).
A harmonic map
is a critical point of the energy functional
which is defined on all maps from to with locally square-integrable derivatives. Here,
is the energy density of , where
Note that the energy depends only on the conformal class of the metric on the domain ; in what follows we shall often drop specifying the choice of such a metric.
The corresponding Euler-Lagrange equation that satisfies is:
where and are the local coordinates on and respectively.
For compact surfaces, we have the following fundamental existence result:
Theorem 2.19 (Eells-Sampson, Hartman, Al’ber, Sampson, Schoen-Yau).
Let and be compact Riemann surfaces with conformal metrics such that is negatively curved. Then there exists a unique harmonic diffeomorphism in the homotopy class of any diffeomorphism, that is a minimizer of the energy functional (3).
Remark. The work of Hamilton and Schoen-Yau also extends this to the case when the surfaces have boundary; namely if the boundary of is convex (alternatively, having non-negative geodesic curvature), there exists a harmonic map in the homotopy class as above, with any prescribed boundary map. (See Theorem 4.1 of [SY78], and also pg. 157-8 of [Ham75].)
Prescribing the Hopf differential
The Hopf differential of such a map is the quadratic differential given by the local expression
and it is well-known that it is holomorphic if and only if is harmonic (see for example, [Sam78]).
The existence and uniqueness theorems for harmonic maps with prescribed Hopf differentials have been proven in much more general settings ( see [TW94], [HTTW95], [WA94]). The statement we shall need, are given below:
Theorem 2.21 (Theorem 3.2 and Proposition 4.6 of [Tw94]).
Let be any holomorphic quadratic differential on . Then there is a harmonic map
that is a diffeomorphism to its image, and has Hopf differential .
Moreover, if and are two such orientation-preserving harmonic diffeomorphisms, then is an isometry from to . In fact, for an isometry .
In fact, as we shall see in the next subsection, one can deduce more about the image in terms of the quadratic differential .
2.5. Geometric estimates
The estimates on image of the harmonic map are in terms of the metric induced by the Hopf differential defined below. The results in this section have been culled from the work of [Han96], [Min92], [Wol91a], [AW05], [HTTW95], [ATW02].
Definition 2.22 (-metric).
The metric induced by a quadratic differential defined on a Riemann surface (also referred to as the -metric) is a conformal metric given by the local expression , which is singular at the zeroes of the quadratic differential . The holomorphicity of then implies that the curvature vanishes away from these singularities, and hence the metric is a singular flat metric.
We also have the following notion:
In any local chart there is a change of coordinates such that the quadratic differential transforms to . The “horizontal” (resp. “vertical”) direction is the -(resp. -) direction in these canonical charts. More intrinsically, they can be defined to be the directions in which the quadratic differential takes real and positive (resp. real and negative) values. A differentiable arc on is said to be horizontal (resp. vertical) if its tangent directions are so. See Figure 3.
Example. For , the quadratic differential on with a pole of order at infinity, the horizontal and vertical directions are obtained by pulling back the horizontal and vertical lines in . In particular, each ray for is horizontal. By a change of coordinate , we see that these are the horizontal rays at the pole at infinity. More generally, the meromorphic quadratic differential of the form (1) can be thought of as a perturbation of this; its horizontal leaves are asymptotic to the above directions at the pole.
Geometric interpretation. A computation shows that the pullback of the metric in the target satisfies:
where is the energy density of (as in Definition 2.18) with respect to the metric induced by . See formula (3.6) in [HTTW95].
Then the horizontal and vertical directions (in the -coordinate chart) are the directions of maximal and minimal stretch of the harmonic map.
We shall state the estimates for a planar domain though the estimates for a harmonic map with Hopf differential work for any Riemann surfaces and , under the assumptions that
the conformal metric on has constant negative curvature , and
the singular-flat -metric induced by the Hopf differential is complete.
Remark. The first requirement above is relaxed in the work in [HTTW95], to encompass all Cartan-Hadamard spaces. The second requirement, that is automatic in the case , is needed to apply the method of sub-and-super solution in the proof of the following estimate.
Proposition 2.24 (Horizontal and vertical segments).
Let be a harmonic map that is a diffeomorphism to its image, with Hopf differential .
Let and on be horizontal and vertical segments in the -metric , each of length and a distance , in the -metric, from the singularities of .
Then their images and have lengths and respectively, where is a universal constant.
Moreover, the image of is an arc with geodesic curvature . In particular, it is a distance away from a geodesic segment, where as .
Sketch of the proof.
The basic analysis concerns the following Bochner equation for the harmonic map :
and using the Laplacian with respect to the -metric, we obtain
The technique of sub- and super-solutions then gives us the following decay estimate on solutions: For any we have
where is the distance from the singularity set in the -metric on , and is an absolute constant.
Moreover by a standard gradient estimate we obtain:
where the gradient is with respect to the -metric.
From a simple calculation, the energy density
so the length estimates follow by considering (5), namely
where we have used (8).
Moreover, by the formula (3.7) of [HTTW95], the geodesic curvature
where the final estimate follows from (9). ∎
The proofs of Lemmas 3.2-4 of [HTTW95] show that:
Proposition 2.25 (Images of horizontal rays).
Let be a horizontal ray in the -metric on , and a harmonic map as before. Then the image of the ray is asymptotic to an ideal boundary point , that is, . Moreover, horizontal rays that are asymptotic to the same direction in have images that limit to the same point in the ideal boundary, and different asymptotic directions give rise to distinct ideal points.
Sketch of the proof.
The first statement follows from the estimate of geodesic curvature (12), and a hyperbolic geometry fact, as in Lemma 3.2 of [HTTW95]. If they are asymptotic to the same direction in , then the vertical distance between them is uniformly bounded, and hence by (11) the -distance between their images tends to zero. For a pair of distinct asymptotic directions in , pick a sequence of horizontal leaves asymptotic to these directions, with increasing vertical height from a basepoint. By Proposition 2.24 the images of these limit to a bi-infinite geodesic line . To show that the limit points are distinct, it suffices to show that is a finite distance away from the basepoint, which in turn follows from Lemma 1.1 of [ATW02] which implies that the image of the vertical segments (from the basepoint to the horizontal leaves) have finite length. ∎
Example: Polynomial quadratic differentials
Theorem 2.26 (Han-Tam-Treibergs-Wan).
Let and let
be a (polynomial) quadratic differential on , where the degree and coefficients for . Then there exists a harmonic map
which is a diffeomorphism to an ideal polygon with vertices in the boundary at infinity, and whose Hopf differential is . Moreover, the harmonic map is unique if three of the ideal vertices are prescribed to be .
Example. The quadratic differential for , has a pole of order at , with residue and the image of the harmonic map above is an ideal quadrilateral with a cross ratio, and metric residue, determined by the imaginary part of . See Figure 4, and [AW05] for details.
In fact, as a consequence of the geometric estimates of §2.5, we have the following picture.
We first define:
Definition 2.27 (Polygonal exhaustion).
Given a meromorphic quadratic differential with a pole of order , a polygonal exhaustion at the pole is a nested sequence of annular regions
such that their union is a neighborhood of the pole, and each boundary is an -sided polygon with sides that are alternately horizontal and vertical segments. (cf. Figure 3.)
We first note the following link between the geometry of the -metric induced by the quadratic differential, and the quadratic analytical residue at the pole (see Definition 2.2):
Lemma 2.28 (Residue and -metric).
Let be a meromorphic quadratic differential with a pole of even order , and let be a polygonal exhaustion, as in the preceding definition, of a neighborhood of the pole.
Then for any the alternating sum of the lengths of the horizontal sides of equals the real part of the (quadratic) residue at the pole (up to sign). That is, we have
The following argument culled from the proof of Theorem 2.6 in [Gup14].
Recall from Defn 2.2 that the quadratic residue
since is a curve homotopic to the puncture.
By a change of coordinates that takes the quadratic differential to , the integral over three successive edges (horizontal-vertical-horizontal) of equals integrating the form on the -plane over a horizontal edge that goes from the right to left on the upper half-plane followed by one over a vertical edge followed by one over a horizontal edge that goes from left to right in the lower half-plane. The integral in (15) over the horizontal sides picks up the horizontal lengths that contributes to the real part of the integral, while the integral over the vertical side contributes to the imaginary part of the integral.
Note that the sign switches over the two successive horizontal edges, and hence we obtain (14). ∎
For the following Proposition, note that the metric residue of an ideal polygon with an even number of sides can be defined like that in the case of a hyperbolic crown (Defn 2.9): namely, it is the real number one obtains by taking an alternating sum of the hyperbolic lengths of the geodesic segments obtained by truncating . (For an ideal polygon with an odd number of sides, the metric residue is defined to be zero.)
Proposition 2.29 (Asymptotic image).
Let be a polynomial Hopf differential of a harmonic map . Let be the ideal polygon with ideal vertices that is the image of (see Theorem 2.26).
Construct a polygonal exhaustion at the pole at infinity (see Definition 2.27) where each boundary has horizontal and vertical sides of length in the -metric such that .
Then the image is -close to the boundary of a truncation of , where as .
Moreover, the metric residue of equals the twice the real part of the analytical residue of at the pole at infinity.
(Throughout the paper, “” shall denote a quantity bounded by a universal constant.)
As the distance from the zeroes of increases, the horizontal sides of the polygonal boundaries are mapped closer to the geodesic sides of , and the vertical sides contract to almost-horocyclic arcs in the cusps, by Proposition 2.24.
By the distance estimates (Prop. 2.24) hold, and the hyperbolic length of the image of any horizontal side differs from twice its length in the -metric by , where the error term depends on the distance of from the zeroes of , and hence can be made arbitrarily small by choosing sufficiently large.
Hence the metric residue of the ideal polygon (cf. Defn.2.9 and the discussion immediately preceding this proposition) differs from twice the alternating sum of the -lengths of the horizontal sides by .
However by Lemma 2.28, this alternating sum of the lengths of the horizontal sides of equals the real part of the quadratic residue at the pole. Letting , we have that this equals , as required. ∎
Remark. Such a picture also holds for harmonic maps to hyperbolic crowns having Hopf differentials with poles at the origin, as we shall exploit in §3. Namely, one can construct a polygonal exhaustion at the pole such that their images are asymptotic to the end of the crown.
For future use, we shall use the following definition from the preceding discussion:
3. Asymptotic model maps
Before we prove the main theorem, we prove the existence of harmonic maps from into a hyperbolic crown that will serve as “model maps” defined on a neighborhood of the puncture on .
Doubling the domain
To be able to use Theorem 2.21 and the estimates of §2.5 (which assume completeness of the domain), we shall consider quadratic differentials defined on the punctured complex plane having an involutive symmetry, namely an invariance under .
Definition 3.1 (Symmetrizing ).
Let . For any quadratic differential of the form (1), we define a differential on as follows:
where . Note that this is the unique choice of coefficients such that
has the involutive symmetry , and
and have the same principal part at the pole at .
Note that conversely, determines uniquely.
We shall consider the space of symmetric differentials arising this way, up to a scaling by positive reals. Since is determined by the coefficients where (see Lemma 2.6), and we have that the space is homeomorphic to .
The main result of this section is:
Theorem 3.2 (Asymptotic Models).
Let , and let . For any and a hyperbolic crown , there exists a harmonic map
that has Hopf differential and is asymptotic to the crown end with boundary cusps. The asymptotic image is independent of rescaling the differential by a positive real, and this assignment defines a homeomorphism
where is the real part of the residue of the principal part .
(Recall that is the space of hyperbolic crowns with boundary cusps and fixed metric residue .)
We shall work in the universal cover.
In what follows , is the upper half plane and
is the universal covering map , with the action of by deck-translations being .
The main step towards proving Theorem 3.2 is:
Proposition 3.3 (Equivariant maps on ).
For fix a . There exists a meromorphic quadratic differential on with a pole of order at the origin and principal part , and a harmonic map
its Hopf differential is ,
its image is a polygonal end in , and
it is equivariant with respect to a -action on domain and range, that is,
for a hyperbolic isometry of the Poincaré disk.
Note that the equivariant harmonic map above on the universal cover of descends to define a harmonic map to a hyperbolic crown.
Such an existence result for harmonic maps with prescribed Hopf differential is known when the domain is (see Theorem 2.21), and the geometric estimates of §2.5 hold when the Hopf differential metric is complete.
We have already taken care of the second requirement by extending the quadratic differential on to defined on the punctured plane . To satisfy the first requirement, we consider the lift of the quadratic differential to the universal cover, namely the differential on , where is the universal covering defined by .
By Theorem 2.21 we have the existence of such a harmonic map
with Hopf differential , which is unique once one specifies a normalization of the image.
To complete the proof of Proposition 3.3, we shall show that the restriction of to the upper half plane
is the desired equivariant map to a polygonal end.
For this, we need to show (a) the image is asymptotic to a bi-infinite chain of geodesics, and (b) the map is equivariant with respect to the -action generated by translation in the domain and a hyperbolic translation in the target as in (18).
Determining the image
We can use the analytical estimates (see §2), since the pullback quadratic differential metric on defines a metric that is complete.
Let be a fundamental domain for the action by deck-translations . Then there are exactly horizontal rays of the -metric on that are asymptotic to the pole at the origin; these pull back to horizontal rays contained in . By Proposition 2.25, the images of these under the harmonic map determine distinct ideal points that we denote . Their images under the deck-translations shall be denoted by for each .
We start with the observation:
The sequence of points defined above are monotonic on the ideal boundary , that is, lie in a cyclic order. In particular, we can define
which are independent of the choice of . These also the limit points of the image of the real axis under , respectively. Moreover, they are distinct ideal boundary points, that is, .
The fact that the sequence of limit points of the boundary cusps are distinct points that are monotonic on the circle, and consequently define limiting points , is a consequence of the fact that is a diffeomorphism to its image (see Theorem 2.21): for details, see the proof of Lemma 3.1 of [ATW02].
For the second statement, consider the polygonal exhaustion at the pole of at the origin (see Definition 2.27) such that for each , the boundary comprises alternate horizontal and vertical segments in the - metric of length , where .
Pulling back this exhaustion to the universal cover gives a collection polygonal bi-infinite lines that are each invariant under the translation . See Figure 5.
Fix any of the lines . Note that the vertical distance in the -metric of from the real axis is uniformly bounded. Then, by the same proof as Proposition 2.25, the limit points of the image of either end of the line are the same as the limit points of the positive and negative real axes.
As the horizontal segments of limit to the horizontal lines of the -metric asymptotic to distinct directions in . Hence by Proposition 2.24 the images of these segments limit to the geodesic lines between the s. In particular, the images of the endpoints of the horizontal segments of are close to in the Euclidean metric, say bounded above by (where as ). Since the limit to , the images of the endpoints of these horizontal segments limit to points -close to . Hence the limit points of the images of are -close to and . Since was arbitrary, and we have already observed that the limit points of the images of coincide with that of the real axis , we conclude that those limit points are precisely .
Finally, to show that the limit points on either end are distinct, recall that is defined on , and hence there is a corresponding sequence of ideal points determined by the image of the restriction of to the lower half-plane. (By the symmetry of the Hopf differentials on the upper and lower half planes are invariant under the conformal involution .)
Assume that . (In what follows we shall denote this point by .)
Since is a diffeomorphism, the (closure of the) image of the real line under is a closed loop starting and ending at the point . By the Jordan Curve theorem, this separates the Poincaré disk into two components, such that one of the components has as the only ideal boundary point in its closure.
Consider a bi-infinite path in intersecting the real axis exactly once and asymptotic to a pair of horizontal rays in the -metric, one in the upper half-plane and one in the lower half-plane. Since is a diffeomorphism, the image of the line is an embedded arc between the two ideal limit points determined by the image of the chosen horizontal rays on either side. Moreover, the arc intersects the image of the real line once. Then, by the above assumption, we have that one of the ideal limit points must be . This is true for any choice of a pair of horizontal rays that is asymptotic to, which contradicts the fact that the limit points for the images of the horizontal rays in the lower half-planes, are distinct (cf. Proposition 2.25). ∎
As a consequence, we obtain:
Suppose we have normalized in (19) such that and the image of the positive vertical (imaginary) axis limits to , the ideal point of cusp “”. Then there are hyperbolic isometries fixing such that
for all .
By Theorem 2.21 any two harmonic maps with identical Hopf differential must differ by a postcomposition with an isometry. The Hopf differential is invariant under the translation and involution . Hence (21) and (22) hold for some isometries and . We need to verify that they are hyperbolic isometries, by showing they have exactly two fixed points .
Since the pullback differential to the universal cover of is invariant under deck-translations, it is preserved under the symmetry . Since the real axis is preserved under this translation, and by the previous Proposition its image has distinct limit points on the ideal boundary, it follows by considering sequences such that and , that (21) holds only if is an isometry which fixes the two points .
By construction, the images under of the translates of the ray by these translations, limit to the points for . Thus, necessarily has to take to its successive limit point for each , and is therefore a hyperbolic isometry translating along an axis with endpoints at .
Similarly, and hence , and which implies . Hence is also a hyperbolic isometry fixing the boundary points . ∎
Remark. The equivariant map determines a shear parameter between the images of the upper and lower half-planes as follows: If the image under of the negative imaginary axis limits a point in the lower semi-circle, then the shear parameter is the (signed) distance between and the foot of the perpendicular from to the geodesic between . (See Figure 6.) This parameter is also equal to the (signed) translation distance of the hyperbolic isometry . This measures the “boundary twist” parameter of the hyperbolic crown obtained in the quotient of the image of the upper half-plane by .
Finally, we have:
Proposition 3.6 (Defining ).
Given , there exists a harmonic map
where is a hyperbolic crown, having Hopf differential . Moreover, the metric residue of is equal to twice the real part of the residue of .
Together with the preceding remark assigning the boundary twist parameter, we have a well-defined map
where is the principal part of .
In the preceding discussion, the equivariant map that is the restriction of to (see 20) descends to the harmonic map .
The statement about metric residue then follows from the final argument in the proof of Proposition 2.29:
Namely, consider a polygonal exhaustion
at the pole in , and its lift to the universal cover .
By the distance estimates, we know that the lengths of the images by of a horizontal segment of length in the -metric, is , where is the distance of the segment from the zeroes of .
Applying this to the horizontal sides of , note that as . Hence for any , we can choose sufficiently large, such that the image of the horizontal sides of have hyperbolic lengths for . Moreover, this image is -close to a truncation of the crown , and hence the metric residue of the crown (see Defn. 2.9) which is independent of differs from the alternating sum of these lengths by .
By Lemma 2.28, this alternating sum of the lengths of the sides of equals the real part of the quadratic residue at the pole, which is also independent of . Letting in the above equalities, we have that this real part equals , as required.
Thus in the universal cover determines a polygonal end in .
3.2. Proof of Theorem 3.2 (Asymptotic Models)
For even, the both spaces and are homeomorphic to and when is odd, both spaces are homeomorphic to . The map is easily seen to be continuous (this follows, for example, by the argument for Lemma 2.2 in [TW94]). Hence to show that the map between them is homeomorphism, it is enough to show that the map is proper and injective.
Asymptotic geometry of the -metric
We begin with the following relation between the principal part and the singular flat geometry near the pole, namely, that relative distances in the induced metric are determined, up to a uniformly bounded error, by the principal part:
Let and let be meromorphic quadratic differentials defined on with the same principal part . Then there exists such that for any pair of points sufficiently close to the origin, we have
where denotes the horizontal distance with respect to the -metric, for . There is a similar bound for differences of vertical distances .
Let where . Recall that the vertical and horizontal distances between and are the corresponding vertical and horizontal lengths of a geodesic path with respect to the -metrics (see Definition 2.23):
The key is to observe that since the principal parts are the same, we have (cf. (2)) :
for some constant which depends on the maximum modulus of the coefficients occurring in an expression of and . Since , we have that the right hand side is .
The same bound holds for the difference of the vertical distances. ∎
Conversely, we have that:
Let and let be meromorphic quadratic differentials defined on having distinct principal parts . Then there exists a sequence of points in as such that
where is a fixed basepoint in .
Suppose and differ in the coefficient of where . Pick , and the sequence to lie in a sector in where and are positive. Then
for a path from to lying . The paths tend to the origin as , and the integral on the right hand side diverges. ∎
The key proposition of this section is to observe that using the preceding results concerning the asymptotic geometry near the pole, together with the estimates on the geometry of the harmonic map in §2.5, we have:
Proposition 3.9 (Bounded distance).
Fix and a principal part . Let be a hyperbolic crown and
be two harmonic maps with Hopf differentials .
Then the maps and are asymptotic in the following sense:
for some constant where is the hyperbolic distance on