Limits of Cubic Differentials and Buildings
Abstract.
In the Labourie-Loftin parametrization of the Hitchin component of surface group representations into , we prove an asymptotic formula for holonomy along rays in terms of local invariants of the holomorphic differential defining that ray. Globally, we show that the corresponding family of equivariant harmonic maps to a symmetric space converge to a harmonic map into the asymptotic cone of that space. The geometry of the image may also be described by that differential: it is weakly convex and a (one-third) translation surface. We define a compactification of the Hitchin component in this setting for triangle groups that respects the parametrization by Hitchin differentials.
1. Introduction
In a pioneering work in the 1980’s and early 1990’s, Hitchin and others developed a beautiful non-Abelian Hodge theory for character varieties of surface groups in higher rank Lie groups. The subject has been intensely studied ever since and while new perspectives, for example more synthetic/algebro-geometric [FG06] or dynamical [Lab06, Gui08], have emerged, there remain some basic questions about how to properly geometrically interpret Hitchin’s original parametrizaton of a principal component of the character variety in terms of the holomorphic data he used. (Indeed, in [Hit92], Hitchin remarks, “Unfortunately, the analytical point of view used for the proofs gives no indication of the geometrical significance of the Teichmüller component.”) The goal of this paper is to relate the holonomy of a representation in a Hitchin component to the local synthetic geometry of the holomorphic differential that Hitchin associates to it, at least in an asymptotic sense.
More precisely, we focus on the Hitchin component of surface group representations into . Hitchin ([Hit92]) parametrizes this component in terms of pairs of quadratic and cubic differentials on a fixed Riemann surface . A parametrization, invariant under the action of the mapping class group, was given independently by Labourie ([Lab07]) and Loftin ([Lof01]): from their perspective, may be seen as the cubic differential bundle over the Teichmüller space .
We study families defined by rays in this parametrization, and in particular the asymptotics. Of course, a ray is defined as multiples , for a fixed cubic differential on a fixed Riemann surface and , and so has holomorphic invariants that are projectively fixed; on the other hand, via the Labourie-Loftin parametrization, the ray defines a family of holonomies of representations . We relate the asymptotics of the holonomies to the holomorphic invariants of the form : we give a formula for the leading term of the holonomy of a curve class in terms of the intersection number of with the form . The class may be represented by a geodesic in the metric : this metric is flat away from the zeroes of , with cone points of total angle at a zero of . The segments, known as saddle connections, between the zeroes are denoted .111Saddle connections in this paper are just Euclidean geodesic segments, not restricted to be horizontal in some way. Of course, in such a zero-free region, the cubic differential has three well-defined cube roots and we may compute intersection numbers of curves against these roots. Let denote the largest of the real parts of the three periods; this is equivalent to the logarithm of the largest eigenvalue of the holonomy of a natural development of that saddle connection into affine space. Then our main results on asymptotic holonomy may be summarized in the theorem below: we comment later on what is elided in the statement as well as some of the subtleties in its statement and hence proof.
Theorem A.
For every curve class , we have
| (1.1) |
where is any submultiplicative matrix norm.
In particular, if denotes the -th largest singular value of the flat geodesic homotopic to with saddle connections , then
| (1.2) |
for .
As suggested previously, the main qualitative result is that the leading term of the holonomy, in this regime of a ray, is visible through the local expressions of the Hitchin holomorphic parametrization. That the right-hand-side of (1.1) is expressed as a sum of maxima is consistent with other “tropical” expressions regarding asymptotics, see for example [Foc98].
There are some nuances. First, we observe the role of the zeros of the holomorphic form . They have no explicit presence in either formula (1.1) or (1.2).
Yet, there is a subtlety in that we add the dominant eigenvalue for each saddle connection, so these must get aligned as the geodesic transitions from a saddle connection coming into a zero. Here, two considerations collide: first we must understand the general form of the unipotents that arise as a path crosses the Stokes lines that emanate from a zero [DW15]. Second, we must show that the matrix permutations defined by these unipotents match up the dominant eigenvalues of saddle connections: this occurs because of the geometry of the unipotents that occur in the limits. The resulting agreement of directions between incoming and outgoing saddle connections is a linchpin of the current work. It is somewhat remarkable that it holds not only generically but also in the special cases where the saddle connections are in the directions of Stokes lines or walls of Weyl chambers.
We also note that, in contrast to some treatments (e.g. [MSWW16], [OSWW20]), we do not restrict to simple zeroes. In general, this present formula might be seen as an extension of the work of the first author in [Lof07]. (See also the extension of that work by Collier-Li [CL17] on more general cyclic Higgs bundles.)
Finally, we relate these considerations to the harmonic map defined via the solution to Hitchin’s equations. Now, equation (1.2) gives asymptotics of the singular values, and we recall that the singular values of an element define the distance in the symmetric space between the origin and , with asymptotics then defining distance in the asymptotic cone. We then study the geometry of the limiting harmonic map to the asymptotic cone, obtained by taking an -limit of , rescaling by the growth of the co-diameter of the image of .
Theorem B.
Upon rescaling by , the family of harmonic maps converges to a Lipschitz harmonic map The corresponding rescaled family of holonomy representations converges to a representation to the isometry group of . The harmonic map is equivariant with respect to . The cubic differential induces a -translation surface structure on the image , which is compatible with the local geometry of .
The structure on involves natural local models , where is a ball and is a conformal harmonic map to the asymptotic cone defined in terms of the cubic differential and its three cube roots . In particular, the image of the punctured ball comprises flat sectors which meet only consecutively on geodesics and to which is defined by
for in a sector of . We prove in Theorems 8.7 and 8.9 that the harmonic map has this local structure, where is a restriction of to .
Now, group actions on buildings have arisen in the work of several authors ([Par12], [BIPP21]) with harmonic maps to these buildings having some prominence ([BIPP21], [KNPS15], [MOT21]). Here one might compare the lower rank constructions of surface group actions on real trees in the context of character varieties: see [Bes88], [Wol89], [Wol95]. In the present context, it is worth focusing on the papers of Katzarkov-Noll-Pandit-Simpson ([KNPS15], [KNPS17]), in which the authors outline an approach to compactifying character varieties of surface group representations in . Of course, the present paper can be seen as demonstrating a part of that program in a real setting.
Moreover, though, in a companion paper ([LTW22]), we prove a uniqueness theorem for conformal equivariant harmonic maps to buildings which applies in our situation. Thus we find that in settings in which the harmonic maps have an image in the asymptotic cone of , those harmonic maps coincide with the maps described in this paper as endpoints of rays. In particular, those maps would be definable in terms of cubic differentials projectively approximated by the Hitchin differentials for the approximating representations. The results of Theorems 8.7 and 8.9, and the uniqueness results in [LTW22], then in some sense unify some of the various approaches to asymptotic holonomy of representations and the limiting buildings.
In the concluding section of this paper, we provide an example of a full compactification in a specific example, that of the Hitchin component of most -triangle groups.
Two features of our technique limit the scope of these results. First, we rely heavily on the cyclic nature of the representations, and the resulting substantial symmetries in the Hitchin system. Indeed, in the present work, that system is but a single scalar equation. That is somewhat less of a limitation than it may seem, as some analogous results are available in the case of ([OT20], [TW19]). Second, here we fix the conformal structure of the domain: considering asymptotics where both the domain Riemann surface and the representation degenerate seems to require an analysis finer than what we present here. Finally, we are greatly aided by the asymptotic cone being two-dimensional, and hence the same dimension as the domain . This restricts the flexibility of the limiting harmonic maps.
Organization.
In the second section, we define our notation and present some background material. Section 3 is devoted to presenting some required analysis of the Hitchin partial differential equation which governs the harmonic maps. In Section 4, we analyze the holonomy near a zero of : we are interested in the holonomy along a generic saddle connection and along a portion of an arc that links the zero that crosses a Stokes line. Section 5 assembles these partial holonomies of individual saddle connections into a preliminary description of the asymptotic holonomy. Then, in Section 6. we discuss the phenomenon that the unipotents that describe the transition between incoming and outgoing saddle connections intertwine the dominant eigenvalues. In Section 7, we collect all of the ingredients from the previous sections and prove the main result on asymptotic holonomy displayed above. Section 8 pivots to describe how the endpoint of a ray is a harmonic map to a building, displaying the local structure and induced metric to , and showing that the image is “weakly convex” in the sense of Parreau [Par21]. Finally, Section 9 displays a corollary of our work in a very special case: the compactification of in the case of a triangle group, where we can provide a somewhat complete account of the compactification using our methods.
Acknowledgements.
The authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 RNMS: GEometric structures And Representation varieties (the GEAR Network). In addition, some of this work was supported by NSF grant DMS-1440140 administered by the Mathematical Sciences Research Institute while the authors were in residence during the period August 12-December 13, 2019 for the program Holomorphic Differentials in Mathematics and Physics. The second author acknowledges support from the U.S. National Science Foundation under grant NSF DMS-2005501. The research of the third author was supported by the National Science Foundation with grant DMS-2005551 and the Simons Foundation. The authors appreciate useful conversations with David Dumas, Michael Kapovich, Anne Parreau, Marc Burger, Alessandra Iozzi and Beatrice Pozzetti.
2. Background material
The geometry relating cubic differentials and representations in is that of a special sort of surface, a hyperbolic affine sphere , in . There are natural affine invariants on a hyperbolic affine sphere, a Riemannian metric called the Blaschke metric , and a cubic differential which is holomorphic with respect to the conformal structure induced by the Blaschke metric, related by the compatibility relation
| (2.1) |
where is the Gauss curvature, and is the pointwise norm of with respect to the metric . Let be a parametrization of conformal with respect to , from a domain . Then is transverse to the tangent plane and we have the following structure equations for the frame in , for a conformal coordinate and
| (2.2) |
Equation 2.2 then gives the connection form of the pullback of the flat connection on to the rank-3 bundle over , for the trivial line bundle. Equation 2.1 and ensure is flat. On a compact Riemann surface, the holonomy of gives a Hitchin representation into .
Alternately, given a holomorphic cubic differential over , we may use this data to define a stable rank-3 Higgs bundle, which then induces an equivariant conformal harmonic map from to the symmetric space . We need not avail ourselves of the details of this construction, as the harmonic map can be constructed directly from the hyperbolic affine sphere. To see this, identify with the space of all metrics (positive-definite quadratic forms) on of determinant 1. The Blaschke lift at a given point on is then the metric on for which
-
•
has norm 1,
-
•
is orthogonal to ,
-
•
the metric restricted to is the Blaschke metric .
To relate the Blaschke lift to the frame , it is useful to use an orthonormal frame for . For , the frame
is orthonormal. Thus
Cheng-Yau ([CY86], [CY77]) proved that any hyperbolic affine sphere with a complete Blaschke metric is asymptotic to a properly convex cone in , which we take to have vertex at the origin. Likewise, for every such convex cone, there is a unique hyperbolic affine sphere asymptotic to the cone invariant under special linear automorphisms of the cone and with complete Blaschke metric. By projecting , we may identify with a properly convex domain in . On a compact Riemann surface of genus at least 2 equipped with a cubic differential , there is a unique solution to Equation 3.1 below (this is the global version of Equation 2.1 above). Thus the universal cover is identified with a convex domain in , and itself admits a quotient of by projective automorphisms. In other words, we have induced a convex structure on . Choi-Goldman ([Gol90],[CG93]) show that convex structures are equivalent to Hitchin representations.
Dumas and the third author [DW15] address the case of polynomial cubic differentials on , and show there is a unique complete Blaschke metric on that plane. The convex domain in this case is a convex polygon with sides, for the degree of the polynomial. Indeed polynomial cubic differentials on are equivalent to convex polygons, up to appropriate equivalences. In this work, we are primarily interested in cubic differentials of the form , which corresponds to the regular convex polygon of sides. In general, the analysis of the boundary of the polygon follows by comparison to the simpler geometry of inscribed and circumscribed triangles around each vertex and edge.
Example 2.1.
The triangle case (for a constant cubic differential on and in which the Blaschke metric is flat) can be worked out explicitly and goes back to Ţiţeica ([Tzi08]). We see that this is a fundamental model for us, and so we describe some of its features. In terms of a natural local coordinate in which the cubic differential , for , there are “Stokes” directions for rays for an integer. In the sectors bounded by , the rays limit on a vertex of the triangle; the rays parallel to those angles fill out the sides of the polygon.
For a general convex polygon, in each sector in between the Stokes directions, the affine sphere is well approximated by an appropriate Ţiţeica surface. Upon crossing a Stokes ray, the approximating Ţiţeica surface changes by the action of a unipotent transformation determined by the geometry of the convex polygon: the unique unipotent transformation in whose projective action on transforms a given inscribed (circumscribed) triangle in the polygon to the subsequent circumscribed (inscribed) triangle by moving a single vertex.
We will also encounter other special directions in , which represent the walls of Weyl chambers at , upon identifying with the maximum torus of the Lie algebra of .
3. Asymptotics of the Blaschke metric
Consider a ray of cubic differentials on a fixed Riemann surface with conformal hyperbolic metric . Let be the Blaschke metric on the associated affine sphere. The real-valued functions are solutions of Wang equation
| (3.1) |
Near each zero of , Nie has studied rescaled limits of the associated convex structure, which converge to a regular polygon [Nie22]. Our approach focuses on precise estimates comparing the corresponding affine spheres.
3.1. Asymptotics far from the zeros
We compare the Blashke metrics with the flat metric with cone singularities . The following results are well-known.
Lemma 3.2 ([OT21]).
The area of the Blaschke metric satisfies
where .
We consider now the quantity
in order to compare and outside the zeros of . Outside the zeros of , the function satisfies the PDE
By Lemma 3.1, and . Hence is subharmonic.
Lemma 3.3 (Coarse bound on ).
Let and let be the radius of a ball around for the flat metric which does not contain any zeros of . Then
Proof.
The ball of radius for the flat metric does not contain any zeros. By subharmonicity of and Jensen’s Inequality, we have
∎
Lemma 3.4 (Error decay).
Let be a point at -distance from the zeros of . Then for any , there is a so that
with as .
Proof.
Consider the ball centered at of radius for the flat metric . Since this ball does not contain any zeros of , we may choose coordinates on the ball so that and is at . Then on this ball satisfies
By Lemma 3.3 and Lemma 3.2, the function is uniformly bounded below by a constant . Then
Similarly, the function is bounded above by a constant on the boundary of the ball. Let be the solution of the system
We know , where is the only radial solution of
It is well-known ([AS64]) that is the Bessel function of the first kind and has asymptotic behavior as . By the maximum principle,
This implies that decays exponentially outside the zeros of . Now, remember that the constant comes from the bound on . From the decay of , we can improve this bound to with as , and the lemma is proved. ∎
Notation.
For simplicity, we denote by the exponent appearing in Lemma 3.4, i.e.
3.2. Estimates around a zero
We now move to the study of the asymptotic behavior of around a zero of the Pick differential. We denote by the Blaschke metric of the affine sphere with polynomial cubic differential on . We want to compare and on the ball . We rewrite the Blaschke metric with respect to the background flat metric on : , where is a solution of the PDE (only defined on the closure of )
| (3.2) |
Thus is a solution of the same equation as , but with different boundary values.
Lemma 3.5.
On , we have as .
Proof.
We know from Lemma 3.4 that
for , where we recall that is the hyperbolic metric on the fixed Riemann surface . Thus it is sufficient to show that
Changing complex coordinates to , the ball can be rewritten as . In these coordinates, with , and . The estimates of ([DW15, Theorem 5.7]) then tell us that
Hence, on ,
∎
Lemma 3.6.
On the ball , we have as .
Proof.
Define . It satisfies the PDE
on . Dividing by , we find, upon setting , that
By the maximum principle,
which gives the desired estimate by Lemma 3.5. ∎
4. Comparison between affine spheres
We denote by the frame field of the affine sphere arising from the data on the surface restricted to the ball . We normalize the affine sphere so that for all . With this choice will be complex-valued, belonging to a subgroup of isomorphic to . Recall that is the solution to the ODE
where are the matrices arising from the structure equations of the affine sphere. Precisely,
| (4.1) |
We denote by the frame field of the (model) affine sphere over with polynomial cubic differential normalized so that . Note solves
We compare and on the ball centered at a zero of order for . Recall that , so we consider
The matrices satisfy the differential equation
when written in the coordinate, where
where was defined in section 3.2.
Lemma 4.1.
On , we have as .
Proof.
We handle the various entries in one-by-one. First of all, the Mean Value Theorem implies for that
for some between and . A straightforward application of the Maximum Principle applied to equation (3.2) on , together with a boundary estimate from Lemma 3.4, then gives
and the same is true for . Then Lemma 3.6 shows
Similarly,
for some between and . By considering the change of coordinate , we see for all
Now is bounded below, as in [DW15, Corollary 5.2], which implies
Lemma 3.6 then shows
The result then follows if we show a decay estimate holds for and . Set . We know from the computations in Lemma 3.6 that
Then is uniformly bounded, because and, using the same notation as in Lemma 3.5 and the subsolution for found in [DW15, Theorem 5.1],
So
and, using the supersolution for in [DW15, Theorem 5.1],
The bounds on and then follow from the Schauder and estimates. ∎
Proposition 4.2.
Let be a neighborhood of the union of the Stokes rays on . For every , there is so that for all and ,
Proof.
Let denote the frame field of the standard Ţiţeica surface (see Example 2.1) with cubic differential on the plane, which can be explicitly be written as
for and some choice of conjugating matrix .
From [DW15, Lemma 6.4], we know that, outside a compact set , we have as , where is a constant matrix that only depends on the sector in the complement of the Stokes rays containing . Here we use because the result of [DW15] is stated in terms of natural coordinates for , instead of the coordinate, say , centered at the zero of . Therefore,
By [DW15, page 1768], we conclude that
where and achieves this maximum value when corresponds to a Stokes direction. In particular, when , there is such that . In the definition of , we may choose both sufficiently close to and sufficiently large so that
for some . Therefore, by Lemma 4.1,
for sufficiently large, independently of . ∎
Corollary 4.3.
There exists such that for all and , we have
Proof.
In particular, this implies that for all ,
Combining this with the fact that , where only depends on the sector in the complement of the Stokes line that contains , we obtain
Corollary 4.4.
For every inside the sector in the complement of the Stokes rays,
Here, of course, we have adapted our choice of to our choice of .
Corollary 4.5.
Let contain one Stokes direction in its interior. Then
where is one of the unipotents introduced in [DW15].
Proof.
The matrices are defined as the limit of as . Thus the result follows from [DW15, Lemma 6.5]. ∎
Theorem 4.6 (holonomy along arcs).
We remark that when the interval contains more than one Stokes direction we can still apply Corollary 4.5 and Theorem 4.6 after splitting the interval into subintervals containing only one Stokes direction and thus satisfying the assumptions of Corollary 4.5. Because the holonomy is multiplicative along concatenation of paths, we will have
where is now a product of unipotents depending on the Stokes directions the arc crosses.
5. Asymptotic holonomy
We want to compute the asymptotic holonomy of the flat connection on the rank-3 bundle along a -geodesic path that may cross some of the zeros of the cubic differential . We say such a geodesic path is regular in that each segment away from the zeros of are
-
•
not in the directions of the walls of a Weyl chamber, so that for . Here is a root of .
-
•
not in the Stokes directions.
We later remove these hypotheses.
It is convenient to work in the universal cover of . Equip with the conformal hyperbolic metric and identify with the strip model of the hyperbolic plane
with metric . The vertical line with arc-length parameter is a geodesic; so a hyperbolic deck transformation can be represented, up to conjugation, by the transformation , where is the translation length.
Remark 5.1.
We want to use this model because it gives a way of defining a frame on which we can use to compute parallel transport. The bundle lifts to a bundle over which we now trivialize using the global frame . This frame is not parallel with respect to . However, because it is globally defined, we can define the holonomy of along an arc as a comparison between the terminal parallel transport of a frame in the fixed basis and the frame at the terminal point. Given , we can assume that the hyperbolic isometry corresponding to is for some . Fix and let . Note that . Let be the -geodesic connecting and with . If we have a matrix representation of the parallel transport along with respect to the frame , then the matrix represents the holonomy of the flat connection between the final and initial points as their frames are identified in the quotient.
Remark 5.2.
We can assume that and are not zeros of , so that the geodesic path starts and ends with a segment not containing any zeros.
The path will in general cross some of the zeros of the cubic differential with multiplicities . In fact, we write as the union of , where each is the straight line path in the flat coordinates for from to . Each does not intersect any zeros of except at its endpoints. We fix and identify a neighborhood of each zero and a conformal coordinate so that on and the -radius of is . Note for small the closures do not intersect. We modify to form a new path by deleting each and replacing it with an arc in so that is continuous and homotopic to the original geodesic. Now consists of a number of line segments in the flat -coordinates. Divide each line segment between and into two segments and , so that each has an incoming line segment and an outgoing one .
In total, is the concatenation of . The basepoint is . We also denote by and the prolongments of and to their forward and backward zero respectively. Since and are homotopic, the holonomies along these paths are the same. Then
We want to find estimates for each factor and arrive at something of the form
with explicit and depending only on and on the geodesic path .
Remark 5.3.
Let be a conformal change of coordinates. For instance, let be a natural coordinate for . The coordinate induces a new frame . There is a diagonal matrix depending on the derivatives of so that . Moreover, if and are two natural coordinates at a point, then and differ by a translation and a multiplication by a third root of unity. If we choose the natural coordinates so that they induce the same frame on the overlaps, we can multiply the matrices representing the parallel transport along consecutive arcs.
Remark 5.4.
If and are two natural coordinate charts that cover a path and overlap at a point , we note that and induce the same frame at if and only if the path makes the same angle with the positive horizontal axis, as seen in the coordinates and .
Let and denote by the parallel transport from to for the lift of the flat connection. Assume and are not zeros of and are in the same natural coordinate . Let be the standard frame induced by . The frame is defined at as well; so we can find a matrix such that
Let be a path connecting and . The parallel transport condition is equivalent to being a solution of the initial value problem
where and are defined in Equation (4.1). The matrix representing the parallel transport with respect to the frames and is then .
In what follows, instead of solving the initial value problem above, we compare with the solution of the initial value problem
where and are the matrices appearing in the structure equations for the affine sphere over with constant cubic differential . We know ([Lof07]) that
where
| (5.1) |
and is the conjugating matrix that appeared e.g. in Proposition 4.2.
Remark 5.5.
Note that solves the same ODE as the frame field of the associated affine sphere. They differ by the value at the initial point.
The first author in [Lof07] considered the case of geodesic paths which do not hit any zeros and determined the asymptotic behavior of the eigenvalues along such paths. We would like to use Proposition 3 of [Lof07], but unfortunately the published statement must be modified to Proposition 5.6 below, as there is a gap in the proof. The final paragraph of the proof in [Lof07] is unsupported. The main theorem of [Lof07] is still true, as follows from the results presented here. The main additional technique needed, which was available at the writing of [Lof07], is the fact that the largest eigenvalue of the holonomy along a path (and the reverse path) is enough to determine all the eigenvalues in . The first author regrets the error.
Thus we have the following proposition. We note that Collier-Li and Mochizuki have proved stronger estimates in a more general setting in the case in which no two eigenvalues are equal [CL17, Moc16].
Proposition 5.6 (Holonomy along rays).
The parallel transports along the segments and with respect to the frame induced by a natural coordinate for are given by the matrices
as , where
are the roots of , and
Remark 5.7.
Note that if we parametrize the path by with for the -length of the geodesic segment between successive zeros of and , then
Hence the position of the largest eigenvalue of the diagonal matrix depends only on the angle that makes with the positive -axis in the chosen natural coordinates.
Remark 5.8.
We now compute the parallel transport along the circular arcs . In the -coordinate centered at a zero so that , the path is parametrized by with . Let be a natural coordinate for . Choose so that the angle the incoming path makes with the positive -axis coincides with the angle we saw in the previous natural coordinate chart.
Proposition 5.9 (Holonomy along arcs).
Assume and do not correspond to Stokes directions. Then the holonomy along satisfies
with respect to the frame induced by the natural coordinate. Here is a product of unipotent matrices depending on which Stokes rays the path crosses.
Proof.
Recall that is the inverse of the matrix which solves the initial value problem
The frame field is a solution of the same ODE with different initial conditions, so
is the solution of the above initial value problem. Now, by Corollaries 4.4 and 4.5 and the subsequent remark,
where is as in the statement. Hence
Because the claim follows. ∎
Combining the holonomy of each subpath (Proposition 5.9 and Proposition 5.6), we obtain
with respect to a frame induced by natural coordinates. Here and are the largest eigenvalues of and respectively.
Remark 5.10.
The holonomy with respect to the global frame will only differ by multiplication on the left and on the right by the change of frame between the global coordinate on and the natural coordinate for . These, however, only grow polynomially in , so they do not influence the estimates that follow.
We can then write
by setting
| (5.3) |
It is worth emphasizing two aspects of the definition above of . First, the definition makes no reference to the radius of the balls cut out around the zeroes. Moreover, the formula for allows for an extension of the formula for , first stated for the modified path to the -geodesic . These points will be essential for computing final holonomy formulas, e.g. in Theorem A, which do not depend on or other constructions around the zeroes.
Lemma 5.11.
Along any regular geodesic, the highest order term in (5.3) is
| (5.4) |
where denotes the elementary matrix with 1 in position and is a non-zero constant. Here and are times the logarithms of the largest eigenvalues of and respectively.
Proof.
Remark 5.12.
Two consecutive terms in the above product have the property that (since ) because the position of the highest eigenvalue only depends on the angle the path makes with the -axis in a natural coordinate and our choices of coordinates keep this angle constant when the coordinate patches cover the same straight path. Note, indeed, that and (with indices intended modulo ) are part of the same straight line segment .
We also give an argument to address the special case in which the the angle of a geodesic segment is in a Stokes direction, extending Lemma 5.11 to the case when Proposition 5.9 does not hold. Define to be the geodesic segment in corresponding to this geodesic arc. Then the estimates of [DW15] fail to hold at its endpoints and . We will modify slightly by moving the endpoints. Recall is the corresponding geodesic segment between the zeros in .
We rewrite (5.3) as
| (5.5) |
Proposition 5.13.
Lemma 5.11 holds for homotopy classes of free loops for which the flat geodesic’s saddle connection segments are all either regular or travel along Stokes rays: in other words, if no saddle connection is contained in a wall of a Weyl chamber.
Proof.
It suffices to address the case of a single saddle connection along a Stokes ray. Each endpoint of the is in the flat coordinate for a Stokes direction. We modify the angle by for a small positive constant to avoid these directions. So define to be the new arc formed by replacing the endpoint by , and similarly define . Define to be the geodesic path between these endpoints of and . By choosing small enough we can ensure the straight line homotopy between and does not cross any other zeros of the cubic differential.
In certain cases we also need to specify the signs . At each zero along the geodesic , the incoming and outgoing rays must make an angle of with respect the flat metric, when measured in clockwise and counterclockwise directions around the zero. Proposition 6.3 below requires that each arc begins and ends away from a Stokes ray and must subtend an angle (and so at least 3 Stokes rays will be transversed by the arc). For each endpoint of choose so that the arcs and both subtend an angle . This is possible since the total angle around a zero of order is . Note that in some cases we are free to choose either or ; then there is a different holonomy matrix along the arc depending on the sign. Lemma 6.5 below shows that this matrix leaves the relevant entries unchanged, in terms of the leading order terms.
Define to be the union of and the two radial paths from the zeros to the endpoints of . See Figure 2. Now by Proposition 5.6 and Remark 5.8, the contribution for in (5.5) is given by
where
since is closed and is homotopic to . This shows as above that the conclusion of Lemma 5.11 holds.
Note we call the entire modified path . It is obtained from by replacing, for each appropriate , by and by , etc. ∎
6. No branching
In this section we exploit the geometry of the convex regular polygon to which a hyperbolic affine sphere with cubic differential project in order to analyze the non-zero entries of the unipotent matrices of the previous section. The key result, Proposition 6.3, asserts that the unipotents in the holonomy formula (5) – that connect the holonomy of segments that come into a zero with the holonomy of segments that leave a zero – have an entry that allows the largest eigenvalues of those holonomies to multiply. This is crucial for the form of the formula (5).
Proposition 6.3 below will be used later to show that the induced map from the Riemann surface to the real building given by the asymptotic cone is locally injective near the zeros of the cubic differential, and thus can have no branching behavior. The corresponding phenomenon in the real tree case is called “folding,” which does occur in some situations (e.g. [DDW00], [Wol07]).
Choose a local coordinate on so that . Consider the corresponding embedding of the Riemann surface into whose image is a hyperbolic affine sphere with Pick differential , and consider the frame of the affine sphere. Let , be the corresponding embedding and frame for a standard Ţiţeica surface (as described in Example 2.1). The osculation map then has limits along rays of angle for respectively ([DW15]). These matrices determine the construction of the convex polygon onto which the affine sphere projects. Let us summarize the main step of the contruction. We label the vertices of as with indices in . Let denote the line connecting and , and let denote the edge of from to . Three successive vertices form an inscribed triangle in around the vertex . Also define points so that are the vertices of a circumscribed triangle of centered around the edge . See Figure 3.
Proposition 6.1.
In the above setting the following holds:
-
(1)
if and only if or .
-
(2)
if and only if or .
Proof.
Statement (1) is obvious from the convexity of .
To prove statement (2), note we need only prove the "only if" part. So assume . As are the vertices of a triangle, we see . To find a contradiction, assume or and that . Recall . By convexity, . Since is a convex polygon, it is the intersection of closed half-planes , each bounded by . Since , we see is a subset of the smaller triangle , which cannot contain both and . This is a contradiction. ∎
The columns of the matrices , project to the vertices of a circumscribed triangle of , whereas the columns of project to the vertices of an inscribed triangle. Moreover, the middle vertex of an inscribed triangle is always obtained as where is the eigenvector corresponding to the highest eigenvalue of . The unipotent matrices arise then by taking the products and similar. We determine these unipotent matrices explicitly in the case is regular. Choose coordinates in so that
We then compute that
while can be computed as , where we view . In other words, in the natural inhomogeneous coordinate chart in , is found by rotating by an angle of .
The paper [DW15] provides a scheme of determining the projective transformations of triangles (across Stokes lines) to form the polygon, as well as the order of the largest eigenvalue corresponding to each vertex. See the tables on pages 1771-1772 in [DW15].
| (6.1) |
Here each refers to crossing a Stokes line, which in standard flat coordinates are at angles , while each vertical line refers to crossing a wall of a Weyl chamber, at angles . The refer to the smallest, medium and largest eigenvalue respectively of the frame .
There is of course more than one projective transformation which takes a triangle to a triangle, but the ones we are interested in are determined by the conditions that they are unipotent and that they fix the vector lifts to of the relevant ’s fixed by the projective transformation. The next proposition shows, at least in the regular case, that these vector lifts of can be done globally, and that the relevant linear transformations corresponding to crossing several Stokes lines near a zero can be computed in terms of simple changes of bases (in contrast to a more complicated scheme of multiplying out several transformations).
Proposition 6.2.
Let be a regular convex polygon as above. There are vector lifts of the points for which the linear transformations lifting the corresponding triangle transformations are all unipotent. The choice of such a set of lifts is unique up to a nonzero multiplicative constant.
Proof.
We may easily check that
satisfy the conditions, with again being defined by rotating by an angle of . ∎
It follows that the matrix represents the change of basis between and where and project to the vertices of inscribed or circumscribed triangles of the regular polygon . Therefore, if is the vector corresponding to the highest eigenvalue of and is the vector corresponding to the highest eigenvalue of , the entry of is nonzero if and only if has a component along in the basis . It is important to note, because of the orientations of the paths near a given zero , that the highest eigenvalue of is the highest eigenvalue of , while the highest eigenvalue of is the lowest eigenvalue of .
As promised above in Lemma 5.11, we now prove
Proposition 6.3.
Consider any convex polygon . If the highest eigenvalue of is in position and the highest eigenvalue of is in position , then the -entry of is not zero.
Proof.
Refer to Figure 3 and Equation 6.1. Because the paths and are part of a geodesic for the flat metric , the angle between and is at least (measured in the singular flat metric) at either side. Since Stokes rays are apart, the interval contains at least three Stokes directions. Crossing a Stokes direction corresponds to “flipping” the initial triangle (i.e., moving to the next triangle in the sequence displayed in (6.1), accomplished geometrically between triangles that share an edge). We assume that the initial basis projects to a circumscribed triangle and thus is of the form . In this case the eigenvector corresponding to the highest eigenvalue can project to either or : it projects to if the direction is in the first half of the interval determined by two Stokes directions and to otherwise. We will explain the argument in detail when the highest eigenvalue is in the direction of ; the case of or when the initial basis projects to an inscribed triangle are analogous and left to the reader. If the direction of the highest eigenvalue is , then after at least three flips in counter-clockwise direction, by Proposition 6.1, the point does not lie in any of the lines generated by the final triangle. Hence all entries in the first column of are non-zero. If we move in clockwise direction instead, then after at least five flips the point never lies on any of the lines generated by the vertices of the final triangle and the claim follows as before. Thus, we only need to check what happens when only three or four Stokes directions are contained in the interval .
If the final triangle is obtained after four flips, then it is circumscribed and the vertex corresponding to the highest eigenvalue of is , i.e. the one vertex outside the polygon. This is because, referencing Equation 6.1 again, we see both that the highest eigenvalue of is the lowest eigenvalue of and that the corresponding eigenvector always projects to the vertex exterior to the polygon. Because (see for example Figure 3), the coordinates of with respect the final basis have a nonzero component along , hence the corresponding entry in is nonzero. Finally, if the final basis is obtained after only three flips, it is easy to see that the highest eigenvalues of and are always in the same position: this follows again by a careful reading of equation (6.1). (We add some details. Recall that we chose the initial point to be in the first half of the region between Stokes lines, i.e. in the region between the Stokes line and the Weyl chamber. For example, we are focused on a point in the region between and ; after three flips, this point lands in the region between and . By inspection of the corresponding bases in (6.1), we see that the initial and terminal basis element, , corresponding to the highest eigenvalue is unchanged, so the corresponding entry in the unipotent is non-vanishing.) In general the chain in Equation 6.1 then shows that the triple is sent to , so the elements on the diagonal are all non-zero.
Finally, we note that in the argument above, all the conditions checked involve only incidences of the given points and lines in , and not the choices of vector lifts in . Thus the proposition holds not only for regular polygons, but for all convex polygons. ∎
Below in Proposition 6.8, we extend Proposition 6.3 to determine the signs of all the relevant entries. These signs will be useful in handling the special cases in which the angle of a geodesic segment in the flat metric is at a Stokes line or wall of a Weyl chamber.
We recall Cramer’s Rule from linear algebra:
Lemma 6.4.
If is a vector in , and is a basis, then , where
Assume for now that the polygon is regular, so that all the lift to vectors as in Proposition 6.2. It is useful to set up some notation for the following results. We let be the eigenvector with the largest eigenvalue of for the incoming ray . The frame for the outgoing ray is , and is the one with the largest eigenvalue for . We can write for some .
Lemma 6.5.
is unchanged if the incoming or outgoing angles vary by crossing a single Stokes line.
Proof.
Refer to equation (6.1). If the incoming angle moves across a single Stokes ray, the largest eigenvector of is unchanged: recall that in equation (6.1), each Stokes ray, denoted by , bisects a Weyl chamber region, whose boundaries are denoted by vertical lines. If the outgoing angle is changed by crossing a single Stokes line, then the smallest eigenvector of is the only vector to change in the frame . Upon replacing with , we see (and indeed the underlying matrix) is unchanged. ∎
Lemma 6.6.
for some . if , and is 0 otherwise.
Proof.
From (6.1) we see , as the eigenvector of the largest eigenvalue for an incoming ray, is of the form for some . Similarly, upon replacing with , we see, upon perhaps performing a cyclic permutation, that for some .
If , it is obvious that . On the other hand, if , then form a counterclockwise-oriented triangle in the plane. By Proposition 6.2, . So this orientation of the triangle implies . ∎
Lemma 6.7.
.
Proof.
In the case of an inscribed triangle, for some . Thus as in the previous lemma, . To pass from an inscribed to a circumscribed triangle, the frame is changed by a unipotent transformation, which leaves the determinant unchanged. ∎
Proposition 6.8.
Let be any convex polygon. Given the notation of Proposition 6.3, the entry of is positive.
Proof.
Proposition 6.3 shows the relevant entry is nonzero. We have checked its positivity for the particular case of regular polygons. Then the general result follows by continuity and the connectedness of the moduli space of convex -gons. ∎
7. Main Theorem - asymptotics of singular values
In this section, we prove the asymptotics of singular values of the holonomy associated to geodesic paths. The singular values naturally give the distance in the symmetric space . First, in Theorem 7.1, we focus on the case of regular geodesic paths (allowing Stokes directions). In these cases, we have shown there is always a unique largest element in each diagonal matrix in (5.5), while the relevant elements linking them together in are positive. Later, in Theorem 7.5, we remove the remaining restrictions to allow any geodesic path without any restriction on the angles of the segments. Finally in Corollary 7.6 we extend the analysis beyond just singular values to include eigenvalues.
Theorem 7.1.
Using the same notation as in Lemma 5.11, for every regular path , and indeed for every path none of whose segments is contained in a wall of a Weyl chamber, we have
where is any submultiplicative matrix norm and is the largest of the .
Proof.
Recall from Section 5 that . Assume for now that ; we return to check this assumption after using it to conclude the theorem. Then
where denotes the elementary matrix. Now,
and
Hence,
where as usual is considered modulo .
We only need to check the condition on the error: . For this estimate we are going to use the -norm . Note, however, that because all matrix norms are equivalent the result holds for any matrix norm. Recall that is equal to
Hence is a sum of terms in which at least one of the factors contains , , or . For instance, in the case where arises once, we will have a term of the form
and so
In this case, of course, we combined diagonal matrices in a convenient way. Yet even when the diagonal matrices do not simplify, we may obtain the same estimate. For instance, consider
Then
using Lemma 5.11, where, we recall that is the largest eigenvalue of . This last expression will be if we can show that . Note that . Now, if with , then with and
So if the largest eigenvalue of is in position , then
Thus, , as required.
The remaining cases are analogous, involving smaller error terms. ∎
In particular, if we choose the submultiplicative matrix norm
where denotes the highest singular value of , in other words the highest eigenvalue of , we obtain
Corollary 7.2.
For every regular path that is the concatenation of saddle connections we have
Proof.
We can also deduce the asymptotics of the other singular values
Corollary 7.3.
Let denote the -th largest singular value. Then
for .
Proof.
We already know that the result holds for by Corollary 7.2. Because
the statement is also true for by applying the previous corollary to the path . Moreover, since , we have
hence the result holds for as well. ∎
We now give an argument extending the above results to the case of flat geodesics that are not regular, in that they contain segments in the wall directions of a Weyl chamber. We begin with the generic situation where each flat geodesic segment has corresponding diagonal holonomy with a unique largest eigenvalue. In that case, as above, we compute the largest asymptotic singular value and then reverse the direction of the path to find the smallest. This determines the asymptotic eigenvalue structure. In particular, the arguments above already suffice to determine the largest asymptotic singular value along any geodesic path in which each segment has a unique largest eigenvalue. We summarize the discussion in the following proposition.
Proposition 7.4.
Now we address the remaining case where paths may contain segments so that at least one has two largest eigenvalues. In this case, Lemma 5.11 becomes
| (7.4) |
where in each sum ranges over one or two indices in : one if there is a unique largest eigenvalue of , and two if there are two largest eigenvalues. Proposition 6.8 then shows that the coefficients are all positive, and thus there are no cancellations. This is enough for the analysis above on submultiplicative matrix norms and singular values to apply.
Thus we have proved
Theorem 7.5.
For every geodesic path , we have
where is any submultiplicative matrix norm.
In particular, if denotes the -th largest singular value of the flat geodesic homotopic to with saddle connections , then
for .
Because is diagonalizable with positive eigenvalues, Theorem 7.5 also implies a similar asymptotic formula for the eigenvalues of .
Corollary 7.6.
For every closed curve , let be the geodesic representative for the flat metric . Assume that is the concatenation of saddle connections . Then
where denotes the spectral radius of .
In particular, Corollary 7.3 holds when replacing singular values with eigenvalues.
Proof.
It is well known that the spectral radius, i.e. the absolute values of the largest (possibly complex) eigenvalue of a matrix , can be computed as
Since has all real and positive eigenvalues, its spectral radius coincides with its largest eigenvalue. Moreover, because and are in the same free homotopy class, we can do this computation for . Now, we know that
with as and
| (7.5) |
for some positive constants . Fix small and let be such that for all we have . Then, for all
with for every integer . Therefore,
for all , where
First we compute the spectral radius of the matrix . From (7.5), we find
as , where
Hence, for sufficiently large,
We then observe that the function is uniformly bounded for , because
which implies that . Therefore,
Now, the saddle connection is the concatenation of and , where indices are intended modulo (see Figure 1), so by Theorem 4.6,
which gives the desired asymptotics of the largest eigenvalue. Repeating the same argument as in Corollary 7.3, the formula actually holds for all eigenvalues of . ∎
8. Harmonic map to the real building
By work of Hitchin ([Hit92]), the Hitchin representation arising from the ray of cubic differentials are constructed along with an associated -equivariant conformal harmonic map to the symmetric space (see Section 2). In this section we study both the asymptotic behavior of around a zero and also describe the geometry of the limiting harmonic map to an -building.
8.1. Generalities on Euclidean buildings
We recall here the definition and main properties of -buildings. We direct the interested reader to [KL97] for a more thorough discussion.
Let denote a finite-dimensional affine Euclidean space. The Tits boundary of is a sphere, denoted by . A subgroup is an affine Weyl group if it is generated by reflections across hyperplanes of , called walls, and its linear part is finite. The pair is a Euclidean Coxeter complex. We denote by the quotient .
An oriented geodesic ray determines a point in . Its -direction is its projection to . A Weyl chamber with tip at is a complete cone with vertex at for which its Tits boundary is a chamber. A germ of a Weyl chamber based at is an equivalence class of Weyl chambers based at for the following equivalence relation: and are equivalent if their intersection is a neighborhood of in both and . The germ of a Weyl chamber is denoted by . We say that two germs and are opposite if one is the image of the other under the longest element in the Weyl group . We say that two Weyl chambers based at are opposite if their germs are.
Definition 8.1.
A Euclidean -building modeled on a Euclidean Coxeter complex is a space that satisfies the following axioms:
-
a)
Each oriented geodesic segment is assigned a -direction . For any pair of oriented geodesic segments and emanating from the same point , the difference of their -directions is smaller than their comparison angle;
-
b)
Given , denote by the finite set given by all of the possible distances between points in their orbit. The angle between any two geodesic segments and lies in the finite set .
-
c)
There is a collection of isometric embeddings that preserve -directions and that is closed under precomposition by isometries in . Each image is called an apartment of . Each geodesic segment, ray and complete geodesic is contained in an apartment.
-
d)
Coordinate charts are compatible in the sense that, when defined, is the restriction of an isometry in the Weyl group .
Remark 8.2.
Many different, though equivalent, sets of axioms of Euclidean -buildings appear in the literature. For a detailed discussion we refer the reader to [BS14].
It follows immediately from the axioms that any two points are contained in a common apartment and the distance between them coincides with the Euclidean distance computed inside . Moreover, we can define a (germ of a) Weyl chamber in as the image of a (germ of a) Weyl chamber in under some chart . The following property will be useful.
Proposition 8.3 ([Par12]).
Two opposite Weyl chambers based at are contained in a unique apartment.
The boundary at infinity of is defined as the set of equivalence classes of geodesic rays, where two rays are equivalent if they remain at bounded distance. Given any and , there is a unique geodesic ray in the equivalence class of starting at .
Given a point , and two geodesic segments such that , the angle between them is the quantity
where denotes the angle of the Euclidean comparison triangle. This induces a distance on the set of equivalence classes of geodesic segments emanating from , where two segments are identified if the angle between them is zero.
8.2. Asymptotic cone of
We denote by the symmetric space endowed with the distance induced by its homogeneous Riemannian metric. The construction of the asymptotic cone of and, more generally, of any metric space relies on the choice of a non-principal ultrafilter.
Definition 8.4.
A non-principal ultrafilter is a finitely additive probability measure on such that
-
(1)
for every ;
-
(2)
for every finite subset .
As we are interested in the behavior as , we only consider non-principal ultrafilters each supported on a countable subset of whose only limit point in is . Non-principal ultafilters allow us to consistently define limits of bounded sequences without passing to subsequences. Precisely, a family of points in a topological space is said to have a -limit , denoted by if for each neighborhood of we have .
Definition 8.5.
Let be a base point in and let be a sequence of scaling factors. Fix a non-principal ultrafilter . The asymptotic cone of is the metric space where
-
(1)
points in are equivalence classes of families such that is bounded. Here, two families are equivalent if ;
-
(2)
the distance between two points and is defined as
By work of [Tho02], the asymptotic cone of the symmetric space is actually, up to isometries, independent of the choice of the ultrafilter (if we assume the continuum hypothesis) and of the base point . Moreover, the asymptotic cone of the symmetric space can also be interpreted as the Gromov-Hausdorff limit of the pointed sequence of metric spaces ([Gro81] [KL97]) and it is a non-discrete Euclidean building modelled on the affine Weyl group of . In particular, we are going to identify the model Euclidean plane with
Then the linear part of the Weyl group consists of reflections across the walls of equation for all . In particular, we identify with the boundary at infinity of the Weyl chamber
Given , we can find an apartment such that and . Then the distance between and is
where are the coordinates of .
Beside the Euclidean distance on an apartment, it is also useful to consider the -valued distance defined by
By a theorem of Parreau ([Par12]), the -valued distance on is the -limit of the analogously defined -valued distance on rescaled by . (Here this distance on relies on finding a flat that contains and .) In other words, if with , then
An apartment in can be obtained as the -limit of a sequence of flats in . Precisely, if are isometric parametrizations of a sequence of maximal flats of with the property that , then the family has an -limit which defines an apartment in . See [KL97] for more details.
A family of isometries of also induces an isometric action on provided that by setting
for any .
8.3. Limiting harmonic map to the building
Given a ray of cubic differentials , we consider the family of conformal harmonic maps that are equivariant under the corresponding Hitchin representations . We fix a non-principal ultrafilter , a base point and the sequence of scaling factors . We can consider the maps to take values in the re-scaled metric spaces .
Proposition 8.6.
The family converges to a Lipschitz equivariant harmonic map . The family of holonomy maps -converges to an isometry of , and is equivariant with respect to .
Proof.
By [DM06, Theorem 1.2], it is sufficient to show that energy of the maps grows as . Since is conformal, this amounts to estimating the area of , where is a compact fundamental domain for the action of . Now, the induced metric on the minimal surfaces can be written in terms of the Blaschke metric as (see [DL19])
| (8.1) |
hence is uniformly bi-Lipshitz to and the result follows from Lemma 3.2.
The -convergence of the holonomy maps follows from Remark 3.19 in [Par12]. The -equivariance of follows from the fact that each is equivariant with respect to . ∎
The behavior of the limiting harmonic map is well-known outside the zeros of the cubic differential .
Theorem 8.7.
For any that is not a lift of a zero of the cubic differential , there is a neighborhood centered at and an apartment with such that
-
i)
the induced distance on is ;
-
ii)
the limiting harmonic map sends inside ;
-
iii)
for any we have
where are the cube roots of (which are well-defined in ).
Proof.
Let be a -disk around that avoids neighborhoods of zeroes of . From Equation 8.1 and Lemma 3.4, we know that the induced metric rescaled by converges to uniformly on . To conclude that sends inside a single apartment, it is sufficient to show that is totally geodesic: this follows from an extendibility feature of flat neighborhoods in buildings (see [AB08, Theorem 11.53]). To this aim we show that for every we have that
| (8.2) |
Since the asymptotic cone is a space, this implies that the unique geodesic connecting and is entirely contained in the image of , hence is totally geodesic inside . We are thus left to prove Equation 8.2. Fix a natural coordinate on and let and be the coordinates of and in this chart. We then parametrize the geodesic connecting and as with so that . Recall that the map is the -limit of the maps that can be expressed as
for some . Indeed, from Section 2, we know that the equivariant harmonic map is simply given by the frame field of the affine sphere whose columns form at each point a real basis of that is orthonormal for (the lift of) the Blaschke metric. The matrices represent the change of frame between a real othonormal basis and the basis induced by the natural coordinate . Therefore,
By Proposition 5.6 (see also [Lof07]) and Corollary 7.3 the singular values of satisfy
where is a reordering of such that . Therefore, using that the distance in the symmetric space from the identity is given as the Euclidean distance to the logarithms of the singular values, we see
Since the proof of Equation 8.2 is complete.
Part is a direct consequence of part and the fact that the coordinates inside an apartment are given by the rescaled limit of the singular values of .
∎
We note, in particular, that outside the zeros of the map is smooth, so the set of its singular points is discrete, and is locally injective. We can also describe how these flats combine outside the zeros.
Proposition 8.8.
Let be a geodesic path which avoids all zeros of and which is not in the direction of a wall of the Weyl chamber. Then there is an apartment such that for all .
Proof.
Let . We want to show that . First, we note that is not empty because there is an apartment containing by axiom in the definition of buildings. Moreover, it is clear that if and then . Let and suppose by contradiction that . Let . By Theorem 8.7, there is a neighborhood and an apartment such that . Up to choosing a smaller , we can assume that for some open interval containing . Let and let . Because , there is an apartment such that . Let denote the Weyl chamber with tip at containing . Since , we can find two opposite Weyl chambers with tip at such that for and for . Note that the germs of the sectors and are opposite because the Weyl chambers and are equivalent and is clearly opposite to . Hence, by Proposition 8.3, there is a unique apartment that contains and . Therefore, we can find with such that . Hence contradicting the fact that . ∎
We intend to complete the description of by studying its behavior in a neighborhood of a zero of the cubic differential. Let us fix a coordinate chart around a zero of order of such that on the ball .
Theorem 8.9.
The image consists of the union of (cyclically ordered) bounded, closed sectors of angle with tip at such that
-
i)
for all (with indices intended modulo );
-
ii)
is a geodesic segment.
Proof.
Let small and denote by for the Stokes directions emanating from . Let be the sector in with tip at bounded by the directions and . The ball can be covered by standard half-planes obtained from the natural coordinates . Note that two such half-planes intersect in a sector of angle and in these coordinates the Stokes directions correspond to the angles and . By Corollary 4.4, we can write
| (8.4) |
Since
by the same argument as in the proof of Theorem 8.7 the -limit of the map sends inside an apartment in . Moreover, the image of a radial path in is a geodesic in the building. Indeed, if we parameterize such a path by with in the natural coordinate , then for all we have as in (8.3)
| (8.5) |
where is a permutation of such that . Because this reordering only depends on , which is constant along a radial path, and we already know that the image of is entirely contained in a flat, we deduce that the image is a straight segment of length
Since is arbitrary and is continuous, we can conclude that the image of each open sector between two consecutive Stokes rays must be contained in a closed sector with tip at of angle .
Now, recalling that for some product of unipotents , we can write
We first use this to show that the interiors of and are disjoint. This immediately implies that is a geodesic segment because is continuous and each is circular sector. The previous computation (8.3) about the behavior of radial paths under shows that we may have for some and only if they are at the same distance from the zero. However, in this present case we compute from the expression and Equation 8.4,
where we use again that the distance in the symmetric space from the identity is given as the Euclidean distance to the logarithms of the singular values. Now, note that in , at least one element on the diagonal is of the form for some : here we use that we can express and in a single coordinate, observing that the matrices are distinct, as well as that the unipotent has but a single off-diagonal nonzero entry. Hence,
and .
The same argument shows that and are disjoint as long as and the natural coordinates and do not coincide. Note that in this case the bound
can be made independent of as the diagonal terms in and never multiply to in the sectors containing and . Hence, in this case and are disjoint.
The only case that remains to be checked is when the natural coordinates on and are the same, when is a multiple of six. This happens when the angle between these sectors for the flat metric is at least . We can then apply Proposition 6.3 to guarantee that if the highest eigenvalue of is in position and the highest eigenvalue of is in position , then the -entry of is not zero. Therefore,
Again the bound can be made independent of , hence and are disjoint in this case as well. ∎
We are then able to describe the global behavior of the harmonic map .
Corollary 8.10.
Let denote the lift of the cubic differential to the universal cover . Let be the path distance induced by on . Then is an isometry.
Proof.
From Equation 8.1 and Lemma 3.4, we know that the induced metric rescaled by converges pointwise to and uniformly on every compact set on the complement of the zeros of . Let be a ball centered at a zero of order as in the setting of Theorem 8.9. Then the induced metric at is singular since the total angle is . Moreover, in the proof of Theorem 8.9 we showed that radial paths from the origin of -length are sent to geodesic arcs in the building of length . We conclude that is isometric to and the statement follows. ∎
Not only is the image of the limiting harmonic map intrinsically a singular flat surface, but inherits from the building a -translation surface structure as well, which allows us to reconstruct the original cubic differential , up to a positive multiplicative constant.
Corollary 8.11.
The image is naturally a -translation surface. This structure is induced precisely by the cubic differential .
Proof.
By Corollary 8.10, we know that is a singular flat surface with metric . This defines on a holomorphic cubic differential up to multiplication by . On the other hand, a neighborhood of a regular point of is contained in an apartment by Theorem 8.7 and the directions of the walls of the Weyl-chambers based at define six special directions. If we identify with
then these directions correspond to the lines for . These can be further divided into two groups, defined in accordance with which pair of the triple of coordinates coincide: if we identify the positive Weyl-chamber with
the two walls are given by and and the orbits of these lines under the Weyl-group divide the six walls into two categories, which we call type I and type II. There is only one choice of so that along the directions of type II the differential is real and positive. ∎
Remark 8.12.
Note that the only scaling factors that guarantee the existence of the harmonic map by Proposition 8.6 are . Different choices of such permissible lead only to homothetic asymptotic cones, hence the projective class of the translation surface is independent of such a choice of .
Finally, we relate the geometry of with the notion of weak-convexity introduced by Anne Parreau ([Par21]). In brief, she considered the -valued distance on and defined -geodesics as those paths such that for all
Theorem 7.5 shows that the image under of a geodesic for the flat metric is a geodesic for the distance . We deduce the following:
Corollary 8.13.
The surface is weakly convex.
The argument proves a conjecture of Katzarkov, Noll, Pandit and Simpson ([KNPS17], Conjecture 8.7) in the context of harmonic maps to buildings arising from limits of Hitchin representations in along rays of holomorphic cubic differentials. (Formally, this conjecture concerns the “Finsler” distance, obtained by taking a maximum of the vector-valued distance described above, and asks that the image be geodesic in that distance. Here the geodesic nature holds for the vector-valued distance by Theorem 7.5, and such “geodesics” for the two vector-valued distance are clearly geodesics for the Finsler distance. The statement for the Finsler distance then follows.)
Remark 8.14.
Note that is not totally geodesic in . One can see this by considering a -geodesic arc which passes through a zero of and makes an angle of more than at that zero. The geodesic connecting the endpoints of that arc lies in an apartment containing the endpoints and is necessarily Euclidean, i.e. has no interior point with an angle between incoming and outgoing directions other than .
9. Epilogue: Triangle groups
We conclude with an example. Of course, for a closed surface of high genus, the Labourie-Loftin parametrization of the Hitchin component is given as the cubic differential bundle over the Teichmüller space of . Thus, even with the results of this paper, an analysis of the limits of this component would require an understanding of the dependence of a diverging sequence of representations on the rays that include them. On the other hand, for triangle groups, we may completely describe the compactification of the Hitchin component.
9.1. The Hitchin component for triangle groups.
Consider the oriented triangle group . It is hyperbolic if . Denote the quotient of by by . Choi-Goldman have shown that the space of convex real projective structures has real dimension 2 for and [CG93]. We call these triangle groups projectively deformable. The general case of the dimension of the Hitchin component for triangle groups was settled by Long-Thistlethwaite [LT19]. Recently, more closely to our point of view, Alessandrini-Lee-Schaffhauser [ALS22] use Higgs bundles to study Hitchin components for orbifolds. In particular, for each projectively deformable group, the space of cubic differentials has complex dimension one, and we can extend our techniques to give a compactification of the Hitchin component in these cases.
We briefly address cubic differentials for oriented triangle groups.
Lemma 9.1.
Let be integers at least 2. The complex dimension of the space of cubic differentials on is 1 if and only if and is 0 if any of is 2.
Proof.
Consider a local coordinate with mapping to an orbifold point of order . Then is a holomorphic coordinate on the orbifold. The condition for a holomorphic cubic differential near to descend locally to the orbifold is that it can be written as a holomorphic cubic differential in away from . Thus descends if and only if and is a multiple of .
The case and leads to a pole of order 2 in , as . The Riemann surface formed by treating the 3 orbifold points of as smooth points has genus zero and 3 distinguished points at the triangle vertices of order . Thus we seek cubic differentials on with poles of order at most 2 at 3 points, thus in a family with one complex parameter. Other values of require lower order poles (or zeros) at the relevant points. There are no nonzero cubic differentials in these cases, in particular when any of is 2. ∎
Proposition 9.2.
For and , the real projective structures on are parametrized by the complex scalings of the nonzero cubic differential .
The fundamental domain of an oriented triangle orbifold consists of two adjacent triangles, each with vertices at the points.
Lemma 9.3.
Let . A singular Euclidean structure on the orbifold is induced by a nonzero holomorphic cubic differential if and only if the triangles with vertices are equilateral.
Proof.
Consider the Euclidean structure given by forcing the given triangles to be equilateral, and let be a flat conformal coordinate on one such triangle. Analytically continue the cubic differential to the other triangle of the orbifold. Then we may check the resulting cubic differential is holomorphic on the orbifold, as any monodromy around the orbifold points amounts to translations and rotations by third roots of unity. Such a scalar multiple , for , is also a holomorphic cubic differential, and together these form the one-dimensional complex vector space of all cubic differentials. ∎
Remark 9.4.
On each of these triangle orbifolds , it is useful to choose a particular representative . Give a consistent orientation to the equilateral triangles forming . We choose so that the sides of the triangles have length 1 and at any given vertex the outgoing edges correspond to the directions on which is real and positive.
Theorem 9.5.
Consider a family of cubic differentials on a closed surface , where and . Let be the corresponding family of Hitchin representations. Then the limiting data in terms of singular values and eigenvalues converge to those determined in the limit by the ray . In particular, Theorems A and B hold in this setting.
Outline of proof.
Assume for simplicity that . For a given free homotopy class of loops in , consider the cubic differential and the piecewise geodesic path considered in Figure 2, modified from a geodesic path along saddle connections in Stokes directions. We compute the holonomy along as in Theorems 7.1 and 7.5.
The main consideration is to ensure that we have uniform estimates in as . Note that the Blaschke metric is independent of , and the connection form in (2.2) depends on only through the cubic differential . The upshot is that varies in Equation 5.1 and thus in terms of the holonomy along linear paths in Proposition 5.6. It is straightforward to show that the error terms in Proposition 5.6 are uniform in as , as can be seen in terms of the proof in [Lof07]. In other words, the error term of the form satisfies
| (9.1) |
uniformly in as .
We also check that the estimates of [DW15] are uniform for near as . Recall, for the path , in the case in which at least one arc has a -Stokes direction as an endpoint, we modify the path to to avoid all such endpoints. Thus the same is true in a neighborhood of as . There the estimates of [DW15] are uniform in compact sets away from the Stokes directions, in terms of the frames needed in Corollary 4.4 and Theorem 4.6. The unipotent terms are also continuous as , as they are determined by the geometry of the limiting (in this case, regular) polygon, which varies continuously.
With these estimates in hand, the quantity we consider is the entry-wise norm of the holonomy matrix, as given in (7.4). Our first concern is that the largest terms in each matrix match up with positive terms in the adjacent unipotent matrices in the product in (5.3). This remains true by continuity of the . If contains any saddle connection along a wall of the Weyl chamber, then there is more than one largest eigenvalue of the holonomy along for . This is no longer true if . We must consider the possibility then that this entry-wise norm may jump by a positive bounded factor, in the case that in (7.4). Fortunately the limit we take in Theorem 7.5, in terms of taking a logarithm and dividing by , is insensitive to multiplication by a positive quantity bounded away from 0 and .
Finally, we must ensure that the largest terms in the product of holonomy matrices, upon taking the limit, are not affected by the various error terms we accumulate. This is exactly the uniformity condition (9.1). ∎
Corollary 9.6.
The previous theorem holds for a closed oriented orbifold of hyperbolic type.
Proof.
Consider a smooth finite orbifold cover and lift the cubic differential to . Then each complex scalar multiple of the cubic differential is also invariant under the orbifold deck transformations. ∎
We next define a compactification of the Hitchin component of representations of a deformable triangle group into .
First we identify the Hitchin component with as in Proposition 9.2. We then consider the map
where denotes the -largest singular value and is the representation corresponding to . By [Kim04, Theorem B], this map is injective.
Next we also embed in as follows. For each , we consider the harmonic map that results as the -limit of the family of harmonic maps parameterized by a ray of cubic differentials . Then for each such map , we compute the Weyl-chamber lengths of a representative a curve class by considering the image in the principal Weyl chamber of , following the prescription in Corollary 8.11. Naturally this defines a map .
This map is also an embedding: to see this, consider the canonical cubic differential defined in Remark 9.4, and a fixed curve class whose whose -geodesic representative makes an angle of with the positive -axis in a fixed natural coordinate chart. (Here we take so that the largest entry in the vector is .) Then for the “rotated” cubic differential , the largest entry changes to . We regard as the boundary of the complex plane in the usual way, and assert that the usual compactification is taken homeomorphically to its image in . This is proved in the next corollary.
Corollary 9.7.
Consider a projectively deformable triangle group. Then the map above provides a compactification of the Hitchin component of representations of .
Proof.
We need to show that there is a subatlas of boundary charts for the compactification comprising images of a segment of the circle and a sector in defined by that range of angles; this amounts to showing that for a family , as and , the images in of the representations associated to converge to those of the harmonic map . This is the content of Theorem 9.5 and that the map defined on the limiting circle defined by Corollary 8.11 depended only on the limit of the representations associated to the ray . ∎
A compactification in terms of harmonic maps is more delicate. Proposition 8.6 associates to a family of harmonic maps parameterized by a ray of cubic differentials a limiting harmonic map , whose image (cf. Corollary 8.11) is a -translation surface, isometrically embedded into . However, if more general diverging families of harmonic maps are considered, corresponding to families with and , their -limit (which exists by the same argument as in Proposition 8.6) are not precluded from depending on the particular family and not only on . We may refer to the more classical Teichmüller theory case, where equivariant harmonic maps are parameterized by a family of quadratic differentials . In that case,
if is a ray, always converges to an equivariant harmonic map to a real tree given by projection onto the leaves of the vertical foliation of ([Wol95]).
What is independent of the family is the projective class of the vertical foliation of the Hopf differential of the limiting harmonic map. In our setting, the harmonic map approach identifies the boundary points of a compactification of the Hitchin component for with projective classes of -translation surfaces.
(The Euclidean triangle group also can studied from this point of view. There is no hyperbolic structure on this orbifold, but there is still a nowhere-vanishing cubic differential, which explicitly leads on the orbifold universal cover to the Ţiţeica example. As the cubic differential scales to infinity, the Ţiţeica scales as well, and the limiting harmonic map into the real building simply covers a single apartment.)
Remark 9.8.
We conclude with an informal remark. In the setting of these triangle groups, the limiting harmonic maps to buildings take on enough of a combinatorial nature that we may display how some of the constructions in Section 8 apply.
The simplest triangle group for this is the group. One can visualize the action of this group on the hyperbolic plane in terms of a triangulation: two of the vertices of each triangle are vertices with a star of 6 triangles, and the remaining, say special, vertex is a vertex with a star of eight triangles. Distinct special vertices of adjacent triangles share the same opposite edge.
For the cubic differential defined in Remark 9.4, the harmonic map takes each triangle to an equilateral triangle in a building. In terms of the local geometry of the map, the six triangles in the image of the star around a non-special vertex will lie in a common apartment in the building, but the eight triangles in the image around a special vertex cannot, due for example to cone angle considerations. We return to this local geometry momentarily.
More globally, we comment a bit on some apartments which meet the image of the harmonic map. Note that a geodesic segment between (images of) special vertices in adjacent triangles – this segment will meet the opposite side orthogonally – may be extended to an infinite geodesic which subtends one of the angles, choose it always to be on the right, at each special vertex of exactly . This infinite (oriented) geodesic is the boundary of a Euclidean strip of parallel geodesics in the image of the harmonic map whose preimages limits on a pair of distinct endpoints. Each of these strips embed isometrically in an apartment which meets the image of the harmonic map in a region that contains the strips. Since there is a geodesic segment between images of special vertices that bisects each triangle in the star of the image of a special vertex, a neighborhood of the image of a special vertex is covered by eight such strips, and two (non-disjoint) strips meet in a rhombus (angles alternately and ) whose vertices are images of special vertices.
Note that each such strip meets four of the triangles in the star around the image of a special vertex so that the apartment containing this strip meets those four triangles, i.e. those four adjacent triangles around the image of a special vertex are in an embedded flat in the building. On the other hand, five adjacent triangles around that image vertex cannot be in an embedded flat (nor apartment) since that flat would then force there to be a sixth triangle with one edge shared with the terminal triangle and one shared with the initial triangle. That sixth triangle would combine with the three remaining image triangles in the image to form an embedded cone in the building of cone angle , which cannot exist in an NPC simplicial complex (e.g. points equidistant but on opposite sides of the cone point are joined by distinct geodesics on opposite sides of the cone point).
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JL: Department of Mathematics and Computer Science, Rutgers-Newark
E-mail address: [email protected]
AT: Department of Mathematics, University of Pisa
E-mail address: [email protected]
MW: School of Mathematics, Georgia Institute of Technology
E-mail address: [email protected]