Chapter 4
The energy spectrum of grafted surfaces
4.1 Introduction
In this chapter we continue the study of the energy spectrum that was introduced in Chapter 3. We will consider the energy spectrum for a particular class of metrics. Namely, the Thurston metrics on grafted surfaces. Such metrics are obtained by cutting a hyperbolic surface along a simple closed geodesic loop and gluing in a Euclidean cylinder (see Section 4.2.3).
We regard this setting as a ‘toy model’ for the study of the energy spectrum of Hitchin representations. We hope to gain some insight into the general properties of the energy spectrum by studying it in a simpler context. The particular question we are interested in is whether the minima of the energy spectrum are coarsely unique. More precisely, does there exists a constantD >0 such that any two minimisers of the energy spectrum of a Hitchin representation are no more thanD apart (measured in the Teichm¨uller distance). We think of this question as a coarse version of Labourie’s conjecture.
With this question in mind we consider the energy spectrum of grafted surfaces. In this setting the minimum of the energy spectrum is always unique (see Lemma 4.3.3). For this reason we look at points in Teichm¨uller space which almost minimise the energy spectrum. Our goal is to prove that such points can not lie very far from the true minimiser.
We consider a surfaceS,σa hyperbolic metric on that surface andγ⊂S a simple closed geodesic loop. Then the grafted surface Grt·γ(σ) is obtained by cuttingSalongγand gluing in a cylinder of heighttand circumference`σ(γ) (a more detailed definition is given in Section 4.2.3). We will examine the energy spectra of the grafted surfaces Grt·γ(σ) fort≥0.
The results of [DK12] show that the underlying conformal structures of the family{Grt·γ(σ)}t≥0 lie within bounded distance from a Teichm¨uller geodesic ray inT(S). Moreover, the length of the curveγ goes to zero as t→ ∞ and the amount of twisting aroundγ that occurs is bounded by a multiple oft. We expect that the points in Teichm¨uller space that almost minimise the energy spectrum of a grafted surface behave similarly. The main result of this chapter, Proposition 4.3.4, confirms that this is indeed the case.
As in Chapter 3 letS be a closed and oriented surface of genusg≥2. We denote byχ(S) = 2−2g its Euler characteristic. IfX ∈ T(S) andγ⊂S is a closed curve, then we denote`X(γ) =`σ(γ), whereσis any hyperbolic metric onS that represents the point X in Teichm¨uller space. Furthermore, we will denote byd:T(S)× T(S)→Rthe Teichm¨uller distance on Teichm¨uller space (see [Hub06, Definition 6.4.4]).
4.2.1 Measured laminations
A reference for the material discussed in this section is [Mar16, Section 8.3].
LetS be equipped with a hyperbolic metric. Ageodesic lamination onS is a closed subset consisting of a disjoint union of complete geodesics. A transverse measure on a geodesic lamination assigns a Borel measure to each arc that is transverse to the lamination in such a way that it is invariant under translations along the lamination. We call a geodesic lamination equipped with transverse measure ameasured lamination.
LetML(S) be the space of measured laminations. It can be equipped with a topology that does not depend on the choice of hyperbolic metric onS. By multiplying the transverse measures by positive real numbers we obtain an action ofR>0 onML(S). When taking the quotient of this action we obtainPML(S), the space of projectivised measured laminations, which is compact.
The set of weighted simple closed geodesics is dense inML(S). If we define the function`X(·) to scale as`X(s·γ) =s·`X(γ), then it extends to a continuous function onML(S). Similarly, EX(·) also extends to a continuous function on ML(S) if we take the scaling to beEX(s·γ) =s2·EX(γ).
4.2.2 Fenchel–Nielson coordinates
We now describe theFenchel–Nielson coordinates on Teichm¨uller space which provide global coordinates forT(S). We begin by choosing a so-called marking of the surface S which consists of two pieces of topological data. First, let {γ1, . . . , γ3g−3}be a collection of pairwise disjoint oriented simple closed curves.
This determines a pants decomposition of the surface S. Secondly, we pick another set{η1, . . . , ηk} of simple closed curves in S, called the seams, such that the intersection of the union of seams with any pair of pants in the pants decomposition consists of three disjoint arcs connecting the boundaries of the pair of pants pairwise.
The Fenchel–Nielson coordinates associated to this choice of marking consist of 3g−3 length parameters and 3g−3 twist parameters. The length parameters of a pointX ∈ T(S) are simply (`X(γ1), . . . , `X(γ3g−3)). We define the twist parameter aroundγi ofX by first selecting a seam ηwhich intersects γi. Each pair of pants in the pants decomposition of S has a hyperbolic metric that is uniquely determined by its boundary lengths. In this metric each pair of boundary curves has a unique shortest geodesic arc connecting them. The curve η is homotopic to a loop consisting of a concatenation of these geodesics arcs and geodesic arcs that run along the curvesγi. Letmbe the signed length in
X of the geodesic arc alongγi (signed according to whether the arc runs along or against the orientation ofγi). The twist parameter forγi is now defined as sX(γ) =m/`X(γi).
4.2.3 Complex projective structures and grafting
Acomplex projective structure onS consists of a maximal atlas of charts that take values inCP1 and whose transition maps are restrictions of M¨obius trans-formations. As M¨obius transformations are in particular holomorphic we see that a complex projective structure determines an underlying complex structure onS.
On a complex projective surface we can define theThurston metricas follows.
The norm of a tangent vectorv∈T S in the Thurston metric is the infimum of the hyperbolic norms of tangent vectorsv0∈TH2for which a complex projective mapf:H2→S exists such thatdf(v0) =v. To compare the Thurston metric with the hyperbolic metric we note that on a surface of genus at least two the hyperbolic metric coincides with the Kobayashi metric. Hence, the hyperbolic metric can be described similarly as the Thurston metric with the modification that we allowf to be any holomorphic map rather than only a projective map.
It immediately follows that the hyperbolic metric is bounded from above by the Thurston metric.
Projective structures onS can be build from hyperbolic structures by cutting along a simple closed curve and gluing in a flat cylinder. This process is called grafting. To make this notion precise considerσa hyperbolic metric onS and letγ⊂(S, σ) be a simple closed geodesic. Fort >0 we look at
A(t) =e {z=r·eiθ∈C|θ∈[π/2, π/2 +t]}
which we will consider as multisheeted ift >2π. The projective cylinder A(t) is obtained as the quotient ofA(t) by the action of the groupe hz7→e`σ(γ)zi ⊂ PSL(2,C). A projective structure is now obtained by cutting (S, σ) alongγ and gluing inA(t) alongγ.
We will denote by Grt·γ(σ) the Thurston metric of the projective structure on the surfaceS that is obtained by the grafting construction. By grt·γ(σ)∈ T(S) we will denote the point in Teichm¨uller space determined by the underlying complex structure.
The Thurston metric on the grafted surface coincides with the original hyperbolic metricσ onS−γ and is Euclidean on the glued cylinder. In this metric the cylinder has circumference`σ(γ) and height t. From the jump in curvature we see that the Thurston metric can not be smooth. It is however of classC1,1 (see [KP94, Section (5.4)]).
The collapsing map of a grafted surface π: (S,Grt·γ(σ)) → (S, σ), which collapses the grafted cylinder onto the geodesic γ, is 1-Lipschitz. From this observation the following lemma follows immediately (see also [Kim99, Lemma 1]).
Lemma 4.2.1. For any closed curve η⊂S we have`Grt·γ(σ)(η)≥`σ(η).
Similarly, because the Thurston metric bounds the Kobayashi metric from above it follows that the identity map id : (S,Grt·γ(σ))→(S,grt·γ(σ)) is also 1-Lipschitz. This observation yields the following result.
Lemma 4.2.2. For any closed curve η⊂S we have`Grt·γ(σ)(η)≥`grt·γ(σ)(η).