ASYMPTOTICALLY POINCARÉ SURFACES IN QUASI-FUCHSIAN MANIFOLDS. DRAFT VERSION 5 August 29, 2018

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1 ASYMPTOTICALLY POINCARÉ SURFACES IN QUASI-FUCHSIAN MANIFOLDS DRAFT VERSION 5 August 9, 018 Abstract. We prove some things 1. Introduction Epstein in [Eps84] describes a map that takes a domain Ω in CP 1 and a conformal metric σ on Ω and gives an Epstein surface Ep σ : Ω H 3 in hyperbolic space. We use this construction to study surfaces in quasi-fuchsian manifolds. Indeed, given a quasi-fuchsian group Γ and a component Ω of the domain of discontinuity, we define the Poincaré surface of Ω to be the Epstein surface for the Poincaré metric on Ω. This surface is Γ-invariant and its quotient in M = H 3 /Γ we call the Poincaré surface of X, where X = Ω/Γ is the Riemann surface at infinity of an end of M with complex structure induced from Ω. Generalizing this, with M and X as above, we say a surface i : S M is an Epstein surface of X if it is covered by an Epstein surface Ω H 3 in the sense that there is a Γ-invariant conformal metric σ on Ω and a diffeomorphism ϕ : Ω/Γ S such that π M Ep σ = i (ϕ π X ). Here π M and π X are the quotient maps. An Epstein surface of X is a surface obtained by any conformal metric while the Poincaré surface is the special case when the conformal metric is the Poincaré metric. When S is an Epstein surface, the given conformal metric on Ω induces a conformal metric on X, which we call the conformal metric at infinity. We say a family (S ɛ ) ɛ (0,1) of essentially Epstein surfaces is an asymptotically Poincaré family if the corresponding conformal metrics at infinity converge projectively to the hyperbolic metric h in C on X as ɛ 0. See Section 4 for a precise definition. The first and second fundamental forms of an asymptotically Poincaré family of surfaces may be thought of as forming two paths in the Teichmüller space of the surface at infinity. We prove the following. Theorem 1.1. In a quasi-fuchsian manifold, let S ɛ, for ɛ (0, 1), be an asymptotically Poincaré family of surfaces with surface at infinity X. If I ɛ and II ɛ are the corresponding first and second fundamental forms then [I ɛ ] [h] and [II ɛ ] [h] as ɛ 0 and the difference of the tangent vectors to these paths at [h] is determined by the holomorphic quadratic differential at infinity φ: [I ɛ ] [II ɛ ] = c Re(φ). Here c is a constant and [g] is the equivalence class of the Riemannian metric g in the Teichmüller space of X. 1

2 This holomorphic quadratic differential at infinity φ on X measures the difference between the complex projective structure on X induced from Ω and its standard Fuchsian projective structure obtained by uniformizing X as a hyperbolic surface. Consider a family of essentially Epstein surfaces that leave an end of M corresponding to X. The conformal metrics at infinity diverge as the surfaces head towards X. One may attempt to rescale them so as to get a convergent family of metrics. The asymptotically Poincaré familes are those whose metrics, once rescaled, converge to the hyperbolic metric. There is actually a distinguished asymptotically Poincaré family of surfaces which consists of surfaces whose metrics at infinity are multiples of the hyperbolic metric. These surfaces are parallel copies of the Poincaré surface of X. The surfaces we are interested in are in a sense asymptotic to this family. Indeed, the surfaces in an asymptotically Poincaré family are asymptotically parallel, as shown in Lemma 4.1. As an application of our results we answer a conjecture posed by Labourie. In [Lab91] he proves that an end of a quasi-fuchsian manifold admits a unique foliation by k-surfaces: surfaces S k, for k in ( 1, 0), such that the Gaussian curvature of S k is identically k. In [Lab9], Labourie notes how k-surfaces may be interpreted as a path in Teichmüller space and he asks what the tangent vectors to the paths [I k ] and [II k ] are at k = 0. He conjectures that they are related to the holomorphic quadratic differential at infinity. We show in Section 5 that these S k form an asymptotically Poincaré family of surfaces. And therefore, Theorem 1.1 proves his conjecture and gives explicitly the relationship: Theorem 1.. Let I k and II k be the first and second fundamental forms of the k-surface. Let φ be the holomorphic quadratic differential at infinity on X. Then, as k 0, the difference of the tangent vectors to [I k ] and [II k ] in Teichmüller space is given by [I k ] [II k ] = Re(φ). We consider another application of Theorem 1.1 in Section 6. The work of [authors] in [ref] show that the ends of a quasi-fuchsian manifold admit foliations by surfaces of constant mean curvature. For mean curvatures 1 + k, with k near zero, we prove that the family of surfaces with this mean curvature forms an asymptotically Poincaré family. Therefore, Theorem 1.1 describes the behavior in Teichmüller space. Theorem 1.3. Let I k and II k be the first and second fundamental forms of the Epstein surfaces with constant mean curvature 1 + k. Let φ be the holomorphic quadratic differential at infinity on X. Then, as k 0, the difference of the tangent vectors to [I k ] and [II k ] in Teichmüller space is given by [I k ] [II k ] = Re(φ).. Preliminaries.1. Quasi-Fuchsian Manifolds. For a discrete subgroup Γ of PSL(, C), let Λ(Γ) denote its limit set and Ω(Γ) = CP 1 Λ(Γ) the domain of discontinuity. If the limit set Λ(Γ) is a Jordan curve then the domain of discontinuity is separated into two domains Ω(Γ) = Ω + Ω. When Ω + and Ω are each Γ-invariant, the subgroup Γ is called quasi-fuchsian and the manifold M = H 3 /Γ is a quasi-fuchsian manifold. The convex hull of Λ(Γ) in hyperbolic space is Γ-invariant and its quotient in M

3 is called the convex core. The complement of the convex core in M consists of two ends diffeomorphic to Ω ± /Γ (0, ), and the surfaces Ω ± /Γ are called the surfaces at infinity for their respective ends. We will be focusing on one end of a quasi-fuchsian manifold with smooth surface at infinity Ω/Γ... Projective Structures and the Schwarzian derivative. A complex projective structure on a surface S is an atlas of charts to CP 1 whose transition functions are restrictions of Möbius transformations. A projective structure induces a complex structure on S since Möbius transformations are holomorphic. The surfaces at infinity Ω/Γ is both a Riemann surface X and a complex projective surface Z. The Riemann surface X also has its standard Fuchsian projective structure Z F coming from the uniformization theorem. The difference between these two projective structures is the quadratic differential φ = Z Z F, which is holomorphic with respect to X (see [Dum09]). We call this the holomorphic quadratic differential at infinity. Here is a more concrete description of φ. Recall that if f is a locally injective holomorphic function on a domain in C or CP 1 then the Schwarzian derivative of f is the holomorphic quadratic differential ( (f ) S(f) = f 1 ( ) ) f f dz. Holomorphically identify the universal cover of X with the upper half plane H. Then φ = S(f), where f : H Ω is a Riemann map, is a Γ-invariant tensor that induces the holomorphic quadratic differential at infinity φ. 3. Epstein Surfaces 3.1. Conformal Metrics. A Riemann surface structure on a surface S distinguishes a conformal class of metrics on S, called conformal metrics. These are metrics σ that, in a complex coordinate chart z, can be written as σ = e η dz, for some real valued function η. This η is called the log density function of σ. The metric σ is of class C k if its log density η is a C k function. For a C conformal metric σ = e η dz, the Gaussian curvature is the function K(σ) = 4e η η z z. For a compact Riemann surface with genus larger than 1, this conformal class contains a unique hyperbolic metric of constant Gaussian curvature 1, which we denote by h. 3.. The Visual Metric. A natural trivialization of the unit tangent bundle of hyperbolic space UH 3 is given by G : UH 3 H 3 CP 1 by G(p, v) = (p, lim t exp p (tv)). We write G p for G(p, ) : U p H 3 CP 1 and using this we can push forward the induced metric on U p H 3 to CP 1. The resulting conformal metric V p on CP 1 is called the visual metric from p. As an example, the visual metric from the origin in the ball model V 0 is just the spherical metric on S, which is identified with CP 1 in this model. In general, if M is a Möbius transformation taking 0 to the point p, then V p = M V 0. 3

4 3.3. The Epstein Map. In [Eps84] Epstein gives a way of constructing surfaces in hyperbolic space from domains in the Riemann sphere. These Epstein surfaces will be our main tool for studying asymptotically Poincaré surfaces. Our exposition of this Epstein construction follows [Dum17] (see also [And98]). Theorem 3.1 (Epstein [Eps84]). Let Ω be a domain in CP 1 and σ a C k conformal metric on Ω, then there exists a unique C k 1 map Ep σ : Ω H 3, called the Epstein map of Ω for the metric σ, such that for all z Ω, V Epσ (z)(z) = σ(z). Moreover, the image of a point z depends only on the 1-jet of σ at z. Epstein s original construction gave a formula for Ep in the ball model of hyperbolic space. Dumas gives an SL(, C)-frame field description of the map as follows. Choose an affine chart z on CP 1 that distinguishes a point 0 Ω and / Ω. Then, on the geodesic in H 3 with ideal endpoints 0 and, there exists a unique point p such that the visual metric from p at 0 is the Euclidean metric of this affine chart, V p (0) = dz. The Epstein map is an SL(, C)-frame orbit of this point. Proposition 3. ([Dum17]). On a domain Ω in CP 1 write σ = e η dz. Define the SL(, C)-frame field Ẽp σ : Ω SL(, C) by ( ) ( ) ( ) 1 z 1 0 e η/ 0 Ẽp σ (z) = 0 1 η z 1 0 e η/, then the Epstein map is given by Ep σ (z) = Ẽp σ(z) p. Even though we call the image an Epstein surface, the Epstein map need not be an immersion. Indeed, if σ is itself a visual metric then the Epstein map for σ is constant. However, the lift of Ep σ from Ω to the unit tangent bundle of hyperbolic space given by Êp σ (z) = (Ep σ (z), G 1 Ep σ (z) (z)) is an immersion. This lift can be thought of as providing a unit normal vector field for the Epstein surface even when the Epstein map is not an immersion. Indeed, this lift agrees with a unit normal vector field when the surface is immersed. Because the Epstein map is unique, it is natural with respect to the action of PSL(, C) in the following sense. Suppose M is a Möbius transformation, then the following diagram commutes: M (Ω, σ) (M(Ω), M σ) Ep Ep H 3 M H 3 That is, Ep M σ(m(z)) = M(Ep σ (z)). See [And98]. This allows us to define Epstein maps on certain quotients. Suppose in general that Γ is a subgroup of PSL(, C) acting freely and properly discontinuously on H 3 CP 1 leaving a domain Ω invariant. Then Ω/Γ inherits a Riemann surface structure. Call this structure X and let σ be a conformal metric on X. Lift this to σ on Ω, which is Γ-invariant. Then Ep σ : Ω H 3 is Γ-equivariant and therefore 4

5 descends to a map Ep σ : X H 3 /Γ. In particular, when Γ is a quasi-fuchsian group and Ω a component of the domain of discontinuity, each conformal metric σ on the surface at infinity X gives rise to a map from X into the quasi-fuchsian manifold M Parallel Surfaces. For a vector field V on H 3 define the time-t flow of V at a point p as Φ t (V )(p) = exp p (tv p ). Using the normal field given by the lift Êp σ, each Epstein surface gives rise to a parallel family by flowing in the normal direction. That is we have the flowed surfaces Φ t Êp σ(ω). In fact, these surfaces are themselves Epstein surfaces corresponding to scalar multiples of σ: Lemma 3.3 (Thurston, see [Eps84]). Let Ω be a domain in CP 1 and σ a conformal metric on Ω. Then Φ t Êp σ = Ep e t σ. That is, flowing the Epstein surface for σ for time t corresponds to taking the Epstein surface for the metric e t σ Schwarzian Derivatives of Conformal Metrics. In general, the Schwarzian derivative of one conformal metric with respect to another on a Riemann surface is the (, 0) part of the Schwarzian tensor defined by Osgood and Stowe in [OS9]. For two conformal metrics σ 1 and σ, if we write u for the log density of σ with respect to σ 1, i.e., σ = e u σ 1, then the Schwarzian tensor is (Hess(u) du du) (,0). In a coordinate chart z write σ i = e ηi dz, the Schwarzian derivative of σ with respect to σ 1 is the quadratic differential B(σ 1, σ ) = ( (η ) zz (η ) z (η 1 ) zz + (η 1 ) z) dz. As opposed to the Schwarzian derivative of a function, this need not be holomorphic. The two are related, though. For f locally injective and holomorphic, we have S(f) = B( dz, f dz ). We also have naturality f B(σ 1, σ ) = B(f σ 1, f σ ) (again, for f holomorphic) and a cocycle property B(σ 1, σ 3 ) = B(σ 1, σ ) + B(σ, σ 3 ). When σ is a conformal metric on a domain in C and B( dz, σ) = 0, there exists a constant a > 0 and A SL(, C) such that aa σ is the restriction of either the hyperbolic metric on the unit disk, the Euclidean metric on C, or the spherical metric on CP 1. Such metrics σ are called Möbius flat metrics and we will denote any one of them by g CP 1. The cocycle property gives us that the Schwarzian derivative of σ = e η dz relative to any Möbius flat metric is B(g CP 1, σ) = (η zz η z)dz. 5

6 3.6. Geometry of Epstein Surfaces. The first fundamental form of the Epstein surface for the metric σ is given by I(σ) = Ep σ(g H 3) for g H 3 the metric of H 3. It is given by I(σ) = 4 σ B(g CP 1, σ) (1 K(σ)) σ + (1 K(σ))Re(B(g CP 1, σ)). The second fundamental form (relative to the normal lift Êp σ) is II(σ) = 4 σ B(g CP 1, σ) 1 4 (1 K(σ) )σ K(σ)Re(B(g CP 1, σ)) (see [Dum17], equations 3. and 3.3). Here K(σ) is the Gaussian curvature of σ and B(g CP 1, σ) the Schwarzian derivative of σ with respect to a Möbius flat metric. With I(σ) and II(σ) we can compute the two principal curvatures; they are κ ± (I(σ)) = K(σ) (1 ± 4 σ B(g CP1, σ) ) (1 K(σ)) 16 σ B(g CP 1, σ). With these we can compute the Gaussian curvature by K(I(σ)) = 1 + κ + κ and the mean curvature by H(I(σ)) = 1 (κ + + κ ). We obtain and K(I(σ)) = 4K(σ) (1 K(σ)) 16 σ B(g CP 1, σ) H(I(σ)) = K(σ) 1 16 σ B(g CP 1, σ) (K(σ) 1) 16 σ B(g CP 1, σ). In the quasi-fuchsian setting, if σ is a Γ-invariant conformal metric on Ω then each term in the above equations is also Γ-invariant. This is maybe less clear for the quadratic differential B(g CP 1, σ) since the Möbius flat metric g CP 1 is not itself Γ- invariant. However, we see that for γ Γ we have γ B(g CP 1, σ) = B(γ g CP 1, γ σ) = B(γ g CP 1, σ), by naturality of Schwarzian derivatives of conformal metrics. The metric γ g CP 1 is still a Möbius flat metric, and so B(γ g CP 1, σ) = B(g CP 1, σ), implying B(g CP 1, σ) is Γ-invariant. Therefore, B(g CP 1, σ) induces a quadratic differential on X, which we will denote by B(σ). In summary of the above, we have the following Gaussian and mean curvatures of the Epstein surfaces in M. Lemma 3.4. The Gaussian curvature for the Epstein surface Ep σ : X M is given by 4K(σ) K(I(σ)) = (1 K(σ)) 16 σ B(σ), and the mean curvature by H(I(σ)) = K(σ) 1 16 σ B(σ) (K(σ) 1) 16 σ B(σ). These are now equations on the compact Riemann surface X. 6

7 4. Asymptotically Poincaré Families of Surfaces We say an embedded closed surface i : S M of genus greater than or equal to two is essentially Epstein if there exists a Γ-invariant conformal metric σ on Ω and a diffeomorphism ϕ : X S such that the diagram ϕ π Ω H 3 S Ep σ commutes. Here the π are the respective quotient maps Ω X and H 3 M. Since σ is Γ-invariant it induces a conformal metric σ on X and the Epstein map Ep σ descends to a map Ep σ : X M such that Ep σ = ϕ i. Hence, given an essentially Epstein surface, there is no loss of generality in assuming the domain of the embedding is X, and that the map is Ep σ for some conformal metric σ on X. Let Ep σ(ɛ) : X M for ɛ in (0, 1) be a family of essentially Epstein surfaces. We call this family asymptotically Poincaré if there exists a function f decreasing to zero and with nonvanishing derivative at zero, so that i f(ɛ)σ(ɛ) h as ɛ 0 in the set of twice continuously differentiable conformal metrics on X (??). Here h is the hyperbolic metric on the surface at infinity. In what follows we will write γ(ɛ) = f(ɛ)σ(ɛ) for the family of metrics which converge. Note that as ɛ tends towards zero, the family of surfaces is leaving the end of the manifold. The following lemma shows that the surfaces in an asymptotically Poincaré family are asymptotically parallel. Lemma 4.1. Suppose Ep σ(ɛ) is an asymptotically Poincaré family of surfaces. Let t > 0. If Φ t is the normal flow operator, then ) d M (Φ t (Êp σ(ɛ)(z)), Ep σ(e t ɛ)(z) 0 as ɛ 0 uniformly in z. That is, the distance between the surface Ep σ(ɛ) flowed for time t and the surface Ep σ(e t ɛ) tends towards zero as ɛ does. Proof. We work with the universal covers. A straight forward computation using the SL(, C)-frame definition of the Epstein map Ω H 3 gives the distance between the image of z under the Epstein map for the metrics σ = e η dz and τ = e λ dz as d H 3 (Ep σ (z), Ep τ (z)) = arctanh ( M π (e η e λ ) + 4 η z λ z (e η + e λ ) + 4 η z λ z where all functions are evaluated at z, which we have suppressed for brevity. Lift σ(ɛ) to σ(ɛ) and γ(ɛ) to γ(ɛ) on Ω. Write σ(ɛ) = e η(ɛ) dz and γ(ɛ) = e λ(ɛ) dz, then η(ɛ) = λ(ɛ) (1/) ln(f(ɛ)). Recall from Lemma 3.3 that Φ t Êp σ(ɛ) = Ep e t σ(ɛ). For ease of notation, let c = e t. Then we have the distance ) 7

8 between Ep c 1 σ(ɛ) and Ep σ(cɛ) can be simplified to arctanh (1 cf(ɛ) f(cɛ) eλ(cɛ) λ(ɛ) ) 4cf(ɛ)e λ(ɛ) λ z (cɛ) λ z (ɛ). (1 cf(ɛ) f(cɛ) eλ(cɛ) λ(ɛ) ) 4cf(ɛ)e λ(ɛ) λ z (cɛ) λ z (ɛ) Since γ(ɛ) converges in C, λ has a C limit. Therefore, the argument of arctanh converges to zero uniformly in z. Since the distance between the surfaces in the universal cover is converging to zero, and since the quotient map H 3 M is a local isometry, we get the Lemma. Next we give a useful condition on a family of conformal metrics giving rise to an asymptotically Poincaré family of surfaces. Proposition 4.. Given a family of conformal metrics σ(ɛ) on X, if there exists a differentiable function f decreasing to zero with nonvanishing derivative at zero, satisfying f(ɛ)σ(ɛ) h then there exists an ɛ 0 > 0 so that for ɛ < ɛ 0, the Epstein map Ep σ(ɛ) is an embedding. Hence, the Epstein surfaces Ep σ(ɛ) : X M, for ɛ < ɛ 0, form an asymptotically Poincaré family. Proof. The proof proceeds in two parts. We first show that for sufficiently small ɛ that the Epstein maps are immersions. Next, since X is compact, it suffices to show that for sufficiently small ɛ, then Epstein maps are injections. Together these prove embeddedness. We begin by showing the Epstein maps are eventually immersions. As the projection π : H 3 M is a local diffeomorphism, the Epstein map Ep σ(ɛ) : X M will be an immersion if and only if Ep π σ(ɛ) : Ω H 3 is an immersion. Hence, it suffices to work in the universal covers. Since the Epstein map is equal to Ẽp σ(ɛ) p, it suffices to check that for small enough ɛ, the Darboux derivative of Ẽp σ(ɛ) does not take values in the kernel of the orbit map of p. Writing σ(ɛ) = e η(ɛ) dz we can compute the Darboux derivative of Ẽp σ(ɛ) to be ( ) Ẽp 1 dẽp = i Im(η(ɛ) z dz) e η(ɛ) dz e η(ɛ) (η(ɛ) zz η(ɛ). z)dz + η(ɛ) z z d z i Im(η(ɛ) z dz) ( ) 0 The kernel of the orbit map for the point p is Asu()A 1 where A is 0 1/. So, the Epstein map for the metric σ(ɛ) will fail to be an immersion if there exists a point z such that the complex 1-form 4e η(ɛ) (η(ɛ) zz η(ɛ) z)dz + (1 K(σ(ɛ))d z restricted to the real tangent bundle T Ω has nontrivial kernel at z. This is equivalent to asking for 4 B(g CP1, σ(ɛ)) = 1 K(σ(ɛ). σ(ɛ) at a point z. But, for small enough ɛ, this is impossible. We have that 4 B(g CP1, σ(ɛ)) B(σ(ɛ), h)) 4 σ(ɛ) h h σ(ɛ) + φ h h σ(ɛ) 8

9 and by Nehari s Theorem the Schwarzian derivative φ = S(f) for the locally injective holomorphic function f is bounded by 3/ in the uniform hyperbolic norm. Consequently, 4 B(g CP1, σ(ɛ)) B(γ(ɛ), h)) f(ɛ)h 4 σ(ɛ) h γ(ɛ) + 3f(ɛ)h γ(ɛ), and this right hand side approaches zero when ɛ 0 since γ(ɛ) h in C and f(ɛ) 0. Contrary to this, 1 K(σ(ɛ)) = 1 f(ɛ)k(γ(ɛ)) 1 as ɛ 0. So, for small and 1 K(σ(ɛ)) to have the same norm. And so, for small enough ɛ the Epstein map for the metric σ(ɛ) is an immersion. We now prove that for sufficiently small ɛ the Epstein maps are injections. It will suffice to prove the principal curvatures are eventually strictly bounded in absolute value by 1, i.e., κ ± (I(σ(ɛ))) < 1. [References] The two principal curvatures of The Epstein surface for the conformal metric σ on the surface at infinity X are enough ɛ it is impossible for 4 B(σ(ɛ)) σ(ɛ) κ ± (I(σ)) = K(σ) (1 ± 4 σ B(σ) ) (1 K(σ)) 16 σ B(σ) For legibility, we will now denote dependence on ɛ by subscripts. For an asymptotically Poincaré family with conformal metrics σ ɛ we have σ ɛ = f(ɛ) 1 γ ɛ where f(ɛ) = 0 and γ 0 = h. Write K ɛ for K(γ ɛ ) and B ɛ for B(γ ɛ ). Then we want the bound κ ± (I(σ ɛ )) < 1 for sufficiently small ɛ. Lemma. For sufficiently small ɛ, κ(σ ɛ ) < 1. Proof. Substituting, we get κ ± (I(σ ɛ )) = f(ɛ) K ɛ (1 ± 4f(ɛ) Bɛ γ ɛ ) (1 f(ɛ)k ɛ ) 16f(ɛ) Bɛ γɛ Notice that the numerator is less than or equal to f(ɛ) Kɛ since something positive is begin subtracted. Also, the denominator is eventually positive since e.g., it is the determinant of I(σ ɛ ) and we have just shown the surfaces are immersed (for small enough ɛ) so that I(σ ɛ ) is positive definite. So only consider ɛ small enough so that this is true. Then κ ± < 1 will follow from which is equivalent to f(ɛ) K ɛ (1 f(ɛ)k ɛ ) 16f(ɛ) Bɛ γ ɛ < 1, f(ɛ) Kɛ < (1 f(ɛ)k ɛ ) 16f(ɛ) B ɛ γɛ f(ɛ) Kɛ < 1 f(ɛ)k ɛ + f(ɛ) Kɛ 16f(ɛ) B ɛ f(ɛ)k ɛ + 16f(ɛ) B ɛ < 1 γ ɛ. γ ɛ 9

10 and this is eventually true since f(ɛ) 0 and since K ɛ and B ɛ and γ ɛ have uniform limits. Lemma. For sufficiently small ɛ, κ ± (I(σ ɛ )) > 1. Proof. Since the denominator is eventually positive, we have that κ ± > 1 is equivalent to κ ± > 1 f(ɛ) Kɛ 1 8f(ɛ) B ɛ 16f(ɛ) B ɛ γ ɛ γɛ > 16f(ɛ) B ɛ γɛ 1 + f(ɛ)k ɛ f(ɛ) Kɛ f(ɛ) Kɛ 8f(ɛ) B ɛ 3f(ɛ) B ɛ f(ɛ)k ɛ > 0 γ ɛ f(ɛ)kɛ 8 B ɛ 3f(ɛ) B ɛ K ɛ > 0. Define ϕ ± (ɛ) : X R by γ ɛ ϕ ± (ɛ) = K ɛ 8 B ɛ γ ɛ γ ɛ γ ɛ + f(ɛ) [K ɛ 3 B ɛ ] and note that κ ± > 1 is equivalent to ϕ ± (ɛ) > 0 for sufficiently small ɛ We have ϕ ± (0) = 4 φ h. So immediately ϕ (0) = + 4 φ h > 0. For the other, we have φ sup X h < 3 by Nehari s Theorem, so ϕ + (0) = φ h > 3 > 0. γ ɛ Thus, ϕ ± (0) > 0, so by continuity there is an interval around ɛ = 0 so that ϕ ± (ɛ) > 0. This is equivalent to κ ± (I(σ ɛ )) > 1 for positive ɛ in this neighborhood. So, κ(i(σ ɛ )) < 1. This implies for sufficiently small ɛ that the Epstein maps are injections. All together we have the Epstein maps are eventually embeddings. Therefore, for sufficiently small ɛ, the family Ep σ(ɛ) forms an asymptotically Poincaré family. We now turn to our main result. In our setting it is natural to work with the Riemannian model of Teichmüller space as developed by [Fischer and Tromba???]. That is, we use T (S) = Met /Diff 0 P, where P is the set of smooth positive functions on S and Diff 0 is the group of smooth diffeomorphisms isotopic to the identity. However, the smooth topology is difficult to work with directly. So, we first work in the Sobolev setting Met s and H s of Sobolev tensors and functions of a fixed regularity s > 3. We then argue our results are independent of the chosen s. 10

11 Proposition 4.3. Suppose ϕ ɛ : S M is an asymptotically Poincaré family of surfaces. Let γ = fσ : [0, 1) Met s be the path of rescaled metrics thought of as taking values in a class of Sobolev metrics for a fixed s > 3. Then the first and second fundamental forms I σ : (0, 1) Met s and II σ : (0, 1) Met s satisfy I ɛ = 4fɛI(σ(ɛ)) h and II ɛ = 4fɛII(σ(ɛ)) h as ɛ 0. Moreover, I ɛ and II ɛ are differentiable at ɛ = 0 and their tangent vectors are given by I ɛ = γ + fh + fre(φ) and II ɛ = γ Proof. We have that γ : [0, 1) Met is continuous and differentiable at ɛ = 0. Therefore, as Met = s>3 Met s we also have that γ is continuous to Met s and differentiable at ɛ = 0. Then, we have I(σ(ɛ)) = I( 1 f(ɛ) γ(ɛ)) is equal to 4f(ɛ) B(γ(ɛ)) γ(ɛ) + 1 4f(ɛ) (1 f(ɛ)k(γ(ɛ))) γ(ɛ) + (1 f(ɛ)k(γ(ɛ)))re(b(γ(ɛ))), which is a smooth tensor independent of s. Since f : [0, 1) R and γ : [0, 1) Met s are differentiable at 0 and since K : Met s H s and B : Conf s Γ s (Σ ) are differentiable at the hyperbolic metric h, we can write f(ɛ) = ɛf + O(ɛ ) K(γ(ɛ)) = 1 + O(ɛ) B(γ(ɛ)) = 1 φ + O(ɛ) Substitution and some simplification gives I(σ(ɛ)) = 1 4ɛf h + 1 4f γ + 1 h + Re(φ) + O(ɛ). Consequently, I ɛ = h + ɛ( γ + fh + 4fRe(φ)) + O(ɛ ). The same reasoning will give The result then follows. II ɛ = h + ɛ γ + O(ɛ ). For each g Met s, there is an L orthogonal decomposition of the tangent space to g as T g Met s = {ġ tr g (ġ) = 0 = δ g (ġ)} {fg + L X g f H s and X Γ(T S)} The second summand is tangent to the Diff 0 s P s orbit of g. The first summand is referred to as transverse traceless tensors and consists of smooth tensors. (See [Ebin or Marsden] for more details). Since the transverse traceless tensors are orthogonal to the group orbit they may be identified with the tangent space T [g] Met s /(Diff 0 s P s ) to the quotient at [g]. Moreover transverse traceless tensors are a set of smooth tensors that are L orthogonal to the Diff 0 s P s orbit for any s. Consequently, they may be identified with the tangent space to the quotient Met /(Diff 0 P ). That is, they may be naturally identified with the tangent space to Teichmuüller space at [g]: T [g] T (S) = {ġ tr g (ġ) = 0 = δ g (ġ)}. 11

12 Under this identification, the action of the derivative of the projection π : Met T (S) at g is given by orthogonal projection onto the transverse traceless tensors T g Met {ġ tr g (ġ) = 0 = δ g (ġ)}. We make one more comment before we prove our main Theorem. Lemma 4.4. Let X be a Riemann surface and h the corresponding hyperbolic metric. If φ is a holomorphic quadratic differential, then Re(φ) is a smooth h- transverse-traceless tensor. Proof. See [?] We now prove our main result, Theorem 1.1 from the introduction. Theorem 4.5. Let ϕ ɛ : (0, 1) M be an asymptotically Poincaré family of surfaces with metrics at infinity σ(ɛ) so that f(ɛ)σ(ɛ) converges to the hyperbolic metric h. Let φ be the holomorphic quadratic differential at infinity. Then in Teichmüller space T (S) we have [I(σ(ɛ))] [h] and [II(σ(ɛ))] [h] as ɛ 0. Moreover, the difference of the tangent vectors in T [h] T (S) is given by [I(σ(ɛ))] [II(σ(ɛ))] = 4fRe(φ). Proof. Using the notation from the Proposition, for each s > 3 we have that I ɛ and II ɛ, which are paths through smooth tensors, converge in Met s to the hypberbolic metric h. Since Met = Met s we know that I ɛ and II ɛ converge to h in Met. Moreover, since the projection Met T (S) is continuous, and since [I(σ(ɛ))] = [I ɛ ] and [II(σ(ɛ))] = [II ɛ ], we have that [I(σ(ɛ))] [h] and [II(σ(ɛ))] [h] as ɛ 0. Since both paths converge to [h] at ɛ = 0 we can extend them to continuous paths [0, 1) T (S). Since [I(σ(ɛ))] and [II(σ(ɛ))] agree with [I ɛ ] and [II ɛ ], respectively, we also have they are differentiable at ɛ = 0. From the Proposition we know that I ɛ II ɛ = fh + 4fRe(φ). Since φ is holomorphic, 4f Re(φ) is trace-free and divergence-free. On the other hand, fh is pure trace and so belongs to the Diff s 0 P s orbit of h. Furthermore, for any s it belongs to the group orbit of h. Hence, the derivative of the projection at h removes this term. We have [I(σ(ɛ))] [II(σ(ɛ))] = [I ɛ ] [II ɛ ] = dπ h ( fh + 4 fre(φ)) = 4 fre(φ), as claimed. 1

13 5. k-surfaces in Quasi-Fuchsian Manifolds 5.1. The k-surface Equation. We now prove that k-surfaces form an asymptotically Poincaré family of surfaces. To do this we derive an equation for a conformal metric that implies its Epstein surface is a k-surface and we show this equation has a unique smooth solution for each k near zero. In the proof of existence of solutions we will see that these conformal metrics satisfy the hypothesis of Proposition 4.. This also gives an alternative proof to Labourie s theorem on the existence of k-surfaces, in this case for quasi-fuchsian manifolds. Finding a metric σ whose Epstein surface has constant Gaussian curvature k is finding a metric σ that solves K(I(σ)) = k, which from above is equivalent to ( (1) 4K(σ) = k (1 K(σ)) 16 ) σ B(σ). We are interested in obtaining solutions to (1) for k near zero. This is hampered by the fact that there are no solutions to (1) when k = 0. Indeed we would be asking for K(σ) = 0, which is impossible on a surface with genus bigger than 1. In an attempt to obtain better asymptotics we consider the case when Γ is Fuchsian. Here we have explicit solutions to the k-surface equation (1). Working in the universal covers, Ep h : Ω H 3 gives a totally geodesic copy of H in H 3 with constant curvature 1. For 1 < k < 0 the k-surfaces are given by equidistant copies of this H. The conformal metric whose Epstein surface is the k-surface is then c(k)h for some function of k satisfying K(I(c(k)h)) = k. Since B(g CP 1, c(k)h) = 0 we get the defining equation of c as or more explicitly c is given by 4K(c(k)h) = k(1 K(c(k)h)), c(k) = k k. Suppose we have solution metrics σ k on X that give Epstein surfaces with constant Gaussian curvature k. As we just saw, in the Fuchsian case we have σ k = c(k)h. In the general quasi-fuchsian case, define f(k) by f(k) = c(k) 1 = k k and τ k by τ k = f(k)σ k, so that in the Fuchsian case, the metric τ k is the hyperbolic metric for all k. Since σ k solves (1), by substitution and some simplification we get that τ = τ k solves the equation () (+k)(1+k(τ)) k ( 1 K(τ) ) ( +16 ) B(τ) 1 + k k τ = 0. In the limiting case k = 0 we see that the hyperbolic metric h on X solves (). Solutions actually exists in a neighborhood of (0, h) as we will show. To apply PDE theory we define the function F : U Conf(X) C (X) given by F (k, τ) = ( + k)(1 + K(τ)) k ( 1 K(τ) ) ( + ) 1 + k k 16 B(τ) τ. 13

14 Here, U is an open interval around 0 small enough not to contain 1 so that F is smooth on its domain. Points (k, τ) where F (k, τ) = 0 are solutions to the scaled equation (); we have already found F (0, h) = Solutions to the k-surface Equation. We will use the Implicit Function Theorem in the Banach space setting to obtain solutions and so we first work with Conf s (S) = {ph p H s (S, (0, )}, where H s (S, (0, )) = W,s (S, (0, )) is a Sobolev space of functions on S taking positive values. With the norm τ s := τ/h s, the set Conf s (S) is naturally identified with an open subset of the Banach space H s (S, R). The Sobolev Embedding Theorem guarantees that weak solutions in H s (S, R), for s 3, are actually strong solutions, being C,α regular (see [Aub8]). We will therefore work with the space H s (S, R) for a fixed s 3 from now on. Recall the function F is defined on U Conf(X) and maps to C (X). Extend F to a function U Conf s (X) H s (X) so that F is then defined on an open subset of a Banach space. We still have F (0, h) = 0 and we can now get solutions near 0 as well. Theorem 5.1. There is a neighborhood V of 0 such that for each k V, there exists a unique τ Conf s (X) such that F (k, τ) = 0. Proof. We have a solution h of k = 0. We obtain solutions near 0 using the Implicit Function Theorem. Note that since the constituent parts of F are smooth on U, F is smooth. We need only show now that D F (0,h) : Conf s (X) H s (X) is an isomorphism, where D F is the partial derivative of F with respect to its second argument. A direct computation of the derivative of F at (0, h) yields DF (0,h) ( k, τ) = 4 DK h ( τ). Note that D 1 F (0,h) = 0 and that D F (0,h) = 4 DK h. The differential of the curvature function DK h is given by a formula of Lichnerowicz: 4 DK h ( τ) = ( h ) τ h, which is an isomorphism Conf s (X) H s (X) (see [Tro9], page 33). Consequently, by the Banach Implicit Function Theorem (see [GT01]) there exists a neighborhood V of 0 and a curve γ : V Conf s (X) with γ(0) = h and F (k, γ(k)) = 0. Moreover, these are the only solutions to F (k, τ) = 0 in V, and γ is smooth since F is Regularity of Solutions. Theorem 5.1 furnishes weak solutions (k, γ(k)) such that F (k, γ(k)) = 0. The Sobolev Embedding Theorem immediately strengthens the regularity of γ(k) to at least C,α, for each k, and so we have strong solutions F (k, γ(k)) = 0. A nonlinear equation A(u) = 0 is said to be elliptic at a u if the derivative of A at u, DA u, is an elliptic linear operator. Solutions u where A is elliptic are smooth [GT01]. In our case, since D F (0,h) = ( h + ) is an elliptic operator we have F (0, τ) = 0 is elliptic at τ = h. Moreover, ellipticity is an open condition, so there exists an open interval ( ɛ, ɛ) such that for each k ( ɛ, ɛ), F (k, τ) = 0 is a 14

15 nonlinear elliptic equation at τ = γ(k). Hence our solutions γ(k) are smooth, for k ( ɛ, ɛ). So, we have smooth metrics γ(k) for k near zero satisfying F (k, γ(k)) = 0. Theorem 5.. There exists an ɛ > 0 such that for ɛ < k < 0 there exists a unique smooth metric σ k whose Epstein surface is a k-surface. Proof. We have a smooth curve γ defined on an interval ( ɛ, ɛ) such that the metric γ(k) satisfies F (k, γ(k)) = 0. This means that γ(k) solves the scaled equation () and so σ k = f(k) 1 γ(k) = k k γ(k), defined for k ( ɛ, 0), solves the k-surfaces equation (1). Thus σ k produces a k-surface: K(I(σ k )) = k. Corollary 5.3. The family of k-surfaces forms an asymptotically Poincaré family of surfaces. Proof. Define f(ɛ) = f( ɛ), σ(ɛ) = σ ɛ, and γ(ɛ) = γ( ɛ) for ɛ (0, 1) and take k = ɛ. Then the Corollary follows from Proposition 4. since f(ɛ) σ(ɛ) = γ(ɛ) = γ(k) h as ɛ A Conjecture of Labourie. A geometrically finite end of a hyperbolic 3 manifold admits a foliation by k-surfaces. Labourie first proved their existence in [Lab91]. This foliation in the quasi-fuchsian case is especially interesting. In [Lab9] Labourie discusses how the conformal classes [I k ] and [II k ] behave as paths in the Teichmüller space of S. When k 1, [II k ] approaches the point at infinity of T (S) corresponding to the measured geodesic lamination on the convex core of the quasi-fuchsian manifold. When k 0, both [I k ] and [II k ] converge to the same point: the complex structure X on the surface at infinity. He conjectures that the tangent vector to these paths is related to the holomorphic quadratic differential at infinity φ. Our Theorem 5. is another proof of the existence of these k-surfaces, at least for k close to zero. By the uniqueness result of Theorem 1.10 in [Lab9], given an end of M and k ( 1, 0), there exists a unique immersed incompressible k-surface. Since essentially Epstein surfaces are incompressible, the k-surfaces produced by our Theorem 5. are the k-surfaces Labourie obtains. Our approach here is more concrete than the methods used by Labourie. In return for sacrificing the generality of Labourie s pseudoholomorphic methods, a proof of Labourie s conjecture follows easily. Corollary 5.3 gives that, for k near zero, these k-surfaces form an asymptotically Poincaré family, and so by Corollary 4.5 we know how the tangent vectors to the families relate to the holomorphic quadratic differential at infinity. Theorem 5.4. Let I k and II k be the first and second fundamental forms of the k-surface. Let φ be the holomorphic quadratic differential at infinity on X. Then, as k 0, the difference of the tangent vectors to [I k ] and [II k ] in Teichmüller space is given by d dk ([I k ] [II k ]) = Re(φ). k=0 15

16 Proof. We take ɛ = k. The derivative f (0) = 1/4 can be computed directly. Moreover, since F (k, γ(k)) = 0 we see that DK h ( γ(0)) = 0 and since the derivative of Gaussian curvature is an isomorphism we get that γ(0) = 0. Theorem 4.5 now gives the theorem. 6. Constant Mean Curvature Surfaces in Quasi-Fuchsian Manifolds We now prove that there exists an asymptotically Poincaré family of surfaces S k for k near zero, such that the mean curvature of the surface S k is 1 + k. This is done similar to the previous section: we derive an equation for a conformal metric that implies its Epstein surface has mean curvature 1 + k and we show this equation has a unique smooth solution for each k near zero. The proof of existence of solutions will show that these conformal metrics satisfy the hypothesis of Proposition 4.. Recall from above that the mean curvature of (X, I(σ)) is given by H(I(σ)) = K(σ) 1 16 σ B(σ) (K(σ) 1) 16 σ B(σ), which, again, is an equation on the compact Riemann surface X. To find a metric whose Epstein surface has constant mean curvature 1 + k we must solve the equation H(I(σ)) = 1 + k, which simplifies to (3) 1 K(σ) 1 + k(1 K(σ)) + ( k) 16 σ B(σ) = 0. As in the k-surfaces case, we scale the equation by assuming σ k solves (3) and defining τ k = f(k)σ k for f(k) = 1 1+k 1+. If σ 1+k k solves (3) then τ k solves the equation k kk(τ) + ( k)(k(τ) 16 τ B(τ) ) = 0. Define G : U Conf(X) C (X) by G(k, τ) = k kk(τ) + ( k)(k(τ) 16 τ B(τ) ) where U is a small enough open set around zero not containing 1 so that G is smooth on its domain. To find solutions to (3) we find solutions to G(k, τ) = 0 and then scale them by f(k) 1. Notice that G(0, h) = 0 is a solution. Theorem 6.1. There exists a neighborhood W of 0 so that for each k W, there exists a unique τ Conf(X) so that G(k, τ) = 0. Proof. Extend G to a map U Conf s (X) H s (X) for s > 3. When k = 0 we have the hyperbolic metric h as solution G(0, h) = 0. The map G is smooth. The derivative of G at (0, h) is given by dg (0,h)) ( k, τ) = 4 φ k + DK h ( τ). Notice that D G (0,h) = DK h, which as in the proof of Theorem 5.1 is an isomorphism Conf s (X) H s (X). Hence, by the Banach Implicit Function Theorem there exists an open set W and a smooth curve γ : W H s (X) such that G(k, γ(k)) = k. Moreover, these are the only solutions to G(k, τ) = 0 in W. h 16

17 The same regularity arguments apply here as in the k-surface case and imply the existence of an ɛ > 0 so that the metrics γ(k) are smooth when k ( ɛ, ɛ). This implies that the mean curvature surfaces form an asymptotically Poincaré family of surfaces. Corollary 6.. There exists an ɛ > 0 so that for k ( ɛ, 0) the family of surfaces S k where S k has constant mean curvature 1 + k forms an asymptotically Poincaré family of surfaces. Proof. Since γ(k) solves G(k, γ(k)) = 0 for k ( ɛ, ɛ), the metric γ(k) solves the scaled mean curvature equation. So, defining σ k = k k γ(k) we see that σ k solves the mean curvature equation (3), which implies that Ep σk have constant mean curvature 1 + k. Moreover, since γ(k) = k k σ k h, the σ k satisfy the hypothesis of Proposition 4. and so the surfaces S k = Ep σk (X) form an asymptotically Poincaré family of surfaces. Consequently, we obtain Theorem 1.3 References [And98] C. Gregory Anderson. Projective structures on Riemann surfaces and developing maps to H 3 and CP n. PhD thesis, University of California, Berkeley, [Aub8] Thierry Aubin. Nonlinear analysis on manifolds. Monge-Ampère equations, volume 5 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 198. [Dum09] David Dumas. Complex projective structures. In Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages Eur. Math. Soc., Zürich, 009. [Dum17] David Dumas. Holonomy limits of complex projective structures. Adv. Math., 315:47 473, 017. [Eps84] Charles L. Epstein. Envelopes of horospheres and weingarten surfaces in hyperbolic 3- space. preprint, [GT01] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 001. [Lab91] François Labourie. Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques. Bull. Soc. Math. France, 119(3):307 35, [Lab9] François Labourie. Surfaces convexes dans l espace hyperbolique et CP 1 -structures. J. London Math. Soc. (), 45(3): , 199. [OS9] Brad Osgood and Dennis Stowe. The Schwarzian derivative and conformal mapping of Riemannian manifolds. Duke Math. J., 67(1):57 99, 199. [Tro9] Anthony J. Tromba. Teichmüller theory in Riemannian geometry. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 199. Lecture notes prepared by Jochen Denzler. 17