On Almost-Fuchsian Manifolds

Almost-Fuchsian manifold is a class of complete hyperbolic three manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1, 1). In such a manifold, the minimal surface is unique and embedded, hence one can parametrize these hyperbolic three-manifolds by their minimal surfaces. In this paper we obtain estimates on several geometric and analytical quantities of an almost-Fuchsian manifold M in terms of the data on the minimal surface. In particular, we obtain an upper bound for the hyperbolic volume of the convex core of M, and an upper bound on the Hausdor? dimension of the limit set associated to M. We also constructed a quasi-Fuchsian manifold which admits more than one minimal surface, and it does not admit a foliation of closed surfaces of constant mean curvature.


Introduction
Foliations of geometrically interesting hypersurfaces of codimension 1 are important objects in the study of differential geometry and mathematical physics. In particular, fibers of constant mean curvature hypersurfaces are highly desired. We are particularly interested in the situation when the ambient space is three dimensional, and the foliations under the consideration of this paper are of closed surfaces: the foliation of evolving surfaces (from a given smooth embedded surface) and the foliation of constant mean curvature surfaces (or CMC foliation).
The topic of existence and/or uniqueness of CMC foliation in three-manifolds has been extensively studied, and such foliation often reflects the global geometry and topology of the ambient space. Even the existence of an incompressible minimal surface can reveal a lot of structures on the three-manifold (see for example [Rub07,SY79]). In the settings of asymptotically Euclidean space, Huisken-Yau ( [HY96]) proved existence of a CMC foliation near infinity. The case of asymptotically hyperbolic manifolds was investigated in the works of Rigger ([Rig04]), and Neves-Tian ( [NT06,NT07]), and more recently Mazzeo-Pacard ( [MP07]) where they showed the existence and uniqueness of a CMC foliation in the ends of any geometrically finite quasi-Fuchsian three-manifold. In [Wan08], the second named author constructed a quasi-Fuchsian manifold which does not admit any CMC foliation.
Throughout this paper, we assume M is a quasi-Fuchsian hyperbolic three-manifold, and our convention of mean curvature is the trace of the second fundamental form, i.e., sum of two principal curvatures. Topologically, it is a product of a closed surface by the real line, i.e., M = S × R, where we always assume S is a closed Riemann surface of genus g ≥ 2. We call such a closed surface incompressible in M if the surface represents the homotopy group of M . We also exclude the situation when M is Fuchisian, in which case, most theorems here are trivial. All surfaces we encounter here will be closed, incompressible, of genus at least 2, unless otherwise stated.
For quasi-Fuchsian three-manifolds, we import a convenient notion introduced by Krasnov and Schlenker ( [KS07]), the almost Fuchsian three-manifolds, i.e., the manifold M which contains a minimal surface whose principal curvatures are in the range of (−1, 1). The almost Fuchsian manifolds form a very special class of quasi-Fuchsian manifolds ( [Tau04]), and the dimension of the almost Fuchsian manifolds are of the same as the quasi-Fuchsian space ( [Uhl83]). Schoen-Yau ( [SY79]) and ) showed the existence of an immersed minimal surface in any quasi-Fuchsian three-manifold M , and Uhlenbeck ([Uhl83]) showed this minimal surface (in the convex core of M ) is the unique embedded minimal surface in M when M is almost Fuchsian.
We wish to consider a larger class of hyperbolic three-manifolds and obtain geometrical information from volume preserving mean curvature flows in M . Almost minimal surfaces are very special in low dimensional topology ( [Rub05]). It is conjectured by Thurston that every complete hyperbolic three-manifold of finite volume has immersed incompressible surface of small curvatures. In M , parallel surfaces defined by equidistant from a given reference surface of small curvatures contains no singularity. This simple fact is of great importance to our work.
The volume preserving mean curvature flow equation generally has the following form: where h(t) = S t Hdµ |St| is the average mean curvature of the evolving surface S t , and all other terms will be made transparent in §2.3. Theorem 1.2. If M is nearly almost Fuchsian, i.e., M contains an incompressible surface S with principal curvatures in the range (−1, 1), and {S(r)} r∈R is the equidistant foliation with S(r) hyperbolic distance r from the reference surface S = S(0). Then (i) each volume preserving mean curvature flow equation (1.1) with initial surface S 0 = S(r) has a long time solution; (ii) the evolving surfaces {S t } t∈R stay smooth and remain as graphs of S for all time; i } converges uniformly in C ∞ to a smooth embedded surface S ∞ of constant mean curvature c(r); (iv) These constants satisfy −2| tanh(r − β)| ≤ c(r) ≤ 2| tanh(r + β)|, for some constant β only depending on the reference surface S.
It is understandably different in hyperbolic geometry whether an immersed closed CMC surface has mean curvature between −2 and 2 or otherwise. In particular, part (iv) improves the inequality |c| < 2 for these constant mean curvature surface in a nearly almost Fuchsian manifold M . Moreover, we showed that, for each prescribed constant c ∈ (−2, 2), there is an embedded incompressible surface of constant mean curvature c ( [HW10]).
We prove our theorems via techniques in volume preserving mean curvature flow developed by Huisken ([Hui86,Hui87b]). Intuitively, such a flow "averages" the mean curvature function on a surface, and obtain a limiting surface of constant mean curvature. This technique has been developed into a powerful tool in understanding surfaces in Riemannian and semi-Riemannian manifolds (see, for example, [Hui87a,HY96,And02,CRM07], and many others). The evolution under the mean curvature flow of the graphs in hyperbolic space behaves quite well, under mild conditions on the initial data. See, for example, [EH89,Unt03].
Here is a (very) simplified version of the proof of the main Theorem: starting with an arbitrary incompressible surface S in M , we "shift" S = S(0) to obtain a family of smooth surfaces S(r) by the means of equidistant from the initial surface S(0); the principal curvatures of this equidistant surfaces can be controlled explicitly (see §2.2); on each fiber of S(r), apply volume preserving mean curvature flow to obtain a limiting CMC surface.
Our use of the volume preserving mean curvature flow is quite different than what we have also considered ( [HW09,HW10]) by using the usual mean curvature flow: here we utilize a parameter family of volume preserving mean curvature flows to obtain surfaces of constant mean curvature simultaneously. We will address the issue of disjointness of these limiting surface in a separate paper.
Acknowledgements: This paper is a project extending parts of the doctoral thesis of the second named author ( [Wan09]). He wishes to thank his thesis advisor Bill Thurston for continuous encouragement and generously sharing many key insights. The research of the first named author is partially supported by a PSC-CUNY grant. We also thank Xiaodong Cao, Ren Guo and Zhou Zhang for many stimulating discussions. Both authors are most grateful to anonymous referees to an earlier version of the paper, for their very helpful comments and suggestions.
2. Background 2.1. Quasi-Fuchsian Three Manifolds. A hyperbolic three-manifold M is a complete Riemannian manifold of constant curvature −1, and we assume M has no boundary. Its universal covering space is H 3 , whose isometry group is given by PSL(2, C). A subgroup of PSL(2, C), denoted by Γ, is called a Kleinian group if it is a finitely generated and torsion free discrete subgroup. Γ is associated to M if M = H 3 /Γ, with π 1 (M ) = Γ. In this paper, we always assume that Γ is parabolic free, i.e., it does not contain any parabolic element.
The limit set of Γ, denoted by Λ Γ ⊂ C = C ∪ {∞}, is the smallest closed Γ-invariant subset of C. The open set Ω Γ = C \ Λ Γ is called the domain of discontinuity. Γ acts properly discontinuously on Ω, and the quotient Ω/Γ are finite union of Riemann surfaces of the finite type.
For any Kleinian group Γ, it is called quasi-Fuchsian if its limit set Λ lies in a Jordan curve. In this case, M is called a quasi-Fuchsian three-manifold. The convex hull of the limit set Λ, denoted by Hull(Λ), is the smallest convex subset in H 3 whose closure in H 3 ∪ C contains Λ. The quotient space C(M ) = Hull(Λ)/Γ is called the convex core of M .
Topologically, M = S × R. Every quasi-Fuchsian three-manifold can be constructed via Bers' simultaneous uniformization theorem ([Ber72]), with a pair of Riemann surfaces in Teichmüller space as the conformal boundaries. The correspondence between Kleinian groups and low dimensional topology is extremely rich and complex, where one can find many references in articles such as [Mar74,Thu82].
We are interested in using minimal surfaces as parameters for the quasi-Fuchsian space. Fundamental results in harmonic maps of [SU82,SY79] can be applied to various hyperbolic manifolds, in particular, quasi-Fuchsian three-manifolds. Dimension three is very special, since the branch points never occur in this case ( [Gul73]). They showed there exists an immersed area minimizing incompressible surface Σ in M . If the principal curvatures of the minimal surface are small in the sense of Definition 1.1, then this minimal surface is unique, and it is an embedded incompressible surface in M ( [Uhl83]). Further understanding of immersed minimal surfaces and constant mean curvature surfaces can also be found in [And82,And83] via approaches of geometric measure theory.
2.2. Geometry of Surfaces in Three Manifolds. In this subsection, we construct a family of parallel surfaces from an incompressible surface S of small curvatures in (M,ḡ αβ ), and collect necessary facts about this family, similar to the construction of Uhlenbeck's equidistant foliation ([Uhl83]) on an almost Fuchsian three-manifold.
Let S be an incompressible surface on the nearly almost Fuchsian manifold (M,ḡ αβ ). The curvature tensor of M is given by and the induced metric on S is then g ij (x) = e 2v(x) δ ij , where v(x) is a smooth function on S, and of the second fundamental form where {e 1 , e 2 } is a basis on S, and ν is the unit normal field on S, and ∇ is the Levi-Civita connection of (M,ḡ αβ ).
Let λ 1 (x) and λ 2 (x) be the eigenvalues of the second fundamental form A(x) of S. We have |λ j (x)| < 1, for j = 1, 2. They are the principal curvatures of S, and we denote H(x) = λ 1 (x) + λ 2 (x) as the mean curvature function of S. We can construct a normal coordinate system in a collar neighborhood of S. More precisely, suppose x = (x 1 , x 2 ) is a coordinate system on S, and choose ε > 0 to be sufficiently small, then the (local) diffeomorphism induces a coordinate patch in M . Let S(r) be the family of parallel surfaces with respect to S, i.e.
The induced metric on S(r) is denoted by g(x, r) = g ij (x, r), and the second fundamental form is denoted by A(x, r) = [h ij (x, r)] 1≤i,j≤2 . The mean curvature on S(r) is thus given by H(x, r) = g ij (x, r)h ij (x, r).
The curvature tensor R αβγδ of (M,ḡ αβ ) has six components, which are not completely independent because of Bianchi identities. In the collar neighborhood of S, these components can be classified into three groups: (i) there are three curvature equations of the form R i3j3 = −g ij , here 1 ≤ i, j ≤ 2, (ii) two of remaining curvature equations have the form R ijk3 = 0, which are called the Codazzi equations, and (iii) the Gauss equation R 1212 = −g 11 g 22 + g 2 12 . This enables us to solve for g(x, r) explicit, and the solution is essentially due to Uhlenbeck ([Uhl83]), though S is a minimal surface in her case. We collect this into the following: Lemma 2.1. Let the isothermal metric on S be given by e 2v(x) I, where v(x) is a smooth function on S and I is the 2 × 2 unit matrix, and let A(x) = [h ij (x)] 1≤i,j≤2 be the second fundamental form of S with respect to the isothermal metric, then the induced metric g(x, r) on S(r) has the form Proof. At each point (x, r) ∈ S(r), direct computation shows that the curvature equations R i3j3 = −g ij has an explicit form Therefore we obtain a first order system of differential equations Since g(x, 0) = e 2v(x) I, we obtain the explicit solution as in (2.1).
A direct consequence is the following: Corollary 2.2. The induced metric on S(r) is non-singular for small ε such that r ∈ (−ε, ε). When the initial surface S has small curvatures, then g(x, r) contains no singularity for all r ∈ (−∞, +∞).
It is clear that these parallel surfaces {S(r)} r∈R form a foliation of incompressible surfaces of M , and this foliation is called the equidistant foliation or the normal flow of M . The formula (2.1) also makes it possible to calculate various curvatures on each fiber S(r), which we will proceed in what follows. These quantities are very crucial to our estimates of the mean curvatures of the CMC surfaces in M .
The following lemma is the well-known Hopf's maximum principle for tangential hypersurfaces which is a powerful tool in Riemannian geometry: Lemma 2.3. Let Σ 1 and Σ 2 be two hypersurfaces in a Riemannian manifold, and intersect at a common point tangentially. If Σ 2 lies in positive side of Σ 1 around the common point, then H 1 < H 2 , where H i is the mean curvature of Σ i at the common point for i = 1, 2.
2.3. Volume Preserving Mean Curvature Flow. Our methods of proving main theorems involve geometric evolution equations, in particular, the volume preserving mean curvature flow developed by Huisken and others. We briefly review our particular set up in this subsection.
As before, we assume M is nearly almost Fuchsian and contains an incompressible surface S whose principal curvatures are denoted by λ 1 (x) and λ 2 (x). S is of small curvatures. Let F 0 : S → M be the immersion of S in M such that the initial surface S 0 = F 0 (S) is contained in the positive side of S and is a graph over S with respect to n, i.e. n, ν ≥ c 1 > 0, here n is the normal vector on S and ν is the normal vector on S 0 and c 1 is a constant depending only on S 0 . For each r ∈ R, we choose an initial surface S r 0 = S(r) to obtain a family of smoothly immersed surfaces in M : x ∈ S} as the evolving surface at time t with initial surface S r 0 . Whenever there is little confusion, we will remove the upper index r from the notation to ease the exposition.
Abusing some notations, we define some quantities and operators on the evolving surface S t : • the induced metric of S t : g = {g ij }; • the second fundamental form of S t : A = {h ij }; • the mean curvature of S t with respect to the normal pointing to S: H = g ij h ij ; • the square norm of the second fundamental form of S t : • the covariant derivative of S t is denoted by ∇ and the Laplacian on S t is given by We add a bar on top for each quantity or operator with respect to (M,ḡ αβ ). We consider the family of volume preserving mean curvature flows: is the average mean curvature of S r t , and ν r (·, t) is the normal on S r t so that −ν r points to the fixed reference surface S.
One easily verifies that the volume of the region bounded by the reference surface S and the evolving surfaces S r t is independent of t. Huisken proved the short-time existence of the solutions to the equation (1.1), and moreover, the blow-up of the square norm of the second fundamental forms if the singularity occurs in finite time.
as t → T .
Our major goal, in next section, is to show the long-time existence of the solutions to the equation (1.1) in the case of M being an almost Fuchsian three-manifold. From above theorem, an essential strategy is to obtain uniform bounds for the second fundamental form of the evolving surfaces.

Proofs of Theorems
3.1. Foliation of Equidistant Surfaces. We will proceed by foliating the nearly almost Fuchsian three-manifold M by an equidistant foliation {S(r)} r∈R , consisting of surfaces equidistant from the surface S = S(0): Theorem 3.1. M admits two foliations which can be described as: (i) a co-dimension two totally geodesic foliation G such that each leaf is a bi-infinite geodesic perpendicular to S; (ii) a co-dimension one foliation F including S as a leaf, such that each leaf S(r) is a parallel surface which has small curvatures.
Proof. We import our notation from §2.2: The isothermal metric on S is given by e 2v(x) I, and A(x) = [h ij (x)] 1≤i,j≤2 is the second fundamental form of S, defined by h ij = − ∇ ν e i , e j , where {e 1 , e 2 } is the basis on S, and ν is the normal vector field on S. By (2.1), the induced metric on S(r) is Since we have λ 1 , λ 2 ∈ (−1, 1), then an elementary calculation shows that Thus F = {S(r) | r ∈ R} is the co-dimension one foliation of M .
For each x ∈ S, the map γ x : R → M defined by r → exp x rν(x) is a bi-infinite geodesic in M , and G = {γ x (R) | x ∈ S} is the codimension two totally geodesic foliation.
The second fundamental form A(x, r) = [h ij (x, r)] of S(r) with respect to the induced metric g(x, r) is We denote the principal curvatures of S(r) by µ 1 (x, r) and µ 2 (x, r) as before. They are the solutions of the equation det[h ij − µg ij ] = 0, which is equivalent to the following characteristic equation: It is easy to see that Tr(e −2v A) = λ 1 + λ 2 and det[e −2v A] = λ 1 λ 2 , and the characteristic equation has the following form Since |λ j | < 1, then |µ j (·, r)| < 1 for all r ∈ (−∞, ∞), j = 1, 2. Each S(r) has small curvatures in the sense of Definition 1.1.
Now we have the equidistant foliation {S(r)} r∈R on M , with S(0) = S, and with principal curvatures µ j (x, r) expressed as in formula (3.2). It is then routine to verify: Proposition 3.2.
(ii) Moreover, the principal curvatures µ j (x, r) and mean curvatures H(x, r) for the fiber S(r) satisfy the following: (a) For fixed x ∈ S, µ j (x, r) is an increasing function of r. Moreover, µ j (·, r) → ±1 as r → ±∞; (b) For fixed x ∈ S, H(x, r) is an increasing function of r. Moreover, H(·, r) → ±2 as r → ±∞.
We also note that, among the fibers S(r), there is one fiber of the least area with the induced metric: Proposition 3.3. Among all parallel surfaces, the one with average mean curvature zero is the one of the least surface area.
It is a simple application of the maximum principle (Lemma 2.3) and {S(r)} r∈R being the equidistant foliation of M with all parallel surfaces have small curvatures, that any CMC surface of M with mean curvature c satisfies that |c| < 2.
3.2. Bounding evolving surfaces. We fix r > 0 for the fiber S(r) for now, let F r 0 : S → M be a smooth embedding such that F r 0 (S) = S(r), and let F r : S × [0, T ) → M be a family of smooth surfaces. Recall from §2.3 the volume preserving mean curvature flow equation F r (·, 0) = F r 0 , where ν r (·, t) is the normal vector on the surfaces S t (r) = F r (·, t)(S), h r (t) is the average mean curvature of S t (r). Under the flow we can find evolution equations for many geometrical quantities on S t (r) such as the metric g ij (·, t), the second fundamental form h ij (·, t), the normal vector field ν(·, t), the mean curvature H(·, t) and the squared norm |A(·, t)| 2 , etc. In what follows we removed up-index r in the formulas for convenience. We list two such evolution equations, which will be standard to use to prove results using mean curvature flows. and where ∆ and ∇ are the Lapalcian and the gradient operators on evolving surface S t , respectively.
Let ℓ(p) = ± dist(p, S) for all p ∈ M , where ± dist(·, ·) means if p is above S then we choose + sign, otherwise − sign: this is a signed distance function. Then we define the height function u(·, t) and the gradient function Θ(·, t) on S t : Here T max is the right endpoint of the maximal time interval on which the solution to (1.1) exists. It is easy to see that the surface S t is a graph over S if Θ(·, t) > 0 on S t .
The evolution equations for u and Θ have the following forms ([Bar84, EH91]): where ∇ is the gradient operator with respect to the hyperbolic metric, div is the divergence on S t , and n(H) is the variation of mean curvature function of S t under the deformation vector field n. Now we assume that T max < ∞. We need the following rough estimate for the height function: Lemma 3.5. If Θ(·, t) > 0, then the height function u(·, t) for the evolving surface S t is uniformly bounded for t ∈ [0, T max ).
The condition Θ(·, t) > 0 will be removed since it's proven to hold in the proof of Theorem 3.7.
Proof. At each t ∈ [0, T max ), let x(t) ∈ S t and y(t) ∈ S t be the furtherest and closest points from S, i.e., u max (t) = max x∈S u(x, t) = u(x(t), t), u min (t) = min y∈S u(y, t) = u(y(t), t).
Since Θ(x, t) > 0 for all (x, t) ∈ S × [0, T max ), together with (3.9), one can see that the part of S t with H > h moves along the negative direction of n and the part of S t with H < h moves along the positive direction of n. Therefore we can assume that u max (t) is increasing and u min (t) is decreasing.
As t → T max , we have the following five cases: Case (i) and (ii) could not happen, since the flow is volume preserving.
For Case (iii), since Θ = n, ν = 1 at F (x(t), t), we have Since {S(r)} r∈R is a foliation on M , there are two fibers S(r 1 ) and S(r 2 ) with r 1 < r 2 such that they are tangent to the surface S t . By Hopf's maximum principle (see Lemma 2.3), we have Similarly, at the point F (y(t), t), we have H.
Thererfore, we have the inequality H.
This lemma bounds the evolving surfaces S t (x, r) by two parallel surfaces S(r 1 ) and S(r 2 ), whose various curvatures can be explicitly calculated. We will refine this estimate later for further applications.
The proof of this lemma also implies: Corollary 3.6. For large enough t ∈ [0, T max ), the average mean curvature of S t satisfies h(t) ∈ (−2, 2).
3.3. Long Time Extistence. With Lemma 3.5, we can bound derivatives of the square norm of the second fundamental form A t (x), which will in turn prove the long-time existence for (1.1): Theorem 3.7. The volume preserving mean curvature flow (1.1) has a long time solution, i.e., T max = +∞. Moreover, each evolving surface S t is a graph over the reference surface S, hence embedded in M .
Proof. Our strategy is to prove by contradiction. If T max < +∞, then we show the flow can be extended beyond the first singular time.
The induced metric on S t can be written in terms of the height function u(·, t) and its derivatives, so the differential equation By [Bar84,EH91], we have the estimates: where C 1 and C 2 are positive constants depending only on the initial surface and the maximal time T max . From Corollary 3.6, |h(t)| ≤ 2 and Θ(·, 0) ≡ 1, we have the following: Since |A| 2 ≥ H 2 /2 and |Θ| ≤ 1, we have the following estimate: Since Θ min (0) = 1 by assumption, we have where C 3 is given by This implies that the gradient function Θ is uniformly bounded from below by a constant Therefore evolving surfaces remain as graphs over the reference surface S along the flow (1.1). Moreover, since Θ = 1/ 1 + |∇u| 2 ([Hui86]), we know that |∇u| is uniformly bounded from above by a constant depending only on the initial data and the maximal time T max . Since the equation (3.12) is quasi-linear parabolic, the standard regularity results in quasilinear second order parabolic equations ( [LSU67,Lie96]) state that for all ℓ = 1, 2, . . ., where {K ℓ } ∞ ℓ=1 is the collection of constants depending only on ℓ, the initial data and the maximal time T max . In particular, |∇Θ| 2 ≤ C 4 < ∞, where C 4 is a constant depending again only on the initial data and the maximal time T max .
The evolution equation (3.6) for |A| 2 is very complicated, not suitable for direct calculation. To absorb positive terms from the right handside of (3.6), for some small σ > 0, we define f σ = |A| 2 /Θ 2+σ and calculate as follows: (3.13) If we assume |A(·, t)| 2 is not uniformly bounded, then |A| 2 max (·, t) → ∞ as t → T , which forces f σ max (·, t) → ∞ as t → T . Since −σC 2+σ 4 f 2 σ dominates the evolution equation (3.13), we obtain that f σ satisfies the inequality for some positive constant C 4 . Therefore, f σ is uniformly bounded. This contradiction shows |A(·, t)| 2 is uniformly bounded. We can then proceed as the Theorem 4.1 in [Hui87b] to obtain uniform bounds for the derivatives |∇ m A| 2 , for all m ≥ 1.
If T = T max < ∞, the limit S T (r) = lim is well defined because of Lemma 3.5, while uniform bounds on the square norm of the second fundamental form and its derivatives imply that S T (r) is a smooth surface ( [Ham82]). Therefore we can consider a new volume preserving mean curvature flow with new initial surface S T (r), and it is standard (see for instance Chapter 6 of [CK04]) that this new flow has a short-time solution, which contradicts the assumption that T is maximal. This implies T max = ∞, long-time existence of the solution.
Theorem 3.7 implies the parts (i) and (ii) of the Theorem 1.2.
3.4. Limiting Behavior. In this subsection, we investigate the limiting behavior of the flow, and obtain a limiting surface of constant mean curvature applying the theorem of Ascoli-Azela. Part (iii) of the Theorem 1.2 is an implication of the following more technical theorem: Theorem 3.8. The mean curvatures of S t approach the average mean curvature: (3.14) sup t→∞ |H(·, t) − h(t)| = 0.
Proof. By Lemma 3.5, surfaces {S t } are bounded in a compact region, hence the area |S t | is uniformly bounded in t. It is easy to see that the area of S t is decreasing along the flow: Therefore the integral and we find that the integral St (H − h(t)) 2 dµ is uniformly bounded. We use the identity St (H(·, t) − h(t))dµ = 0, and (3.5), to compute the t−derivative of above integral: The absolute value is easily seen to be bounded from above by the uniform bounds for |A(·, t)| 2 , H(·, t) as well as their derivatives, and the uniform bound for St (H − h(t)) 2 dµ. Therefore we have So we obtain the L 2 estimate: This L 2 -estimate in conjunction with uniform bound on |∇(H − h(t))| allow us to apply the standard interpolation argument to show the L ∞ bound, i.e., (3.14).
Proof. (of (iii) of Theorem 1.2) Now we can denote as a limiting surface of the flow. It is well defined since {S t } 0≤t<∞ are contained in a bounded domain of M (Lemma 3.5), and uniform bounds on |A(·, t)| 2 , H(·, t) and their derivatives enable us to apply us to apply the Theorem of Ascoli-Azela to obtain a subsequence of evolving surfaces which converges smoothly to a limiting surface. Therefore S ∞ (r) is a smooth surface. This also implies that |∇u| must be bounded as t → ∞, so Θ must have a positive lower bound as t → ∞. This means that S ∞ (r) is a graph over the reference surface S. The limiting surface S ∞ (r) is of constant mean curvature by the Theorem 3.8. The constant is clearly a function of r.
We call a CMC surface Σ is strictly stable if the stability operator on Σ Proof. Let λ ∞ be the lowest eigenvalue of the stability operator L on the surface S ∞ , and it is positive by our assumption. Let λ t be the lowest eigenvalue of the stability operator on the evolving surface S t , and by Theorem 3.8, when t is large enough such that for any ǫ we have both Then we proceed from (3.17) to find which implies the exponential convergence, and unique limit.
The unique limit is a consequence of the assumption on the stability of the surface. Therefore this proposition is about the behavior of the convergence. We do expect all limiting surfaces, for r ∈ R, are strictly stable.

Mean Curvature Estimates
In this section, we prove the part (iv) of the Theorem 1.2. These mean curvature estimates are considered applications of our method, and most of them will be quite difficult to generalize to higher dimensions. 4.1. First Estimate. CMC surfaces (including minimal surfaces) are classical topics in differential geometry. In hyperbolic geometry, it is a consequence of Gauss-Bonnet theorem and Schwarz inequality that if Σ be a closed CMC surface of genus g > 2 and mean curvature c in a hyperbolic manifold N , then (1 − c 2 4 )|Σ| ≤ 2π(2g − 2). In our setting of M being nearly almost Fuchsian, we have a rough estimate |c| < 2.
Clearly, this is a positive constant, only depending on S.

4.2.
Close to Infinities. In this subsection, we consider these volume preserving mean curvature flow all at once: in space direction, we have r ∈ R, while in time direction, we have t ∈ [0, ∞). Note that at time t = 0, we have the equidistant foliation {S 0 (r) = S(r)} r∈R . We provide a finer estimate on the height function u(·, t), especially for r → ∞.
Theorem 4.2. If the height function for the r-th volume preserving mean curvature flow is given by u(·, t, r), then there is a positive constant β, only depending on S such that (4.6) r − 2β ≤ u(·, t, r) ≤ r + 2β, Proof. We use notations from the proof of Theorem 4.1. Constants α and β are given as in formulas (4.2) and (4.3), and parallel surfaces S(r 1 ) and S(r 2 ) are tangent to the evolving surface S t , with r 1 < r 2 . With this set-up, we see that r 1 < r < r 2 since the flow is volume preserving. Moreover, we have max St (u(·, t, r)) = r 2 and min St (u(·, t, r)) = r 1 . Now we apply inequalities (4.4) and (4.5) to find that tanh(r 2 − β) ≤ tanh(r 1 + β), which implies r 2 − r 1 ≤ 2β and hence (4.6).
Therefore, as r → ∞, the height function as goes to infinity. Combining with estimates in (4.4) and (4.5), we have Corollary 4.3. There are closed, incompressible, embedded CMC surfaces arbitrarily close to both conformal infinities, with mean curvatures arbitrarily close to ±2.
In the opposite direction, we, in [HW10], for any constant c in (−2, 2), can construct a closed, embedded incompressible surface of constant mean curvature equal to c.