Existence and uniqueness of complete constant mean curvature surfaces at infinity of H^{3} ^{∗}

John A. Velling ^{†}

December 1992

Abstract

A set of conditions are given, each equivalent to the constancy of mean curvature of a surface in H^{3}. It is shown that analogs of these equivalences exist for surfaces in S_{∞}^{2}, the bounding ideal sphere of H^{3}, leading to a notion of constant mean curvature at infinity of H^{3}. A parametrization of all complete constant mean curvature surfaces at infinity of H^{3} is given by holomorphic quadratic differentials on Ĉ, C, and D.

1 Introduction

Let H^{3} denote the usual hyperbolic 3-space, with the upper half-space model {(X, Y, T) ∈ R^{2} × R_{+}} or {(Z, T) ∈ C × R_{+}} and metric line element given by the expression

ds^{2} = \frac{dX^{2} + dY^{2} + dT^{2}}{T^{2}} = \frac{|dZ|^{2} + dT^{2}}{T^{2}} . (1.1)

We will denote the corresponding inner product by ⟨·, ·⟩, and note that H^{3} is a symmetric space with constant sectional curvature −1. There are other models of H^{3} which may facilitate calculations or geometric observations, but for convenience the upper half-space model will be used exclusively.

∗Key words and phrases: Hyperbolic space, Constant mean curvature, Quasiconformal deformations

∗Mathematics Subject Classification: 53C42, 30C62 †Partially supported by PSC-CUNY grant #661365.

In various computations throughout the paper we will use the summation convention implicitly, so that if g_{ij}, h^{kl} ∈ C for i, j, k, l ∈ 1, ⋯, n then g_{ij}h^{jk} means ∑_{j=1}^{n} g_{ij}h^{jk}.

This paper concerns an apparently new use of quasiconformal analysis in surface theory, and a criterion the satisfaction of which is equivalent to a surface Σ_{0} ↪ H^{3} having constant mean curvature, H(Σ_{0}) = c. Briefly:

For a smooth surface Σ_{0} ↪ H^{3}, consider unit speed normal geodesic flow. This gives a deformation of Σ_{0} in H^{3}, with the image of Σ_{0} under the flow denoted by Σ_{t} after time t. Let N_{t} : Σ_{0} → Σ_{t} denote this map. For the moment let us assume that Σ_{t} is smooth. In this case both Σ_{0} and Σ_{t} inherit metrics from H^{3}, and therewith conformal structures. The conformal deformation by the map N_{t} is measured from a conformal parameter Z on Σ_{0} by the Beltrami differential

μ_{t} = \frac{∂Z̄N_{t}}{∂_{Z}N_{t}} (1.2)

on Σ_{0}.

The space of Beltrami differentials on Σ_{0} is a vector space, and μ_{t} is a point in this space. Thus there is natural a map [−∞, +∞] → (Beltrami differentials) ∪{∞} on Σ_{0}, where t ↦ μ_{t}. (The images of ±∞ are obtained from the hyperbolic Gauss maps G_{±} : Σ_{0} → S_{∞}^{2} = ∂H^{3}.) As N_{0} = id, μ_{0} ≡ 0, and the curve of realized Beltrami differentials passes through the origin. It turns out that H(Σ_{0}) = c if and only if this curve is a radial segment, i.e. if and only if there is some Beltrami differential μ′ so that μ_{t} = f(t)μ′ with f : [−∞, +∞] → R ∪ {∞} injective.

This is the first point of the paper, and is presented in §5. In §2–4 background is presented to make this precise: normal flow in H^{3}, harmonic maps between surfaces, the hyperbolic Gauss maps, and several related topics. Much of the background material is a recapitulation of unpublished results of C. L. Epstein. One consequence of this development is that several apparently new characterizations of minimal surfaces are given in the main theorem (theorem 1) of §5. One example: Σ_{0} is minimal if and only if this radial segment is symmetric with respect to 0. Though much of the development herein holds for R^{3} or S^{3}, the treatment is restricted to H^{3}.

The second major point of the paper is that the characterizations of constant mean curvature surfaces in H^{3} (from §5) can be used to give a notion of constant mean curvature surfaces immersed in S_{∞}^{2}. A surface Σ_{+∞} immersed in S_{∞}^{2} with constant mean curvature must necessarily satisfy H(Σ_{+∞}) = ±1. This is expected: if one considers spheres concentric about x ∈ H^{3}, S_{ρ}(x) of radius ρ, their mean curvatures are ± coth ρ, which tend to ±1 as ρ → ∞. Similarly, all horospheres Σ_{0} ⊂ H^{3} have H(Σ_{0}) = ±1. The development of the local theory of constant mean curvature surfaces at infinity is found in §6, the point (theorem 4) of which is to show that theorem 1 holds in this boundary case. When this boundary equivalence is satisfied we say that Σ_{+∞} has constant mean curvature at infinity. One example of this equivalence: considering a family {Σ_{t}}_{t∈R} of parallel surfaces in H^{3}, as above, and letting N_{+∞}^{t} : Σ_{t} → Σ_{+∞} denote the forward Gauss map from Σ_{t}, let N_{t}^{+∞} = (N_{+∞}^{t})^{−1} : Σ_{+∞} → Σ_{t}. This is a potentially multivalued map. If the maps N_{+∞}^{t} are sufficiently nice, we can give Σ_{+∞} a conformal structure (the limit of the structures on the Σ_{t} as t → ∞). Letting Z now be a complex parameter on Σ_{+∞}, we look at

μ_{t}^{+∞} = \frac{∂Z̄N_{t}^{+∞}}{∂_{Z}N_{t}^{+∞}} (1.3)

as a Beltrami differential on Σ_{+∞}. The constancy of mean curvature for Σ_{+∞}is equivalent to the map t ↦ μ_{t}^{+∞}, given by [−∞, +∞] → (Beltrami differentials on Σ_{+∞} ∪ {∞}), having image on a radial line.

Examples of such surfaces are known. They are precisely the images of D = {z ∈ C : |z| < 1} by locally univalent maps, up to Möbius transformation, with the infinitesimal form of the metric on each sheet being the pushforward of the Poincaré metric on D. The corresponding family of parallel surfaces Σ_{t} interpolate between this possibly multisheeted conformal image of D and a generally not conformal (and often not even smooth) image of D obtained by a construction of Ahlfors and Weill. These are presented briefly in §7. In §8 an apparently new class of such surfaces is given, generalizing to flat structures the above interpolation and the Ahlfors–Weill construction.

Now if Σ_{0} is a complete surface of constant mean curvature, its second fundamental form pulls back to its universal cover Σ̃_{0}, furnished with a complete metric. This form splits and gives a holomorphic quadratic differential on Σ̃_{0} which is invariant under the deck transformations of the covering. It is natural to ask to what extent this quadratic differential, plus the completeness condition, characterize the surface Σ_{0}. This has been considered by other authors, and is discussed in the beginning of §9. Much of the motivation for this paper comes from this question. The essential existence/uniqueness issues in this setting remain unresolved.

It turns out that for constant mean curvature surfaces at infinity the second fundamental form again splits and gives a holomorphic quadratic differential. With notions of both completeness and second fundamental form for surfaces immersed in S_{∞}^{2} (given in §4), the same sorts of existence and uniqueness questions as were asked above may be asked in this case. Here much more satisfactory answers are obtained. A full classification of complete constant mean curvature surfaces at infinity of H^{3} is established.

To put it simply, the space of complete constant mean curvature surfaces at infinity of H^{3}, up to ambient isometry, is parametrized by the holomorphic quadratic differentials on one of S^{2} = Ĉ, C, or D. The parametrization is given by solving the Schwarzian differential equation arising on one of the given spaces from a given quadratic differential, and one can recover the quadratic differential by reading off the holomorphic part of the second fundamental form of our surface at infinity (sense must be made of this). In this way it is seen that the examples of §7 and 8, plus S_{∞}^{2} as the limit of a family of concentric spheres in H^{3}, are precisely the complete examples arising. This terminates the mathematical content of the paper, which is intended to be relatively self-contained and as such includes some proofs of known results.

The author thanks Dennis Sullivan for his interest and support as this paper developed. IHES is also thanked for their wonderful hospitality while the author visited Bures–sur–Yvette. This work also benefitted from the regular comments of Isaac Chavel and Józef Dodziuk.

In closing, a special word of thanks is extended to the author’s thesis adviser, Newton S. Hawley, who died of cancer in early 1988. He was irascible and witty, feisty and gentle. He was a spellbinding instructor who breathed life into the content of his lectures. He pursued his interests with unbounded childlike energy, pulling those who knew him along with him. But most of all he was the author’s dear friend. This paper is dedicated to his memory.

2 Normal flow

Let Σ_{0} ⊂ H^{3} be a smooth, embedded oriented surface patch parametrized by Ω ⊂ R^{2}. Let n⃗ be a continuous unit normal vector field on Σ_{0}. This induces a directed geodesic field on Σ_{0}, with γ_{p} the geodesic in H^{3} passing through p with n⃗(p) a unit tangent to γ_{p} at p. Define a positive direction for each γ_{p} using n⃗(p), and define the maps N_{t} : Σ_{0} → H^{3} by

N_{t}(p) = exp_{p}(tn⃗(p)), (2.1)

for p ∈ Σ_{0}.

Denote by Σ_{t} the image N_{t}(Σ_{0}), and henceforth consider N_{t} as a map from Σ_{0} to Σ_{t}. Note that although the Σ_{t} may not always be manifolds the singularities occurring in this deformation are rather well understood [14]. In an unpublished document [7], C. Epstein studied the geometry of the Σ_{t} in terms of the geometry of Σ_{0}. The family of {Σ_{t}}_{t∈R} will be called a family of parallel surfaces, even though some of the Σ_{t} may be singular or degenerate. This paper makes liberal use of Epstein’s treatment and, as [7] has not seen print, some of that work will first be reviewed.

If X_{1}, X_{2} are standard positively oriented smooth coordinates on Ω, and ∂_{1}, ∂_{2} the corresponding unit vector fields on Σ_{0}, the metric on Σ_{0} induced by its embedding is denoted by (g_{0})_{ij}, with inverse (g_{0})^{ij} = (g_{0})_{ij}^{−1}. Let ∇_{0} denote covariant differentiation on Σ_{0}, and ˜∇ that on H^{3}. The Weingarten map on Σ_{0} is thus (II_{0})_{j}^{i} = (g_{0})^{ik}(II_{0})_{kj} (2.2)

where (II_{0})_{ij} = ⟨ ˜∇_{j}∂_{i}, n⃗⟩ ( ˜∇_{i} = ˜∇_{∂_{i}}). (2.3)

is the second fundamental form.

Note in passing that if k_{0}^{1}(p), k_{0}^{2}(p) are the two eigenvalues of (II_{0})_{j}^{i}(p), i.e. the principal curvatures of Σ_{0} at p, the Gauss and mean curvatures, K(p) and H(p), are defined by

\begin{aligned}K(p) & = & k_{0}^{1}(p)k_{0}^{2}(p) − 1 \\ H(p) & = & \frac{1}{2}(k_{0}^{1}(p) + k_{0}^{2}(p)) = \frac{1}{2}(II_{0})_{i}^{i}. \end{aligned} (2.4)

Using N_{t} we define the forms (g_{t})_{ij}(p), (II_{t})_{ij}(p), and (II_{t})_{j}^{i}(p) on Σ_{0}:

(g_{t})_{ij}(p) = ⟨(N_{t})_{∗}(∂_{i}(p)), (N_{t})_{∗}(∂_{j}(p))⟩,

(II_{t})_{ij}(p) = (g_{t})(∂_{i}(p), ˜∇_{(∂_{j}(p))}n⃗(p)),
(II_{t})_{j}^{i}(p) = (g_{t})^{ik}(p)(II_{t})_{kj}(p).

These are the pullbacks via N_{t} to Σ_{0} of the metric, second fundamental form, and Weingarten map on Σ_{t}, and they evolve via the differential equations

\begin{aligned}\frac{d(g_{t})_{ij}}{dt} & = & −2(II_{t})_{ij}, \\ \frac{d(II_{t})_{j}^{i}}{dt} & = & (II_{t})_{k}^{i}(II_{t})_{j}^{k} − δ_{j}^{i}, \\ \frac{d^{2}(II_{t})_{ij}}{dt^{2}} & = & 4(II_{t})_{ij} \end{aligned} (2.5)

which one solves explicitly by

(g_{t})_{ij} = (g_{0})_{ik} · [(II_{0})_{l}^{k} sinh t − δ_{l}^{k} cosh t]

· [(II_{0})_{j}^{l} sinh t − δ_{j}^{l} cosh t] , (2.6)

etc. We see that both (g_{t})_{ij} and (II_{t})_{ij} are finite for all time, with (g_{t})_{ij} always non-negative definite. Furthermore, Epstein shows that the only singularities in (II_{t})_{j}^{i} arise when (g_{t})_{ij} is degenerate. In doing so, he considers the vector fields ˜∂_{i} obtained from parallel translation of ∂_{i}(0) along the geodesic field, i.e. ˜∂_{i}(0) = ∂_{i}(0) and ˜∇n⃗ ˜∂_{i} = 0. Then if ∂_{i}(0) is a characteristic vector of II_{m}^{l}(0) corresponding to the i-th characteristic value k_{0}^{i}, i.e. ∂_{i}(0) is one of the directions of principal curvature, then so is ∂_{i} for all t ∈ R and

∂_{i}(t) = \frac{1}{2}[(1 − k_{0}^{i})e^{t} + (1 + k_{0}^{i})e^{−t}]˜∂_{i}. (2.7)

It is this equation which tells us how lengths deform under N_{t}, hence how the geometries of the Σ_{t} evolve.

Note 1. How the Σ_{t}’s may degenerate can be seen from (2.5). This has been analyzed thoroughly by Epstein. In particular, if Σ_{0} is a smooth surface patch with directional curvatures k_{0}^{i} satisfying |k_{0}^{i}| ≤ 1 then Σ_{t} is smooth for all t ∈ R. If, however, p ∈ Σ_{0} with some directional curvature k_{0}^{i}(p) > 1 then for t > 0, Σ_{t} is degenerate at γ_{p}(t) only for t = coth^{−1} k_{0}^{i}, and similarly if k_{0}^{i}(p) < −1. Thus Σ_{t} can degenerate at γ_{p}(t) only for at most two values of t. It follows that if in a sufficiently small neighborhood of p the directional curvatures of Σ_{0} are bounded away from ±1 then Σ_{t} are smooth for |t| sufficiently large.

At any rate, it will be useful to have at our avail the corresponding expression in complex notation. Thus if Z = X_{1} + iX_{2}, and Z̄ = X_{1} − iX_{2}, we will use the standard notation for complex derivatives

∂_{Z} = \frac{1}{2}(∂_{1} − i∂_{2}),
∂Z̄ = \frac{1}{2}(∂_{1} + i∂_{2}).

Similarly will ∇_{Z} and ∇Z̄ denote \frac{1}{2}(∇_{1} −i∇_{2}), and \frac{1}{2}(∇_{1} +i∇_{2}), respectively, etc. These live in the complexified tangent bundle of Σ_{0} and if necessary in the complexified tangent bundle of either Σ_{t} or H^{3}.

At times it will be convenient to write these forms and other geometric entities on Σ_{t} in terms of these complex parameters. To this end we make a couple of purely formal observations.

If E dX^{2} + 2F dX dY + G dY^{2} denotes the symmetric quadratic form

( \begin{matrix}E & F \\ F & G \end{matrix} )

on Σ_{t}, then the corresponding hermitian form is A dZ^{2} + 2B dZ dZ̄ + Ā dZ̄^{2}, i.e.

( \begin{matrix}B & A \\ Ā & B \end{matrix} )

where A = \frac{1}{4}(E − 2iF − G), and B = \frac{1}{4}(E + G). Thus, for example, 4(B^{2} − AĀ) = EG − F^{2} tells us how determinants of the real and hermitian forms are related. Similarly, if the quadratic form is a square, i.e.

( \begin{matrix}E & F \\ F & G \end{matrix} )^{2}

in real coordinates, in complex notation it becomes

2 ( \begin{matrix}B & A \\ Ā & B \end{matrix} )^{2}

with A and B as above. In the same fashion the form E dX ∂_{X} +2F (dX ∂_{Y} + dY ∂_{X}) + G dY ∂_{Y} becomes, in complex notation, 2A dZ ∂_{Z} + 2B (dZ ∂Z̄ + dZ̄ ∂_{Z}) + 2Ā dZ̄ ∂Z̄ with A and B as above.

We will denote by h_{t}, (Π_{t})_{j}^{i} and (Π_{t})_{ij} the hermitian forms of g_{t}, (II_{t})_{j}^{i} and (II_{t})_{ij} respectively. If we have h = A dZ^{2} + 2B dZ dZ̄ + Ā dZ̄^{2}, then we will say A = h_{ZZ}, B = h_{ZZ̄}, etc., so that

h = ( \begin{matrix}h_{ZZ̄} & h_{ZZ} \\ hZ̄Z̄ & h_{ZZ̄} \end{matrix} ) .

Finally we note that Σ_{0} ↪ H^{3} inherits a conformal structure, i.e. a notion of an infinitesimal circle, via the immersion. In particular, the Gauss equations for isothermal coordinates (conformal coordinates) on Σ_{0} are

dξ = \frac{1}{λ\sqrt{g_{11}}} (g_{11} dx_{1} + g_{12} dx_{2})

dη = \frac{\sqrt{g_{11}g_{22} − g_{12}^{2}}}{λ\sqrt{g_{11}}} dx_{2}

as is well-known [10, §17]. In ξ, η–coordinates the metric is ds^{2} = λ^{2}(dξ^{2} + dη^{2}). If ζ = ξ +iη, then the hermitian form h is, with respect to this complex parameter, λ^{2} dζ dζ̄. These coordinates will be used presently.

3 Harmonic maps and the deformation of surfaces

This section reviews several phenomena associated with differentiable maps from a smooth surface (with either a Riemannian or a conformal structure) into Riemannian manifolds. Most of this is well known, but we recount it here for the sake of completeness. It will be useful but not required herein to keep in mind the example of normal flow of surfaces discussed in the preceding section.

We consider

1. harmonicity of maps from a surface,

2. harmonicity of quadratic differentials, and

3. quasiconformal deformation and surface maps.

Each of these will be considered in turn.

(1) Harmonic maps. Given Riemannian manifolds (M, g_{M}) and (N, g_{N}) and a smooth map

f : (M, g_{M}) → (N, g_{N}),

suppose X_{i} are coordinates on M. By f^{∗}(g_{N}) (f^{∗}(h_{N}) respectively) will be meant the quadratic form (hermitian form) which is the pull-back by f of g_{N} (h_{N} - the Hermitian form corresponding to g_{N}). When f is understood we will use (g_{N})^{∗} ((h_{N})^{∗}) for f^{∗}(g_{N}) (f^{∗}(h_{N}) respectively). We let ∇ denote the metric covariant differentiation on M. This is defined for quadratic forms [15, ch.6].

The map f is said to be harmonic if and only if

−∇_{k}(g_{M})^{ij}(g_{N})_{ij}^{∗} + 2(g_{M})^{ij}∇_{i}(g_{N})_{jk}^{∗} = 0

for all k [6, §3]. Since ∇_{i}(g_{M})^{ij} = 0 we may rewrite this as

(g_{M})^{ij}(−∇_{k}(g_{N})_{ij}^{∗} + 2∇_{i}(g_{N})_{jk}^{∗}) = 0.

In the case when M is a surface and g_{M} is given in conformal parameters, we have (g_{M})_{ij} = λ^{2}δ_{ij} and the equations for harmonicity become

−∇_{k}(g_{N})_{ii}^{∗} + 2∇_{i}(g_{N})_{ik}^{∗} = 0 (3.1)

for k = 1, 2.

If we rewrite these as

\begin{aligned}∇_{1}((g_{N})_{11}^{∗} − (g_{N})_{22}^{∗}) + 2∇_{2}(g_{N})_{12}^{∗} & = & 0 \\ ∇_{2}((g_{N})_{22}^{∗} − (g_{N})_{11}^{∗}) + 2∇_{1}(g_{N})_{12}^{∗} & = & 0 \end{aligned}

and let Z = X_{1} + iX_{2}, etc., we find that these are the Cauchy–Riemann equations for (h_{N})_{ZZ}^{∗}, i.e. the above equations are the real and negative imaginary parts of the expression

∇Z̄(h_{N})_{ZZ}^{∗} = 0.

Thus we have the following standard.

Lemma 1. If M is a surface, the map f : (M, g_{M}) → (N, g_{N}) is harmonic only if the ZZ–part of the induced hermitian form (h_{N})^{∗} is holomorphic.

(2) Harmonic Quadratic Differentials. There are various notions of harmonicity for tensor fields over manifolds with certain features – like a Riemannian metric or a Kähler structure, etc. Here we have a conformal metric and call a symmetric 2–tensor Q = A dZ^{2} + 2B dZ dZ̄ + Ā dZ̄^{2}, written in conformal coordinates, a quadratic differential. We will follow Kodaira and say that Q is harmonic if and only if

∇_{Z}∇Z̄(A dZ^{2}) + (∇_{Z}∇Z̄ + ∇_{Z}∇Z̄)(B dZ dZ̄) + ∇Z̄∇_{Z}(Ā dZ̄^{2}) = 0.

The explicit covariant derivatives of quadratic differentials in conformal coordinates are precisely

\begin{aligned}∇Z̄(A dZ^{2}) & = & ∂Z̄A dZ^{2} \\ ∇_{Z}∇Z̄(A dZ^{2}) & = & λ^{4} ∂_{Z}(λ^{−4}∂Z̄A) dZ^{2} \\ ∇_{Z}(B dZ dZ̄) & = & λ^{2} ∂_{Z}(Bλ^{−2}) dZ dZ̄ \end{aligned} (3.2)

and their conjugates. Thus A dZ^{2} + 2B dZ dZ̄ + Ā dZ̄^{2} is harmonic if and only if

∂_{Z}(λ^{−4}(∂Z̄A)) = ∂_{Z}∂Z̄(λ^{−2}B) = 0.

We sum this up in lemma 2.

Lemma 2. If Q = A dZ^{2} + 2B dZ dZ̄ + Ā dZ̄^{2} is a harmonic quadratic differential on (M, g) in conformal coordinates, with the hermitian form of the metric h given by h = λ^{2}|dZ|^{2}, then ∂_{Z}(λ^{−4} ∂Z̄A) = 0 and Bλ^{−2} is a harmonic function on M. In particular, if A is holomorphic and B is a constant multiple of the metric, the quadratic differential is harmonic.

Lemma 3. Given Q, (M, g_{M}) as in the preceding lemma, with Q the hermitian form of the Weingarten map for some isometric immersion M↪^{ι}(N, g_{N}) into a Riemannian manifold N of dimension ≥ 3 then Q is harmonic if and only if Bλ^{−2} is constant, or equivalently if and only if A is holomorphic.

Proof: The Codazzi–Mainardi equations for the image of M are ∇Z̄(A dZ^{2}) = ∇_{Z}(B dZ dZ̄) and its conjugate. Thus the holomorphicity of A is equivalent to the constancy of Bλ^{−2} and, as noted in the preceding lemma, both imply the harmonicity of Q.

Now assume Q is harmonic. Covariant differentiation of the first Codazzi– Mainardi equation with respect to Z plus the fact that ∇Z̄∇_{Z}(B dZ dZ̄) = 0 gives that ∇_{Z}(B dZ dZ̄) = cλ^{2} dZ dZ̄ where c is globally constant. From (3.2) we deduce that ∂_{Z}(Bλ^{−2}) = c. Conjugation gives the conclusion that ∂Z̄(Bλ^{−2}) = c̄. Thus Bλ^{−2} = cZ + c̄Z̄ + d, i.e. is real affine, for any choice of conformal coordinate, whence c = 0 and ∇Z̄(A dZ^{2}) = ∇_{Z}(B dZ dZ̄) = 0. The lemma follows. □

(3) Quasiconformal Deformations. Let M again be a surface with Riemannian structure and assume we have conformal coordinates thereon. Let f : (M, g_{M}) → (N, g_{N}) be a map to a general Riemannian manifold N. If the hermitian form induced by f has the matrix form

(h_{N})^{∗} = ( \begin{matrix}β & α \\ ᾱ & β \end{matrix} )^{2} (3.3)

then the Beltrami differential of f is easily expressed by the entries of this matrix. The proof is a straightforward and formal computation.

Lemma 4. The Beltrami differential of f is μ = \frac{ᾱ}{β}.

Assuming (3.3), we make the following definition.

Definition: The Beltrami differential μ of f is given by

μ = \frac{ᾱ}{β}.

We will sometimes call this the “conformal deformation” of M by f.

Definition: The space of Beltrami differentials on M will be denoted by B(M) = F(M) ⊗ T^{0,1}(M) ⊗ T^{−1,0}(M), where F(M) is the vector space of measurable functions on M. Similarly we define B^{∞}(M) = L^{∞}(M) ⊗ T^{0,1}(M) ⊗ T^{−1,0}(M). We give B^{∞}(M) the topology inherited from L^{∞}(M).

If g_{M} is not given in terms of the conformal structure, one lets

h_{M} = ( \begin{matrix}β_{M} & α_{M} \\ ᾱ_{M} & β_{M} \end{matrix} )^{2}

be the metric hermitian form on M and (h_{N})^{∗} the induced hermitian form from f of (3.3). It is readily checked that if

( \begin{matrix}β & α \\ ᾱ & β \end{matrix} )^{2} = ( \begin{matrix}β_{M} & α_{M} \\ ᾱ_{M} & β_{M} \end{matrix} )^{−1} ( \begin{matrix}β_{N} & α_{N} \\ ᾱ_{N} & β_{N} \end{matrix} )^{2} ( \begin{matrix}β_{M} & α_{M} \\ ᾱ_{M} & β_{M} \end{matrix} )^{−1}

then

μ = \frac{ᾱ}{β}.

Note 2. It is important that lemmas 1 and 4 make sense even if N is not a manifold. All that is necessary is that N carries a symmetric quadratic form g_{N} so that the expression (g_{N})^{∗} makes sense. We thus make the following definition.

Definition: The map f : (M, g_{M}) → (N, g_{N}) is said to be harmonic if and only if ∇Z̄(h_{N})_{ZZ}^{∗} = 0. The Beltrami differential of f is given by μ = \frac{ᾱ}{β}.

Note 3. The notions of harmonic maps, holomorphic differentials and conformal deformation are determined by the conformal structure on M, and up to scalar multiplication of the appropriate quadratic forms.

Lemma 1 and the above definition can be generalized a bit if N is also a surface.

Proposition 1. Let (M, g_{M}) and (N, g_{N}) be surfaces. Suppose that f : M → N is a local diffeomorphism. The map f is harmonic if and only if the ZZ–part of (g_{N})^{∗} is holomorphic on M.

Proof: Let Z (W) be a conformal parameter on M (N) such that at x ∈ M (f(x) ∈ N) the metric g_{M} (g_{N}) is given by |dZ|^{2} (|dW |^{2}), and the metric connection vanishes. The rest of the proof now takes place at x ∈ M. The form |dW |^{2} on N pulls back to |W_{Z} dZ + WZ̄ dZ̄|^{2} = W_{Z} \overline{W}_{Z} dZ^{2} + (W_{Z} W̄Z̄ + WZ̄\overline{W}_{Z}) dZ dZ̄ + WZ̄\overline{W} Z̄ dZ̄^{2} on M. The ZZ–part of this is W_{Z} \overline{W}_{Z}.

Now if f is harmonic then W_{ZZ̄} = \overline{W}_{ZZ̄} = 0 so that ∂Z̄(W_{Z} \overline{W}_{Z}) = 0 as desired. In the other direction, if ∂Z̄(W_{Z} \overline{W}_{Z}) = 0 we suppose W_{ZZ̄} ≠ 0. Since f is a local diffeomorphism at every point either W_{Z} or WZ̄ is nonzero.

We’ll assume that W_{Z} ≠ 0, as if WZ̄ ≠ 0 the argument will be identical. As \overline{W}_{ZZ̄} = \overline{(W_{ZZ̄})} we have W_{ZZ̄}\overline{W}_{Z} + W_{Z} \overline{W}_{ZZ̄} = 0 implies

\frac{\overline{W}_{Z}}{W_{Z}} = −\frac{\overline{W}_{ZZ̄}}{W_{ZZ̄}}

is of unit modulus. The Jacobian of f, |W_{Z}|^{2} − |WZ̄|^{2}, must therefore vanish, contradicting the fact that f is a local diffeomorphism. Thus indeed W_{ZZ̄} = 0. □

This argument is due to Newton Hawley [11].

4 Flow to infinity and envelopes of horospheres

In §2 we introduced for each t ∈ (−∞, +∞) a nonnegative definite quadratic form (g_{t})_{ij} (with inverse (g_{t})^{ij} when extant) such that

(g_{t})_{ij} = (g_{0})_{ik}[II_{l}^{k}(0)II_{j}^{l}(0) sinh^{2} t − 2II_{j}^{k}(0) sinh t cosh t + δ_{j}^{k} cosh^{2} t].

This is the pullback by N_{t} of the metric on the possibly singular variety Σ_{t}. As such we will use g_{t} both for the metric on Σ_{t} and its pullback to Σ_{0}. We may for all t ∈ (−∞, +∞) determine whether or not the map N_{t} is harmonic. As was mentioned in note 3, the harmonicity of N_{t} is determined by the definition of g_{t} up to a multiplicative constant. Thus if we give Σ_{t} the metric ĝ_{t} = 4e^{−2t}g_{t}, the map N̂_{t} : Σ_{0} → (Σ_{t}, ĝ_{t}), agreeing pointwise with N_{t}, is harmonic precisely when N_{t} is.

We define the maps N_{±∞} as

N_{+∞}(p) = lim_{t→+∞} exp_{p} (tn⃗(p)) ∈ S_{∞}^{2},

with n⃗(p) as in §2. Similarly for N_{−∞}. The metrics g_{t} diverge as t → ±∞. Thus to consider whether the map N_{+∞} is harmonic or not it is more convenient to note that the ĝ_{t} to converge to a limiting quadratic form ĝ_{+∞} as t → +∞.

Epstein showed [7] that the limiting metric ĝ_{+∞} is nondegenerate if and only if the map N_{+∞} is a local diffeomorphism to S_{∞}^{2}, or equivalently if and only if the Σ_{t} converge at +∞ to a surface immersed in S_{∞}^{2}. Whether or not ĝ_{+∞} is degenerate, the (possibly degenerate) surface on which it lives will be called Σ_{+∞}, and we note that

(ĝ_{+∞})_{ij} = lim_{t→+∞} 4e^{−2t}(g_{t})_{ij} = (g_{0})_{ik}[(II_{0})_{l}^{k}(II_{0})_{j}^{l} − 2(II_{0})_{j}^{k} + δ_{j}^{k}].

We make a similar definition when t → −∞.

This suggests a definition.

Definition: We say that N_{+∞} is harmonic if and only if the map N̂_{+∞} : Σ_{0} → (Σ_{+∞}, ĝ_{+∞}), agreeing pointwise with N_{+∞}, is harmonic. Similarly for N_{−∞}.

Note 4. Epstein [7] has shown that where Σ_{+∞} is nondegenerate, its conformal structure is precisely that inherited from S_{∞}^{2}. This is intuitively clear as if X_{1}, X_{2} form an orthonormal basis for T_{p}Σ_{0} then parallel translation along γ_{p} to N_{t}(p) gives an orthonormal basis for T_{N_{t}(p)}Σ_{t}. However if we orient such an orthonormal basis and project out to S_{∞}^{2}, one of the orientations is the same as that of S_{∞}^{2}, while the other is different. Our convention will be to give the Σ_{t} orientations opposite those induced by H^{3} with normal n⃗(p, t) on Σ_{t} (see §2). Thus Σ_{+∞} is given an inward pointing normal so that its orientation is that induced by S_{∞}^{2} = Ĉ. In the same fashion Σ_{−∞} inherits an outward pointing normal so that its orientation is opposite that induced by S_{∞}^{2} = Ĉ.

Note 5. In a family of parallel surfaces, there is no natural choice of Σ_{0}. These families are naturally parametrized by R as a metric space with its Euclidean metric, not as a vector space. Thus for a different parametrization of the family τ = t − t_{0} we have

(ĝ_{τ=+∞})_{ij} = e^{2t_{0}}(ĝ_{t=+∞})_{ij}.

We see then that the quadratic form on Σ_{+∞} is determined up to multiplication by a positive constant by the family of parallel surfaces. Hence whether or not the maps N_{t_{2}}^{t_{1}} = N_{t_{2}} ∘ (N_{t_{1}})^{−1} : Σ_{t_{1}} → Σ_{t_{2}} (for any choice of Σ_{0}) are harmonic is intrinsic to the family of surfaces, even when the t_{i} are ±∞.

Furthermore, the completeness of a Riemannian manifold (M, g) is equivalent to the completeness of (M, cg) for any c ∈ R_{+}. Thus we are led to the following definition.

Definition: We say that Σ_{+∞} is complete if ĝ_{+∞} is positive definite and the Riemannian manifold (Σ_{+∞}, ĝ_{+∞}) is complete. Since ĝ_{+∞} is determined up to a constant, it follows that the completeness of Σ_{+∞} is determined by our family of parallel surfaces.

We can represent geometric information on Σ_{+∞} in terms of that on Σ_{0}. It is not far fetched to suppose we can reverse this. We can construct each Σ_{t} from Σ_{0} by considering the envelope of spheres of radius t about points on Σ_{0}, we construct Σ_{0} from Σ_{t} in the same fashion. Now let us assume that Σ_{+∞} is immersed in S_{∞}^{2}. Heuristically, letting t → +∞, we construct Σ_{0} from Σ_{+∞} as an envelope of horospheres with centers on Σ_{+∞}. Thus, we ask, given a domain Ω ⊂ S_{∞}^{2} and a choice of horosphere at each point θ ∈ Ω, how do we reconstruct Σ_{0}; that is, what sort of conditions on the family of horospheres allow us to integrate to get Σ_{0}, and, given that we can indeed integrate, how do we extract geometric information on Σ_{0} from the family of horospheres?

These questions have also been answered by Epstein [7]. Picking a basepoint x ∈ H^{3} and if θ ∈ S_{∞}^{2} is the center of the horosphere ν_{θ}, we say that ν_{θ} has radius ρ ∈ R with respect to x if ±d_{H}(x, ν_{θ}) = ρ, where we take − if x is interior to ν_{θ} and + if not. Thus if we have a distribution of horospheres ν_{θ} on Ω ⊂ S_{∞}^{2}, on choosing a base point x ∈ H^{3} we can give criteria for the integrability of ν_{θ} in terms of the radius function ρ(θ). In fact Epstein shows that this distribution is integrable if ρ is C^{1}. Clearly this does not depend on the choice of x. Of course to extract geometric information (metric form, curvature, etc.) on integral surfaces from knowledge of ρ we need more derivatives. Let us assume as many derivatives as necessary to make the computations work. In general ρ being C^{4} suffices.

Fixing x ∈ H^{3}, we have a well defined metric on S_{∞}^{2} with constant curvature ≡ +1 by visual identification with the sphere of unit tangent vectors at x. Identifying S_{∞}^{2} with Ĉ via stereographic projection, so that θ ∈ S_{∞}^{2} is 0 ∈ C, and taking z to be a standard planar coordinate in a neighborhood of 0 ∈ Ω, we can express the metric and second fundamental form on Σ_{t} in a rather straightforward fashion.

The hermitian form of the metric on Σ_{t} is given by Epstein [7] as

h_{t}|_{z=0} = \frac{1}{2}e^{2(ρ+t)} ( \begin{matrix}(∂ ¯∂ρ − 1)e^{−2(ρ+t)} + 1 & (∂^{2}ρ − (∂ρ)^{2})e^{−2(ρ+t)} \\ (¯∂^{2}ρ − (¯∂ρ)^{2})e^{−2(ρ+t)} & (∂ ¯∂ρ − 1)e^{−2(ρ+t)} + 1 \end{matrix} )^{2} (4.1)

while

(Π_{t})_{j}^{i}|_{z=0} = D^{−1} [( \begin{matrix}0 & 2A \\ 2Ā & 0 \end{matrix} ) + ωId] (4.2)

where

\begin{aligned}A & = & ∂^{2}ρ − (∂ρ)^{2} \\ D & = & e^{2(ρ+t)} [|1 + (∂ ¯∂ρ − 1)e^{−2(ρ+t)}|^{2} − |Ae^{−2(ρ+t)}|^{2}] \\ ω & = & e^{2(ρ+t)} [((∂ ¯∂ρ − 1)e^{−2(ρ+t)})^{2} − |Ae^{−2(ρ+t)}|^{2} − 1] . \end{aligned} (4.3)

Evidently at z = 0 we get ĥ_{+∞} = lim_{t→+∞} e^{−2t}h_{t} to be

ĥ_{+∞}|_{z=0} = \frac{1}{2}e^{2ρ} ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )^{2} .

Thus at the center of our stereographic coordinate system, z = 0 on Σ_{+∞}, the length element is e^{2ρ}|dz|^{2}. Since the surface Σ_{0} determines the metric ĥ_{+∞}, we can read off from h_{0} and Π_{0} the support function ρ. This leads to the following theorem.

Theorem. [7] The support function ρ for Σ_{0} on Σ_{+∞} determines the metrics h_{t} (or ĥ_{t} = e^{−2t}h_{t}) for all t ∈ [−∞, +∞]. Conversely, any of the h_{t} determine the support function ρ.

Also at z = 0 we get (Π_{±∞})_{j}^{i} = ∓id. Thus the mean curvatures of the surfaces Σ_{t} approach ∓1 pointwise as t → ±∞. However, it appears that the off diagonal parts of (Π_{±∞})_{j}^{i} vanish.We wish to note that infinitesimally these terms persist in (Π_{t})_{j}^{i} as t → ±∞. We do this by letting τ = e^{−2t} and taking the 1–jet at τ = 0. Since

lim_{t→+∞} \frac{[(zz–term of (Π_{t})_{j}^{i})]}{2Aτ} = e^{−2ρ},

for example, we use the following definition.

Definition: The second fundamental form of Σ_{+∞} is defined by

(Π_{+∞})_{j}^{i}|_{z=0} = ( \begin{matrix}−1 + 2(∂ ¯∂ρ − 1)e^{−2ρ} dτ & 2Ae^{−2ρ} dτ \\ 2Āe^{−2ρ} dτ & −1 + 2(∂ ¯∂ρ − 1)e^{−2ρ} dτ \end{matrix} ) .

(4.4)

The zz-term of (Π_{+∞})_{j}^{i} is now defined as the infinitesimal Beltrami parameter 2A dτ. In the same way we may obtain the expression for (Π_{−∞})_{j}^{i}|_{z=0}.

The intrinsic curvature of Σ_{t} is readily computed from (Π_{t})_{j}^{i}. One checks that det(Π_{t})_{j}^{i} = det(II_{t})_{j}^{i} so that

K_{t} = det(Π_{t})_{j}^{i} − 1.

Thus to compute the intrinsic curvature of (Σ_{+∞}, ĥ_{+∞}) we use the fact that K(cg) = c^{−1}K(g) for any metric g, and obtain

\begin{aligned}K̂_{+∞}|_{z=0} & = & K(ĥ_{+∞})|_{z=0} \\ & = & lim_{t→+∞} \frac{e^{2t}}{4} (det(Π_{t})_{j}^{i} − 1) \\ & = & (1 − ∂ ¯∂ρ)e^{−2ρ} \end{aligned} (4.5)

We record this as the following proposition.

Proposition 2. For a family of parallel surfaces Σ_{t} (t ∈ [−∞, +∞]) such that Σ_{0} has support function ρ from Σ_{+∞} for the choice of origin x ∈ H^{3}, the curvature of (Σ_{+∞}, ĥ_{+∞}) is precisely K̂_{+∞} = (1 − Δ_{S^{2}}ρ)e^{−2ρ}.

This entire discussion can be made independent of basepoint on Σ_{+∞}. We allow for fractional linear transformations by letting ρ′ = ρ − log (1 + |z|^{2}) in (4.1) – (4.4). For example, we find that

h_{t} = \frac{1}{2}e^{2(ρ′+t)} ( \begin{matrix}(∂ ¯∂ρ′)e^{−2(ρ′+t)} + 1 & (∂^{2}ρ′ − (∂ρ′)^{2})e^{−2(ρ′+t)} \\ (¯∂^{2}ρ′ − (¯∂ρ′)^{2})e^{−2(ρ′+t)} & (∂ ¯∂ρ′)e^{−2(ρ′+t)} + 1 \end{matrix} )^{2} . (4.6)

At t = +∞ we get

ĥ^{+∞} = \frac{1}{2}e^{2ρ′} ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )^{2} (4.7)

and

(Π_{+∞})_{j}^{i} = ( \begin{matrix}−1 + 2(∂ ¯∂ρ′)e^{−2ρ′} dτ & 2Ae^{−2ρ′} dτ \\ 2Āe^{−2ρ′} dτ & −1 + 2(∂ ¯∂ρ′)e^{−2ρ′} dτ \end{matrix} ) . (4.8)

with K̂_{+∞} = −e^{−2ρ′}(∂ ¯∂ρ′). We repose proposition 1 in these terms as follows.

Proposition 3. For a family of parallel surfaces Σ_{t} (t ∈ [−∞, +∞]) such that Σ_{0} has support function ρ from Σ_{+∞} for the choice of origin x ∈ H^{3}, the hermitian metric ĥ_{+∞} is given by e^{ρ′}|dz|^{2} with the curvature K̂_{+∞} = −e^{−2ρ′}(∂ ¯∂ρ′).

It is nice to note that the Codazzi–Mainardi equations hold in this boundary setting.

Proposition 4. The Codazzi–Mainardi equations hold for (Π_{+∞})_{ij}, i.e. if

(Π_{+∞})_{ij} = ( \begin{matrix}e^{2ρ′}(−1 + 2(∂ ¯∂ρ′)e^{−2ρ′} dτ) & 2A dτ \\ 2Ā dτ & e^{2ρ′}(−1 + 2(∂ ¯∂ρ′)e^{−2ρ′} dτ) \end{matrix} )
= ( \begin{matrix}B & A \\ Ā & B \end{matrix} )

then

∇Z̄(A dZ^{2}) = ∇_{Z}(B dZ dZ̄).

Proof: From (3.2) we must establish the equality

¯∂A = e^{2ρ′}∂(Be^{−2ρ′}).

But this follows from the equality

¯∂(∂^{2}ρ′ − (∂ρ′)^{2}) = e^{2ρ′}∂((∂ ¯∂ρ′)e^{−2ρ′}). □

5 Constant mean curvature.

We now have the following theorem.

Theorem 1. Let Σ_{0} ⊂ H^{3} be an oriented smooth surface with metric λ^{2}dZdZ̄. The following are equivalent:

1. Σ_{0} has constant mean curvature;

2. (Π_{0})_{ij} is harmonic; 3. the ZZ–part of (Π_{0})_{ij} is holomorphic on Σ_{0};

4. N_{t} : Σ_{0} → Σ_{t} is harmonic for all t ∈ [−∞, +∞];

5. there exists t ∈ [−∞, 0) ∪ (0, +∞] such that N_{t} : Σ_{0} → Σ_{t} is harmonic;

6. μ_{t}(p) = f(t) · g(p) where f : [−∞, +∞] → R ∪ {∞} is injective and g ∈ B(Σ_{0});

7. μ_{t}(p) = f(t) · g(p) where f(t) = \frac{tanh t}{1 − c tanh t}, with c a constant (c

= H(Σ_{0})) and g ∈ \overline{Q(Σ_{0})} × λ^{−2}.

Proof: The equivalence of (1), (2) and (3) is immediate from lemma 3.

That (1) is equivalent to (4) and (5) is straightforward, though we give a sketch for the sake of completeness. First we discuss harmonicity of surface maps in our setting of normal flow.

As for fixed t

(g_{t})_{ij} = (g_{0})_{il}[(II_{0})_{m}^{l}(II_{0})_{j}^{m} sinh^{2} t − 2(II_{0})_{j}^{l} sinh t cosh t + δ_{j}^{l} cosh^{2} t],

we use the definition of harmonicity given by (3.1). It follows that

−∇_{k}(g_{0})^{ij}(g_{t})_{ij} + 2(g_{0})^{ij}∇_{i}(g_{t})_{jk}

= −∇_{k} ((g_{0})^{ij}(g_{0})_{ip}[(II_{0})_{m}^{p}(II_{0})_{j}^{m} sinh^{2} t − 2(II_{0})_{j}^{p} sinh t cosh t])

+2(g_{0})^{ij}∇_{i} ((g_{0})_{jp}[(II_{0})_{m}^{p}(II_{0})_{k}^{m} sinh^{2} t − 2(II_{0})_{k}^{p} sinh t cosh t])

−δ_{p}^{j} [((II_{0})_{m;k}^{p}(II_{0})_{j}^{m} + (II_{0})_{m}^{p}(II_{0})_{j;k}^{m}) sinh^{2} t − 2(II_{0})_{j;k}^{p} sinh t cosh t]

+2δ_{p}^{i} [((II_{0})_{m;i}^{p}(II_{0})_{k}^{m} + (II_{0})_{m}^{p}(II_{0})_{k;i}^{m}) sinh^{2} t − 2(II_{0})_{k;i}^{p} sinh t cosh t]

= −((II_{0})_{m;k}^{j}(II_{0})_{j}^{m} + (II_{0})_{m}^{j}(II_{0})_{j;k}^{m}) sinh^{2} t + 2(II_{0})_{j;k}^{j} sinh t cosh t

+2((II_{0})_{m;i}^{i}(II_{0})_{k}^{m} + (II_{0})_{m}^{i}(II_{0})_{k;i}^{m}) sinh^{2} t − 4(II_{0})_{k;i}^{i} sinh t cosh t

= 0,

and by using the Codazzi–Mainardi equations, i.e. (II_{0})_{j;k}^{i} = (II_{0})_{k;j}^{i}, we are able to reduce this to

2 sinh t [(II_{0})_{k}^{p} sinh t − δ_{k}^{p} cosh t] (II_{0})_{i;p}^{i} = 0.

We consider this as a system of equations in (II_{0})_{i;p}^{i} for p = 1, 2, i.e.

\{ \begin{matrix}[(II_{0})_{1}^{1} sinh t − cosh t] (II_{0})_{i;1}^{i} + (II_{0})_{1}^{2} sinh t (II_{0})_{i;2}^{i} & = & 0 \\ (II_{0})_{2}^{1} sinh t (II_{0})_{i;1}^{i} + [(II_{0})_{2}^{2} sinh t − cosh t] (II_{0})_{i;2}^{i} & = & 0. \end{matrix} (5.1)

We wish to show that this system has only the trivial solution. To have non-trivial solutions it is necessary that the discriminant is zero, i.e. that

((II_{0})_{1}^{1}(II_{0})_{2}^{2} − (II_{0})_{2}^{1}(II_{0})_{1}^{2}) sinh^{2} t − ((II_{0})_{1}^{1} + (II_{0})_{2}^{2}) sinh t cosh t + cosh^{2} t = 0.

This means that coth t is everywhere an eigenvalue of (II_{0})_{j}^{i}. Hence at least one of the principal curvatures is fixed if (5.1) is to have non-trivial solutions. We will now see that the other is also fixed, so that indeed no non-trivial solutions can exist.

We may assume (II_{0})_{1}^{1} = coth t, (II_{0})_{2}^{2} ≠ coth t and (II_{0})_{j}^{i} is diagonal, i.e. k_{1} ≠ k_{2}. We wish to show that k_{2} is constant, whence (II_{0})_{i}^{i} is constant. From (5.1) we have [(II_{0})_{2}^{2} − coth t](II_{0})_{i;2}^{i} = 0, (5.2)

whence (II_{0})_{i;2}^{i} = 0. But we already have (II_{0})_{1;2}^{1} = 0, so that (II_{0})_{2;2}^{2} = 0. We also have (II_{0})_{2;1}^{2} = (II_{0})_{1;2}^{2} = 0. Thus indeed (II_{0})_{2}^{2} is constant, whence (II_{0})_{i}^{i} is also constant, and (5) implies (1).

That (1) implies (4) is evident from (5.1), and of course (4) implies (5). To establish (6) and (7), we will prove (6) and note that f : [−∞, +∞] →

[−∞, +∞] and g ∈ L^{∞} ⊗ T^{1,−1}(Σ_{0}) determine the constant mean curvature (up to sign) and are determined thereby (up to multiplicative constants). Furthermore we will show that g must be anti-holomorphic, establishing the theorem.

As mentioned earlier, the metric on Σ_{t} is given in terms of the metric, (g_{0})_{ij}, and the Weingarten map, (II_{0})_{ij}, on Σ_{0}. If we have a conformal coordinate Z = X_{1} + iX_{2} on Σ_{0}, so that with respect to X_{1} and X_{2} we have

(g_{0})_{ij} = ( \begin{matrix}λ & 0 \\ 0 & λ \end{matrix} )^{2} and (II_{0})_{ij} = ( \begin{matrix}L & M \\ M & N \end{matrix} ) ,

then the metric g_{ij}^{t} on Σ_{t} is

(g_{t})_{ij} = \{[( \begin{matrix}L & M \\ M & N \end{matrix} ) sinh t − ( \begin{matrix}λ^{2} & 0 \\ 0 & λ^{2} \end{matrix} ) cosh t] ( \begin{matrix}λ^{−1} & 0 \\ 0 & λ^{−1} \end{matrix} )\}^{2} .

In complex coordinates the corresponding hermitian forms are thus

h_{0} = \frac{1}{2} ( \begin{matrix}λ & 0 \\ 0 & λ \end{matrix} )^{2}

and

h_{t} = \frac{1}{2} ( \begin{matrix}\frac{(L + N)}{2λ} sinh t − λ cosh t & \frac{(L − N − 2iM)}{2λ} sinh t \\ \frac{(L − N + 2iM)}{2λ} sinh t & \frac{(L + N)}{2λ} sinh t − λ cosh t \end{matrix} )^{2} .

From this we compute the “change of complex structure” matrix to be

\sqrt{h_{0}} (h_{t})^{−1} \sqrt{h_{0}} = ( \begin{matrix}β_{t} & α_{t} \\ ᾱ_{t} & β_{t} \end{matrix} )^{2}

= 4 ( \begin{matrix}\frac{(L + N)}{2λ^{2}} sinh t − cosh t & \frac{(L − N − 2iM)}{2λ^{2}} sinh t \\ \frac{(L − N + 2iM)}{2λ^{2}} sinh t & \frac{(L + N)}{2λ^{2}} sinh t − cosh t \end{matrix} )^{2} .

It follows that

μ_{t} = \frac{ᾱ_{t}}{β_{t}} = \frac{(L − N + 2iM) tanh t}{(L + N) tanh t − 2λ^{2}}
μ_{+∞} = \frac{(L − N + 2iM)}{(L + N) − 2λ^{2}}

and

\frac{μ_{+∞}}{μ_{t}} = 1 − \frac{2λ^{2}(1 − tanh t)}{((L + N) − 2λ^{2}) tanh t}.

Thus \frac{μ_{+∞}}{μ_{t}} depends only on t if and only if \frac{2λ^{2}}{(L + N) − 2λ^{2}} does, i.e. if

and only if \frac{(L + N)}{2λ^{2}} is independent of z and z̄ (x and y, respectively). That is, if and only if the trace of (II_{0})_{j}^{i}, the mean curvature, is constant. This establishes the equivalence of (1) and (6).

We have (7) by noting in the above that if

c = \frac{(L + N)}{2λ^{2}}

then

μ_{t} = −\frac{(L − N + 2iM)}{2λ^{2}} · \frac{tanh t}{(1 − c tanh t)}.

Since the Z, Z–part of Π, i.e. Ȧ, is precisely

Ȧ = −\frac{(L − N − 2iM)}{2}

and is holomorphic, the proof is concluded. □

Corollary 1. If Σ_{0} has constant mean curvature, then Σ_{t_{0}} (t_{0} ≠ 0) does also if and only if either

1. tanh t_{0} = \frac{1}{c} (so that c ≠ 0) and H(Σ_{t_{0}}) = −H(Σ_{0}), or

2. every surface Σ_{t} has constant mean and constant intrinsic curvatures.

In the second case, the surfaces Σ_{t} are all either pieces of metric spheres, horospheres, cylinders equidistant from a geodesic, or surfaces equidistant from a totally geodesic plane.

Proof: Given that Σ_{0} has constant mean curvature c, we have

h_{t} = \frac{1}{2}λ^{2} ( \begin{matrix}c sinh t − cosh t & Ȧλ^{−2} sinh t \\ ¯Ȧλ^{−2} sinh t & c sinh t − cosh t \end{matrix} )^{2} .

We note that

\sqrt{h_{t_{0}}} (h_{t})^{−1} \sqrt{h_{t_{0}}} = \frac{1}{det(h_{t_{0}})} ( \begin{matrix}D & C \\ C̄ & D \end{matrix} )^{2} .

where

C = Ȧλ^{−2}(sinh t_{0} cosh t − cosh t_{0} sinh t)
D = (c^{2} − |Ȧλ^{−2}|^{2}) sinh t_{0} sinh t
−c(sinh t_{0} cosh t + cosh t_{0} sinh t) + cosh t_{0} cosh t.

The conformal deformation of N_{t}^{t_{0}} = N_{t} ∘ (N_{t_{0}})^{−1}, as noted in §3, is thus

μ_{t}^{t_{0}} = \frac{¯Ȧλ^{−2}(tanh t_{0} − tanh t)}{(c^{2} − |Ȧλ^{−2}|^{2}) tanh t_{0} tanh t − c(tanh t_{0} + tanh t) + 1}.

and similarly for N_{+∞}^{t_{0}} = N_{+∞} ∘ (N_{t_{0}})^{−1},

μ_{+∞}^{t_{0}} = \frac{¯Ȧλ^{−2}(tanh t_{0} − 1)}{(c^{2} − |Ȧλ^{−2}|^{2}) tanh t_{0} − c(tanh t_{0} + 1) + 1}.

This yields

(5.3) \frac{μ_{+∞}^{t_{0}}}{μ_{t}^{t_{0}}} = \frac{(c^{2} − |Ȧλ^{−2}|^{2}) tanh t_{0} tanh t − c(tanh t_{0} + tanh t) + 1}{(c^{2} − |Ȧλ^{−2}|^{2}) tanh t_{0} − c(tanh t_{0} + 1) + 1} × (\frac{tanh t_{0} − 1}{tanh t_{0} − tanh t}) = tanh t + \frac{(c tanh t_{0} − 1)(tanh t − 1)}{(c^{2} − |Ȧλ^{−2}|^{2}) tanh t_{0} − c(tanh t_{0} + 1) + 1} × (\frac{tanh t_{0} − 1}{tanh t_{0} − tanh t}) ,

which depends only on t if and only if either

1. tanh t_{0} = \frac{1}{c}, or

2. |Ȧλ^{−2}| is constant.

Case 1: If tanh t_{0} = \frac{1}{c}, then

μ_{t}^{t_{0}} = \frac{¯Ȧλ^{−2}(1 − c tanh t)}{−|Ȧλ^{−2}|^{2} tanh t}
= \frac{(1 − c tanh t)}{−Ȧλ^{−2} tanh t},
μ_{+∞}^{t_{0}} = \frac{¯Ȧλ^{−2}(1 − c)}{−|Ȧλ^{−2}|^{2}}
= \frac{(1 − c)}{−Ȧλ^{−2}} .

It follows that

\frac{μ_{+∞}^{t_{0}}}{μ_{t}^{t_{0}}} = \frac{(1 − c) tanh t}{(1 − c tanh t)}.

Letting t = τ + t_{0} we have

μ_{t}^{t_{0}} = \frac{¯Ȧλ^{−2}(1 − c^{2})}{−|Ȧλ^{−2}|^{2}} \frac{tanh τ}{(c tanh τ + 1)} = \frac{(1 − c^{2})}{−Ȧλ^{−2}} \frac{tanh τ}{1 + c tanh τ}

and by equivalence 7 we have H(Σ_{t_{0}}) = −c.

Case 2: |Ȧλ^{−2}| being constant implies that every μ_{t} is constant, whence, as

|μ_{t}| = \frac{tanh t}{1 − c tanh t} · \frac{|k_{1} − k_{2}|}{2},

we deduce that k_{1} and k_{2} are both constant.

The final conclusion follows from a result of Cartan [5]. He completely classified surfaces in H^{3} for which H and K are constant. These are called isoparametric surfaces.

Theorem. [5] If Σ_{0} is a surface in H^{3} for which H and K are constant then one of the following cases must hold:

1. Σ_{0} is a piece of a metric sphere;

2. Σ_{0} is a piece of a horosphere;

3. Σ_{0} is a piece of a surface equidistant from a hyperbolic geodesic; or

4. Σ_{0} is a piece of a surface equidistant from a totally geodesic plane in H^{3}.

This completes the proof of the corollary. □

Corollary 2. Σ_{0} is minimal if and only if μ_{t}(p) = f(t) · g(p) with f : [−∞, +∞] → R ∪ {∞} odd. In this case f(t) = tanh t.

Proof: This is immediate from equivalences 1, 6, and 7 of theorem 1. □

Corollary 3. If H(Σ_{0}) = c and the N_{t} are not all conformal, we can recover c by knowing only the conformal deformations to ±∞, i.e. by knowing μ_{±∞}.

Proof: From equivalence (7),

μ_{−∞}(p) = \frac{−1}{1 + c}g(p)  and  μ_{+∞}(p) = \frac{1}{1 − c}g(p).

If g ≢ 0 then

M = \frac{μ_{−∞}}{μ_{+∞}} = \frac{c − 1}{c + 1}

so that

c = \frac{M + 1}{1 − M}. □

Corollary 4. If Σ_{0} ↪ H^{3} and H(Σ_{0}) = 1, let Ȧ = ZZ–part of Π on Σ_{0}. Then N_{+∞} is anti-conformal and N_{−∞} has conformal distortion −\frac{1}{1 + c} ¯Ȧλ^{−2}.

Proof: If H(Σ_{0}) = 1 then

μ_{t} = \frac{tanh t}{1 − tanh t} ¯Ȧλ^{−2}

so that μ_{+∞} is infinite. Thus N_{+∞} is anticonformal and μ_{−∞} = \frac{1}{2} ¯Ȧλ^{−2} as desired. □

Surfaces Σ_{0} ↪ H^{3} with constant mean curvature H(Σ_{0}) = 1 have been considered by Bryant [4].

Corollary 5. If Σ_{0} has constant mean curvature, then the map μ : [−∞, +∞] → B(Σ_{0}) actually maps into B^{∞}(Σ_{0}), and is continuous if and only if |k_{1} − k_{2}| is bounded, i.e. if and only if the directional curvatures of Σ_{0} are bounded.

Proof:

|μ_{t}| = \frac{tanh t}{1 − c tanh t} · \frac{|k_{1} − k_{2}|}{2},

as in general

|μ_{t}| = \frac{|\frac{k_{1}−k_{2}}{2} || tanh t|}{|1 − \frac{k_{1}+k_{2}}{2} tanh t|}. □

6 Constant mean curvature surfaces at infinity, the local theory

We assume now that Σ_{+∞} is the embedded image of a surface patch in S_{∞}^{2}. Since a family of parallel surfaces in H^{3} is determined by a metric on Σ_{+∞} which induces the same conformal structure on Σ_{+∞} as S_{∞}^{2} (note 4, §4), we make the following definition.

Definition: Σ_{+∞} will be called a surface patch at infinity of H^{3} if it is the conformally embedded image in S_{∞}^{2} of a metric surface patch. A surface at infinity will refer to the conformally immersed image in S_{∞}^{2} of a metric surface.

We observed in §4 that if ρ is the support function for Σ_{0} from any x ∈ H^{3} then the hermitian form h^{t} is given by

h^{t} = \frac{1}{2}e^{2(ρ′+t)} ( \begin{matrix}(∂ ¯∂ρ′)e^{−2(ρ′+t)} + 1 & (∂^{2}ρ′ − (∂ρ′)^{2})e^{−2(ρ′+t)} \\ (¯∂^{2}ρ′ − (¯∂ρ′)^{2})e^{−2(ρ′+t)} & (∂ ¯∂ρ′)e^{−2(ρ′+t)} + 1 \end{matrix} )^{2}
= ( \begin{matrix}β_{t} & α_{t} \\ ᾱ_{t} & β_{t} \end{matrix} )^{2}

where ρ′ = ρ − log (1 + |z|^{2}). We will now need the expressions introduced at the end of §4 for ρ′, h_{t}, (Π_{t})_{j}^{i}, ĥ_{+∞}, and (Π_{+∞})_{j}^{i}.

First we make several preliminary observations. We have the following proposition.

Proposition 5. A surface Σ_{+∞} at infinity of H^{3} must have H(Σ_{+∞}) = ±1.

Proof: This is immediate from the observation following the theorem of §4. □

Following our definition of (Π_{+∞})_{j}^{i} we’ll take as our normalization that Σ_{+∞} has mean curvature then this H(Σ_{+∞}) ≡ −1.

Secondly, we measure the conformal distortion from Σ_{+∞} to Σ_{t} by considering N_{t}^{+∞} = N_{t} ∘(N_{+∞})^{−1} : Σ_{+∞} → Σ_{t}. The conformal deformation μ_{t}^{+∞} for N_{t}^{+∞} is thus

μ_{t}^{+∞} = \frac{∂z̄N_{t}^{+∞}}{∂_{z}N_{t}^{+∞}} ,

where z is a conformal parameter on Σ_{+∞}. We take z to be a standard coordinate on S_{∞}^{2}, and compute the conformal deformation from Σ_{+∞} to be

μ_{t}^{+∞} = \frac{ᾱ_{t}}{β_{t}} = \frac{(¯∂^{2}ρ′ − (¯∂ρ′)^{2})e^{−2(ρ′+t)}}{(∂ ¯∂ρ′)e^{−2(ρ′+t)} + 1} .

Consequently

μ_{−∞}^{+∞} = \frac{¯∂^{2}ρ′ − (¯∂ρ′)^{2}}{∂ ¯∂ρ′} ,

and

\frac{μ_{−∞}^{+∞}}{μ_{t}^{+∞}} = 1 + e^{2t} \frac{1}{e^{−2ρ′}(∂ ¯∂ρ′)}.

This yields, from the definition of K̂_{+∞} in §4, the following theorem. Theorem 2. The image of μ_{( )}^{+∞} : [−∞, +∞] → B(Σ_{+∞}) lies in a radial line in B(Σ_{+∞}) if and only if K̂_{+∞} is constant.

Hence when the image μ_{[−∞,+∞]}^{+∞} lies in a radial line in B(Σ_{+∞}) we have

μ_{t}^{+∞} = \frac{(¯∂^{2}ρ′ − (¯∂ρ′)^{2})e^{−2(ρ′+t)}}{1 + ce^{−2t}} (6.1)

for some c ∈ R.

Finally, we have the following proposition. Proposition 6. If μ_{t_{0}}^{+∞} ≡ 0 for some t_{0} ∈ [−∞, +∞) then this is so for all t ∈ [−∞, +∞]. Proof: If N_{t_{0}}^{+∞} is conformal then we see from (6.1) that ¯∂^{2}ρ − (¯∂ρ)^{2} ≡ 0. Thus μ_{t}^{+∞} ≡ 0 in t. □

Thus when the equivalence of theorem 2 holds we have four possible cases:

1. μ_{t}^{+∞} ≡ 0;

2. μ_{t}^{+∞} = f(t)·μ_{0}^{+∞} with μ_{0}^{+∞} ≢ 0 and f(t) is bounded and never infinite.

3. μ_{t}^{+∞} = f(t) · μ_{0}^{+∞} with μ_{0}^{+∞} ≢ 0 and |f(t)| → ∞ as t → −∞; or

4. μ_{t}^{+∞} = f(t) · μ_{t_{0}}^{+∞} with μ_{t_{0}}^{+∞} ≢ 0, f(t) is infinite for some finite value of t and bounded as t → −∞.

The image μ_{[−∞,+∞]}^{+∞} ⊂ B(Σ_{+∞}) ∪ ∞ in these four cases are shown in the adjoining figures 1—4. We treat these separately.

Case 1: This should be compared to case 2 of corollary 1, §5. As there, μ_{t}^{+∞} ≡ 0 implies that N_{t}^{t_{0}} = N_{t}^{+∞} ∘ (N_{t_{0}}^{+∞})^{−1} is conformal for all t_{0}, t. Thus all Σ_{t} are either pieces of spheres, horospheres, cylinders about a geodesic, or surfaces equidistant from a totally geodesic plane.

Case 2: Here e^{−2ρ′}(∂ ¯∂ρ′) is a nonzero constant. In fact it is a positive constant c > 0, as otherwise μ_{t_{0}}^{+∞} would be infinite for some value of t_{0} ∈ [−∞, +∞), contrary to the assumption in this case. Thus we see that ρ′ satisfies

∂ ¯∂ρ′ = ce^{2ρ′},

which is precisely the equation for constant negative curvature. As was shown in §4, we may rescale the metric on Σ_{+∞} by a constant and thus assume K̂_{+∞} ≡ −1. Thus, we must have

ρ = log (|ψ′| · \frac{1 + |z|^{2}}{1 − |ψ|^{2}} )

μ_{t}^{+∞} ≡ 0 Figure 1:

[figure: μ_{−∞}^{+∞}] Figure 2:

[figure: |μ_{−∞}^{+∞}| ≡ ∞] Figure 3:

[figure: μ_{−∞}^{+∞}] Figure 4: where ψ : Σ_{+∞} → D. In words, we have shown that in this case the metric on Σ_{+∞} is the pull–back of the complete constant −1 curvature hyperbolic metric on D via some locally univalent map ψ : Σ_{+∞} → D.

Note that

μ_{t}^{+∞} = \frac{(¯∂^{2}ρ′ − (¯∂ρ′)^{2})e^{−2(ρ′+t)}}{1 + e^{−2t}} .

We quickly check that for no t_{0} ∈ [−∞, +∞) is Σ_{t_{0}} a surface of constant mean curvature.

Using the formula for the transformation of Beltrami parameters [1, §1C] we obtain

μ_{t}^{t_{0}} = s · [\frac{μ_{t}^{+∞} − μ_{t_{0}}^{+∞}}{1 − μ_{t}^{+∞}\overline{μ_{t_{0}}^{+∞}}} ∘ N_{+∞}^{t_{0}}]

= s · [\frac{(e^{−2t} − e^{−2t_{0}})(¯∂^{2}ρ′ − (¯∂ρ′)^{2})e^{−2ρ′}}{(1 + e^{−2t})(1 + e^{−2t_{0}}) − e^{−2(t+t_{0})}|(¯∂^{2}ρ′ − (¯∂ρ′)^{2})e^{−2ρ′}|^{2}} ∘ N_{+∞}^{t_{0}}]

where s is a t independent function of modulus 1 on Σ_{t_{0}}. The ratio \frac{μ_{+∞}^{t_{0}}}{μ_{t}^{t_{0}}}

is dependent only on t (see (5.3)) if and only if ¯∂^{2}ρ′ − (¯∂ρ′)^{2} ≡ 0, reducing to case 1. One may see in the same fashion that μ_{[−∞,+∞]}^{−∞} does not lie in a radial line of B(Σ_{−∞}) ∪ ∞.

Case 3: Evidently μ_{−∞}^{+∞} ≡ ∞ if and only if ∂ ¯∂ρ′ ≡ 0. By equation (6.1) we see that K̂_{+∞} ≡ 0, or (Σ_{+∞}, ĝ_{+∞}) is flat. As N_{−∞}^{+∞} is anticonformal and Σ_{+∞} has its normal pointing into the H^{3} while Σ_{−∞} has its normal pointing out of H^{3}, if we change the normal on Σ_{−∞} to point inward we give it the oriented conformal structure induced by Ĉ. This makes N_{−∞}^{+∞} holomorphic. It was already noticed by Bianchi [3] (see §497, equation (64) in particular) that if f : Ω ⊂ C → C is holomorphic then the family of geodesics joining θ ∈ Ω to f(θ) is orthogonally integrable (for a discussion orthogonal integrability in this setting see [9]) in the sense that there is a family of parallel surfaces Σ_{t} orthogonal to these geodesics and the maps N_{t}^{t_{0}} are obtained by unit speed flow along the geodesics. The author thanks Peter Doyle for pointing out this reference.

As in case 2, we may see that for no t ∈ (−∞, +∞) is Σ_{t} a constant mean curvature surface in H^{3}. But now μ_{[−∞,+∞]}^{−∞} does lie in a radial line of B(Σ_{−∞}) ∪ ∞ (compare this with case 1 in corollary 1, §5). To see this we note that

μ_{t}^{+∞} = (¯∂^{2}ρ′ − (¯∂ρ′)^{2})e^{−2(ρ′+t)}

and compute μ_{t}^{−∞} as in case 2. The Beltrami parameter for the map N_{t}^{−∞} = N_{t}^{+∞} ∘ (N_{−∞}^{+∞})^{−1} is

μ_{t}^{−∞} = s · [\frac{μ_{t}^{+∞} − μ_{−∞}^{+∞}}{1 − μ_{t}^{+∞}\overline{μ_{−∞}^{+∞}}} ∘ N_{+∞}^{−∞}] = s · [\frac{1}{∂^{2}ρ′ − (∂ρ′)^{2}} e^{2(ρ′+t)} ∘ N_{+∞}^{−∞}] ,

with s a t independent function of modulus 1 on Σ_{−∞}. Thus \frac{μ_{+∞}^{−∞}}{μ_{t}^{−∞}} is indepen-

dent of z and Σ_{−∞} is also a (possibly singular) surface of constant curvature 1 at infinity of H^{3}. Away from N_{−∞}^{+∞}(θ) where θ is a zero of ∂^{2}ρ′ − (∂ρ′)^{2}, Σ_{−∞} is regular, and the singularities of Σ_{−∞} are precisely at these zeroes.

Note that the above equation shows that if A_{+∞} (A_{−∞}) denotes the zz– part of the second fundamental form of Σ_{+∞} (Σ_{−∞}, respectively), then A_{+∞} and A_{−∞} are inverse conjugates in the sense that Ā_{−∞} = \frac{1}{A_{+∞}}.

We now collect the information in this case.

Proposition 7. If Ω ⊂ Ĉ is a connected domain and f : Ω → Ĉ is a holomorphic function then there exists a family of parallel surfaces Σ_{t}, t ∈ [−∞, +∞], such that the limiting surfaces are precisely Σ_{+∞} = Ω and Σ_{−∞} = f(Ω).

Proposition 8. If Σ_{+∞} is a surface at infinity of H^{3} for which K̂_{+∞} = 0 (i.e. ∂ ¯∂ρ′ = 0) then Σ_{−∞} is also a (possibly singular) surface at infinity satisfying this same condition. The metrics on Σ_{±∞} are all flat away from possible singularities of Σ_{−∞}, i.e. pulled back from C by locally univalent functions ψ_{±∞} : Σ_{±∞} → C. The zz–parts of their respective second fundamental forms are inverse conjugates of each other. The singularities of Σ_{−∞} occur at the images under N_{−∞}^{+∞} of the zeros of the zz–part of the second fundamental form of Σ_{+∞}.

Case 4: Here μ_{t_{0}}^{+∞} ≡ ∞ for some finite t_{0}, so that by equation (6.1) we must have c < 0. Again rescaling, as in §4, we may assume that we have (∂ ¯∂ρ′)e^{−2ρ′} = −1, or

∂ ¯∂ρ′ = −e^{−2ρ′}.

This is the equation for constant curvature K̂_{+∞} ≡ 1. In this case the metric on Σ_{+∞} is pulled back by some locally univalent map ψ : Σ_{+∞} → Ĉ with a spherical metric.

Among the surfaces Σ_{t} there is another surface Σ_{t_{0}} (t_{0} is finite) of constant mean curvature. If c = 1 then t_{0} = 0. This may be seen as in cases 2 and 3. Since N_{+∞}^{t_{0}} is anticonformal we see that H(Σ_{t_{0}}) ≡ 1 as in corollary 4, §4.

We summarize in

Theorem 3. If Σ_{+∞} is a surface patch in S_{∞}^{2} for which K̂_{+∞} is constant then one of the following must hold:

1. The Σ_{t} are all pieces of spheres, horospheres, cylinders about a geodesic, or surfaces equidistant from a totally geodesic plane.

2. The metric on Σ_{+∞} is obtained by pulling back a hyperbolic metric on D via some locally univalent map ψ : Σ_{+∞} → D. No other surface in the family Σ_{t} has constant mean curvature.

3. The metric on Σ_{+∞} is obtained by pulling back a flat metric on C via some locally univalent map ψ : Σ_{+∞} → C and the map N_{−∞}^{+∞} is anticonformal. The possibly singular surface Σ_{−∞} is also a flat constant mean curvature surface patch in S_{∞}^{2}.

4. The metric on Σ_{+∞} is obtained by pulling back a spherical metric on Ĉ via some locally univalent map ψ : Σ_{+∞} → Ĉ. In the family Σ_{t} there is a finite t_{0} for which the possibly singular Σ_{t_{0}} has constant mean curvature.

Compare this with corollary 1, §5.

Now compare the following with theorem 1.

Theorem 4. Let Σ_{+∞} ⊂ S_{∞}^{2} be a surface patch at infinity of H^{3}. The following are equivalent:

1. K̂_{+∞} is constant;

2. (Π_{+∞})_{ij} is harmonic; 3. the zz–part of (Π_{+∞})_{ij} (i.e. the quadratic differential (∂^{2}ρ′ − (∂ρ′)^{2}))

is holomorphic on Σ_{+∞};

4. N_{t}^{+∞} : Σ_{+∞} → Σ_{t} is harmonic for any t ∈ [−∞, +∞];

5. there exists t ∈ [−∞, +∞) such that N_{t}^{+∞} : Σ_{+∞} → Σ_{t} is harmonic;

6. μ_{t}^{+∞}(p) = f(t) · g(p) where f : [−∞, +∞] → R ∪ {∞} is injective and g ∈ B(Σ_{+∞});

7. μ_{t}^{+∞}(p) = f(t) · g(p) where f(t) = 0, \frac{1}{1 + e^{−2t}}, e^{−2t}, or \frac{1}{1 − e^{−2t}}, and

g ∈ \overline{Q(Σ_{+∞})} × e^{−2ρ′}.

This suggests a definition.

Definition. If a surface Σ_{+∞} at infinity of H^{3} satisfies the above equivalences it will be called a surface of constant mean curvature at infinity of H^{3}.

Proof: That (1) and (6) are equivalent was already seen in theorem 2.

The equivalence of (1) and (2) is a repetition of the proof of lemma 3, §3, noting from proposition 4, §4, that we have the Codazzi–Mainardi equations satisfied in this setting.

We show now that (1) is equivalent to (3). Differentiating e^{−2ρ′}(∂ ¯∂ρ′) = c with respect to z gives

0 = ∂ [e^{−2ρ′}(∂ ¯∂ρ′)] = e^{−2ρ′} ¯∂(∂^{2}ρ′ − (∂ρ′)^{2}),

so that indeed ∂^{2}ρ′ − (∂ρ′)^{2} is holomorphic. Reversing the same equality, as well as its conjugate, gives that

0 = ∂ [e^{−2ρ′}(∂ ¯∂ρ′)] = ¯∂ [e^{−2ρ′}(∂ ¯∂ρ′)]

if ∂^{2}ρ′ − (∂ρ′)^{2} is holomorphic, yielding the desired equivalence.

To see now the other equivalences, we note that if e^{−2ρ′}(∂ ¯∂ρ′) = c then letting A = ∂^{2}ρ′ − (∂ρ′)^{2} we have

h^{t} = \frac{1}{2}e^{2(ρ′+t)} ( \begin{matrix}1 + ce^{−2t} & Ae^{−2(ρ′+t)} \\ Āe^{−2(ρ′+t)} & 1 + ce^{−2t} \end{matrix} )^{2} ,

(Π_{+∞})_{ij} = ( \begin{matrix}e^{2ρ} & 2A dτ \\ 2Ā dτ & e^{2ρ} \end{matrix} )

and

μ_{t}^{+∞} = \frac{e^{−2t}}{1 + ce^{−2t}} Āe^{−2ρ′}.

The zz–part of h^{t} is e^{−2t}(1 + ce^{−2t})A which is holomorphic, implying (4) and (5), while (7) is evident from the third expression above.

Clearly (7) implies (1), and (4) implies (5), so the proof will be complete when we show that (5) implies (1). The proof is essentially a repeat of the proof that (5) implies (1) in theorem 1. Assume N_{t_{0}}^{+∞} is harmonic. Since

h^{t} = \frac{1}{2}e^{2(ρ′+t_{0})} ( \begin{matrix}(∂ ¯∂ρ′)e^{−2(ρ′+t_{0})} + 1 & Ae^{−2(ρ′+t_{0})} \\ Āe^{−2(ρ′+t_{0})} & (∂ ¯∂ρ′)e^{−2(ρ′+t_{0})} + 1 \end{matrix} )^{2} ,

the zz–part is ((∂ ¯∂ρ′)e^{−2(ρ′+t_{0})} + 1)A. We assume that this is holomorphic, to wit

0 = ¯∂ [((∂ ¯∂ρ′)e^{−2(ρ′+t_{0})} + 1)A] = (∂Ā)Ae^{−2t}e^{−2ρ′} + ∂ ¯∂ρ′(¯∂A)e^{−2t}e^{−2ρ′} + ¯∂A.

Writing out the real and imaginary parts of this, with A = u + iv, the following system of equations in u_{x} − v_{y} and u_{y} + v_{x} results:

\begin{aligned}(∂ ¯∂ρ′e^{−2ρ′} + u + e^{2t})(u_{x} − v_{y}) & +v(u_{y} + v_{x}) & = 0 \\ v(u_{x} − v_{y}) & +(∂ ¯∂ρ′e^{−2ρ′} − u + e^{2t})(u_{y} + v_{x}) & = 0 \end{aligned} .

Here, either there is only the trivial solution or there are nontrivial solutions. In the first case, ¯∂A = ∂K_{+∞} = 0 and its conjugate, whence K_{+∞} is constant. In the second case, −\frac{e^{2t}}{2} is everywhere an eigenvalue of Π_{+∞}. As in the proof of theorem 1, one checks that the other eigenvalue is also constant, whence K_{+∞} is again constant. In either case we have the desired conclusion. □

7 The Ahlfors–Weill examples

Naturally occurring examples of constant mean curvature surfaces at infinity have already been observed [18], although not in these terms. What follows is a brief description of these examples.

Let A ∈ Q(D) and consider the linear ODE

f′′ + Af = 0 on D. (7.1)

Let f_{1}, f_{2} be independent solutions such that

( \begin{matrix}f_{1} & f′_{1} \\ f_{2} & f′_{2} \end{matrix} )_{z=0} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) . (7.2)

It is easily seen that the function F_{∞,A} : D → Ĉ given by F_{∞,A} = \frac{f_{1}}{f_{2}} is locally univalent and satisfies the Schwarzian differential equation

\{F_{∞,A}, z\} = 2A (\{f, z\} = \frac{f′′′}{f′} − \frac{3}{2} (\frac{f′′}{f′} )^{2}).

Nehari showed [12] that if ∥A∥_{∞} = sup_{z∈D} |A(z)·(1−|z|^{2})^{2}| ≤ 1 then actually F_{∞,A} is globally univalent. Ahlfors and Weill expanded on this result to show that if ∥A∥_{∞} < 1 then F_{∞,A} extends to S^{1} = ∂D and F_{∞,A}(S^{1}) is a quasicircle. Furthermore, they produced a quasireflection of F_{∞,A}, call it F_{0,A}, which is a quasiconformal map F_{0,A} : D ∪ S^{1} → Ĉ agreeing with F_{∞,A} on S^{1} and has image F_{0,A}(D) = Ĉ \ F_{∞,A}(D ∪ S^{1}). Letting 1/D = {z ∈ C : 1/z ∈ D}, the map

G_{A}(z) = \{ \begin{matrix}\overline{F_{∞,A}(\frac{1}{z̄})} & z ∈ 1/D \\ \overline{F_{0,A}(z)} & z ∈ D ∪ S^{1} \end{matrix}

is a quasiconformal homeomorphism of Ĉ holomorphic on 1/D. This extension F_{0,A} is given explicitly in terms of the f_{1}, f_{2} of (7.2) by

\overline{F_{0,A}(z)} = \frac{z\overline{f_{1}(z)} + (1 − zz̄)\overline{f′_{1}(z)}}{z\overline{f_{2}(z)} + (1 − zz̄)\overline{f′_{2}(z)}} (7.3)

and satisfies

μ_{A}(z) = \frac{∂z̄G_{A}}{∂_{z}G_{A}} = \{ \begin{matrix}0 & z ∈ 1/D \\ \overline{A(z)}(1 − zz̄)^{2} & z ∈ D \end{matrix} . (7.4)

In [18], the author showed that the phenomena described and analyzed by Ahlfors and Weill are special boundary cases of more general phenomena. To be specific, let P SU(1, 1) act as orientation preserving Möbius transformations on D, and P SL(2, C) as isometries on H^{3} in our upper half-space model as usual:

( \begin{matrix}a & b \\ c & d \end{matrix} ) · (Z, T) = (\frac{(aZ + b)\overline{(cZ + d)} + ac̄T^{2}}{|cZ + d|^{2} + |c|^{2}T^{2}} , \frac{T}{|cZ + d|^{2} + |c|^{2}T^{2}} ) .

It was shown in [18] that the ODE (7.1) generates a monodromy map Mon_{A} : P SU(1, 1) → P SL(2, C) given by

\begin{matrix}Mon & A & [\frac{1}{(1 − zz̄)^{1/2}} ( \begin{matrix}1 & z \\ z̄ & 1 \end{matrix} ) ( \begin{matrix}e^{−iθ} \\ & e^{iθ} \end{matrix} )] \\ = \frac{1}{(1 − zz̄)^{1/2}} ( \begin{matrix}f_{1}(z) & z̄f_{1}(z) + (1 − zz̄)f′_{1}(z) \\ f_{2}(z) & z̄f_{2}(z) + (1 − zz̄)f′_{2}(z) \end{matrix} ) ( \begin{matrix}e^{iθ} \\ & e^{−iθ} \end{matrix} ) . \end{matrix} (7.5)

So by means of Mon_{A} we can consider the action of P SU(1, 1) on H^{3} as

γ : (Z, T) ↦ Mon_{A}(γ) · (Z, T).

If stab(x) denotes the stabilizing group of x as a subgroup of the appropriate ambient group, we observe that the definition of Mon_{A} yields Mon_{A}(stab(0)) ⊂ stab(0, T) for all T ∈ R_{+}. Hence for all T ∈ R_{+} we have an induced map F_{T,A} : D → H^{3} via the factored diagram

P SU(1, 1) ⟶^{Mon_{A}} P SL(2, C)
↓stab(0)⟶  ↓stab(0, T)
D  ⟶^{F_{T,A}}  H^{3}.

The F_{T,A} have been given explicitly in terms of the f_{1}, f_{2} of (7.2) as

F_{T,A}(z) = Mon_{A} ( \begin{matrix}1 & z \\ z̄ & 1 \end{matrix} ) · (0, T)
= (\frac{num}{denom}, \frac{(1 − zz̄)T}{denom} ) ,

(7.6)

where

num = (z̄f_{1}(z) + (1 − zz̄)f′_{1}(z))\overline{(z̄f_{2}(z) + (1 − zz̄)f′_{2}(z))}
+f_{1}(z)\overline{f_{2}(z)}T^{2},
denom = |z̄f_{2}(z) + (1 − zz̄)f′_{2}(z)|^{2} + |f_{2}(z)|^{2}T^{2}.

We may also extend this action continuously to T = 0 and T = ∞ to get the F_{0,A} and F_{∞,A} of Ahlfors and Weill, from the beginning of this section.

It was shown in [18] that the maps F_{T,A} quasiconformally interpolate between the F_{∞,A} and the F_{0,A} in that the Beltrami differential of F_{T,A} is

μ_{T,A} = −\frac{\overline{A(z)} · (1 − zz̄)^{2}}{(T^{2} + 1)} .

Thus if t = log T, the images Σ_{t} = F_{T,A}(D) form a family of surfaces such that the maps M_{t,A} = F_{T,A} ∘ F_{∞,A}^{−1} : Σ_{+∞} → Σ_{t} have Beltrami differentials

ν_{t,A} = \frac{∂z̄M_{t,A}}{∂_{z}M_{t,A}} = \frac{1}{(e^{2t} + 1)} (−\overline{A(z)}(1 − zz̄)^{2}) (7.7)

which are a radial family. To deduce that Σ_{+∞} is a surface of constant mean curvature +1 we still need to show that the Σ_{t} are parallel. But this follows from the fact that

∥∂_{T} F_{T,A}(z)∥_{H^{3}} = \frac{1}{T}

which is independent of z ∈ D. Hence these Σ_{+∞} are examples of the precise phenomena desired.

For the record, we note the directional, mean and intrinsic curvatures for F_{T,A}(D). Letting m = |A(z) · (1 − |z|^{2})^{2}| we have

\begin{aligned}k_{1} & = & \frac{m^{2} + (T^{4} − 1) + 2T^{2}m}{m^{2} − (T^{2} + 1)^{2}}, \\ k_{2} & = & \frac{m^{2} + (T^{4} − 1) − 2T^{2}m}{m^{2} − (T^{2} + 1)^{2}}, \\ H & = & \frac{k_{1} + k_{2}}{2} = \frac{m^{2} + (T^{4} − 1)}{m^{2} − (T^{2} + 1)^{2}} \\ K & = & k_{1}k_{2} − 1 = \frac{4T^{2}}{m^{2} − (T^{2} + 1)^{2}} . \end{aligned} (7.8)

As we observed in theorem 3, Σ_{+∞} is endowed with a metric of constant negative curvature. We observe this here by letting T = e^{t} and computing K̂_{+∞} as in §4. We obtain

K̂_{+∞} = lim_{t→+∞} \frac{e^{2t}}{4} \frac{4e^{2t}}{m^{2} − (e^{2t} + 1)^{2}} = −1.

Two notes are in order here —

Note 6. The Σ_{+∞} are locally univalent images of D. If we think of Σ_{+∞} not as the image of D but rather as an immersion of D, we can push the hyperbolic metric on D (thought of as H^{2}) forward onto Σ_{+∞} locally. Thus one point in S_{∞}^{2} may have many sheets of Σ_{+∞} containing it, and correspondingly many different metrics, one for each sheet. Here the map F_{0,A} is thus an isometry from H^{2} to Σ_{+∞}, so that Σ_{+∞} is complete. We will discuss this further in the next two sections.

Note 7. There is a nice geometric interpretation of the Σ_{t} as envelopes of horospheres which can be deduced from [7] and [18]. We sketch this.

These examples are obtained locally by considering a surface patch Ω ⊂ S_{∞}^{2} as part of the image of D under a locally univalent conformal map f : D → S_{∞}^{2}. In this case we take ρ to be

ρ = log (\frac{1}{|f′|} · \frac{1 + |f(z)|^{2}}{1 − |z|^{2}} ) = log (\frac{ds_{H^{2}}(p)}{ds_{S^{2}}(f(p))}).

This means that for any p ∈ D, precompose by a Möbius transformation so that we may assume p = 0, and then normalize f so that f(z) = z + O(z^{3}) by Möbius transformations. Then ν_{0∈S_{∞}^{2}} is the horosphere going through (0, 1) ∈ C × R_{+} (as following note 5 of §4).

In these cases one checks that ĝ_{ij}^{∞} is, up to a constant, the Poincaré metric on Ω obtained by pushing forward that on D = H^{2}. This is possible as f is locally univalent.

In closing this section it is important to note that in neither the geometric envelopes of horospheres treatment [7] nor the more computational treatment [18] was it necessary to have Σ_{+∞} embedded in S_{∞}^{2}. At no time in the computations of [18] was ∥A∥_{∞} < 1 used, and the discussion in [7] is strictly local. Hence we observe that the entire discussion of this section holds for any A ∈ Q(D).

8 Ahlfors–Weill: the flat case

For any A ∈ Q(C), we can construct complete constant mean curvature surfaces at infinity of H^{3} in a fashion completely analogous to the construction of the preceding section. Again let f_{1}, f_{2} be independent solutions of

f′′ + Af = 0 on C.

such that

( \begin{matrix}f_{1} & f′_{1} \\ f_{2} & f′_{2} \end{matrix} )_{z=0} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) .

We denote by Isom(1, C) the group of isometries of C. Here the monodromy map Mon_{A} : Isom(1, C) → P SL(2, C) given by

\begin{matrix}Mon & A & [( \begin{matrix}e^{iθ} \\ & e^{−iθ} \end{matrix} ) ( \begin{matrix}1 & −z \\ 0 & 1 \end{matrix} )] \\ = ( \begin{matrix}f_{1}(z) & f′_{1}(z) \\ f_{2}(z) & f′_{2}(z) \end{matrix} ) ( \begin{matrix}e^{iθ} \\ & e^{−iθ} \end{matrix} ) . \end{matrix} (8.1)

Again by means of Mon_{A} we consider the action of Isom(1, C) on H^{3} as

γ : (Z, T) ↦ Mon_{A}(γ) · (Z, T).

Since Mon_{A}(stab(0)) ⊂ stab(0, T) for all T ∈ R_{+}, we have an induced map F_{T,A} : C → H^{3} via the factored diagram

Isom(1, C) ⟶^{Mon_{A}} P SL(2, C)
↓stab(0)⟶  ↓stab(0, T)
C  ⟶^{F_{T,A}}  H^{3}.

We have the following explicit form for the F_{T,A} in terms of f_{1} and f_{2}:

\begin{aligned}F_{T,A}(z) & = & Mon_{A} ( \begin{matrix}1 & −z \\ 0 & 1 \end{matrix} ) · (0, T) \\ & = & (\frac{f′_{1}(z)\overline{f′_{2}(z)} + f_{1}(z)\overline{f_{2}(z)}T^{2}}{|f′_{2}(z)|^{2} + |f_{2}(z)|^{2}T^{2}} , \frac{T}{|f′_{2}(z)|^{2} + |f_{2}(z)|^{2}T^{2}} ) , \end{aligned} (8.2)

and again extend this action continuously to T = 0 and T = ∞ to get

F_{∞,A}(z) = \frac{f_{1}(z)}{f_{2}(z)}  and  F_{0,A}(z) = \frac{f′_{1}(z)}{f′_{2}(z)}. (8.3)

Since F_{0,A} is conformal, \overline{F_{0,A}} is anticonformal with conformal deformation μ_{0,A} ≡ ∞. As F′_{0,A} = \frac{−A}{(f′_{2})^{2}} , we see that the singularities of F_{0,A} come from

precisely the zeroes of A.

If we compute the deformations of the F_{T,A} we have

μ_{T,A} = −ĀT^{−2}.

Again, for completeness, the curvatures of the Σ_{t} are included. We have

\begin{aligned}k_{1} & = & \frac{|A| + T^{2}}{|A| − T^{2}} \\ k_{2} & = & \frac{|A| − T^{2}}{|A| + T^{2}} \\ H & = & \frac{|A|^{2} + T^{4}}{|A|^{2} − T^{4}} \\ K & = & 0. \end{aligned} (8.4)

Clearly we also have K̂_{+∞} ≡ 0. This can be checked as was done in the hyperbolic case at the end of §7.

The relationship between entire functions and Q(C), i.e. families of parallel surfaces generated by a complete flat constant mean curvature 1 surface at infinity of S_{∞}^{2}, is easily understood. We have the following evident theorem.

Theorem 5. The vector space Q(C) is naturally isomorphic to (entire functions)/ (affine transformations of C), where the affine transformations act on the space of entire functions as (az + b) · f(z) = a^{2}f(az + b).

One consequence is that, for a ∈ R, the entire function A_{1}(z) = a^{2}A(az) reparametrizes the family of parallel surfaces generated by the complete constant mean curvature surface at infinity arising from A. This is seen through formula (8.2) as follows:

If f_{1}, f_{2} are our normalized solutions to f′′ + Af = 0, and g_{1}, g_{2} are those for g′′ + A_{1}g = 0, then g_{i}(z) = a^{−1/2}f_{i}(az). Thus

\begin{aligned}F_{T,A_{1}}(z) & = & Mon_{A_{1}} ( \begin{matrix}1 & −z \\ 0 & 1 \end{matrix} ) · (0, T) \\ & = & (\frac{g′_{1}(z)\overline{g′_{2}(z)} + g_{1}(z)\overline{g_{2}(z)}T^{2}}{|g′_{2}(z)|^{2} + |g_{2}(z)|^{2}T^{2}} , \frac{T}{|g′_{2}(z)|^{2} + |g_{2}(z)|^{2}T^{2}} ) \\ & = & (\frac{a^{−1}f′_{1}(az)\overline{f′_{2}(az)} + af_{1}(az)\overline{f_{2}(az)}T^{2}}{a^{−1}|f′_{2}(az)|^{2} + a|f_{2}(az)|^{2}T^{2}} , \\ \frac{T}{a^{−1}|f′_{2}(az)|^{2} + a|f_{2}(az)|^{2}T^{2}} & ) \\ & = & (\frac{f′_{1}(az)\overline{f′_{2}(az)} + f_{1}(az)\overline{f_{2}(az)}T^{2}}{|f′_{2}(az)|^{2} + |f_{2}(az)|^{2}(aT)^{2}} , \\ \frac{aT}{|f′_{2}(az)|^{2} + |f_{2}(az)|^{2}(aT)^{2}} & ) \\ & = & F_{aT,A}(az). \end{aligned} (8.5)

We note the following for general interest. The proof was suggested by A. Hinkkanen.

Proposition 9. There are locally univalent conformal maps F : C → S_{∞}^{2} for which the image is the whole sphere and each θ ∈ S_{∞}^{2} has infinitely many preimages.

Proof: Let A be a polynomial of odd degree n on C, and consider the ODE

f′′ + Af = 0.

If f is a nonzero solution to this ODE it has fractional order \frac{n + 2}{2}, and so has infinitely many zeroes [16]. It follows that if f_{1} and f_{2} are independent solutions then F = \frac{f_{1}}{f_{2}} is locally univalent and has infinitely many zeroes and poles. To see that F assumes the value c ∈ C \ {0} infinitely often, consider that F − c = \frac{f_{1} − cf_{2}}{f_{2}} and f_{1} − cf_{2} vanishes infinitely often. □

9 Existence and uniqueness of complete constant mean curvature surfaces at infinity

Suppose that Σ_{0} has its metric given conformally by ds^{2} = λ^{2}|dZ|^{2}, that

(II_{0})_{ij} = ( \begin{matrix}L & M \\ M & N \end{matrix} )

and that H(Σ_{0}) = \frac{L + N}{2λ^{2}} ≡ c ∈ (−1, 1). Then A = \frac{L − N − 2iM}{2} is holomorphic. The intrinsic curvature of Σ_{0} is given by

K = k_{0}^{1}k_{0}^{2} − 1
= \frac{LN − M^{2}}{λ^{4}} − 1
= c^{2} − AĀλ^{−4} − 1
≤ −(1 − c^{2}) < 0,

whence the universal cover of Σ_{0} is conformally D, with coordinate z. In this case, (Π_{0})_{ij} pulls back to give the quadratic differential (Π̃_{0})_{ij} = Adz^{2} + 2cλ^{2}|\frac{dZ}{dz} |^{2}dzdz̄+ Ādz̄^{2} on D. The author considers this of particular interest in the case where Σ_{0} is a complete surface, in which case the following issues arise naturally:

Question: To what extent does the holomorphic quadratic differential A, the mean curvature c, and the completeness of the metric λ^{2}|dZ|^{2} determine the surface Σ_{0}? Or more precisely, given a holomorphic quadratic differential A on D, and a constant c ∈ (−1, 1), does there exist a complete surface Σ_{0} ↪ H^{3} with H(Σ_{0}) = c and Π̃_{zz} = A? Given (A, c) for which this problem is solvable, to what extent is a solution unique?

Example. If A ≡ 0 and c ∈ (−1, 1) then one readily sees that any surface Σ_{0} inducing the trivial quadratic differential on D satisfies k_{1} = k_{2} = c globally, i.e. Σ_{0} is a piece of a surface parallel to a totally geodesic plane. The completeness of Σ_{0} implies that it is precisely a surface parallel to a totally geodesic plane. Hence for A ≡ 0 we have both existence and uniqueness up to ambient isometry.

Karen Uhlenbeck [17] addressed this issue. She showed that if A is a sufficiently small quadratic differential on D then there exists a complete Σ_{0} when c = 0 inducing A. In [2], Anderson proved that, for example, if C is a quasicircle in S_{∞}^{2} then there is a stable complete minimal embedding of D as Σ_{0} ↪ H^{3} with ∂_{∞}Σ_{0} = C. By work of Epstein [8], one may see that when C is sufficiently ‘close’ to a circle (we won’t discuss what this means) then the quadratic differentials arising in Uhlenbeck’s work parametrize the minimal surfaces produced by Anderson.

Bennett Palmer [13] asked the same question for space-like surfaces immersed in symmetric constant curvature k three dimensional Lorentzian manifolds L_{k}. He established existence and uniqueness given that the holomorphic quadratic differential A on D is automorphic with respect to a cocompact Fuchsian group. In fact, Palmer also considered when the other conformal simply connected Riemann surfaces (C, Ĉ) immerse as space-like constant mean curvature surfaces in L_{k}.

To date, little progress has been made toward the resolution of the above question. We do note, however, that the case of constant mean curvature c ∈ (−1, 1) is equivalent to the case c = 0. This follows immediately from Uhlenbeck’s work.

Corollary 6. If ∥Aρ^{−2}∥_{∞} < (1−c^{2})^{− \frac{1}{2}} then there is an immersed complete surface Σ_{0} ↪ H^{3} of constant mean curvature c so that Π̃_{zz}(Σ_{0}) = A.

Proof: To see this, note that the Gauss equation for λ in the case of c ∈ (−1, 1) is

Δ log λ = 2AĀλ^{−2} + 2(1 − c^{2})λ^{2},

and letting λ = \frac{1}{2}e^{\frac{1}{2} ϕ} yields

Δϕ = 16AĀe^{−ϕ} + (1 − c^{2})e^{ϕ}.

If we now let ψ = ϕ + log (1 − c^{2}) we get

Δψ = 16AĀ(1 − c^{2})e^{−ψ} + e^{ψ},

i.e. the Gauss equation for the metric of a minimal surface with Π̃_{zz} = \sqrt{1 − c^{2}}A. □

It is worth noting that as c → 1 this is solvable for more quadratic differentials, and when c = ±1 we get exactly the Liouville equation for constant Gauss curvature −1.

We now examine a variant of the question above, and give a parametrization of complete constant mean curvature surfaces at infinity of H^{3}. As was observed in §6, these must necessarily have mean curvature ±1. We begin with two lemmas.

Lemma 5. If Σ_{+∞} is a surface patch at infinity of H^{3}, and if N_{t_{0}}^{+∞} : Σ_{+∞} → Σ_{t_{0}} is conformal for some t_{0}, then H and K are constant on Σ_{t_{0}}, or equivalently the directional curvatures k_{1} and k_{2} on Σ_{t_{0}} are constant.

Proof: If N_{t_{0}}^{+∞} : Σ_{+∞} → Σ_{t_{0}} is conformal, then at least locally so is N_{+∞}^{t_{0}}. Thus by theorem 2, μ_{+∞}^{t_{0}} ≡ 0 which implies that μ_{t}^{t_{0}} ≡ 0 for all t and we are in the second case of corollary 1. This also follows from proposition 6 and the expression for (Π_{t})_{j}^{i} in §4. □

Lemma 6. Under the assumptions of lemma 5, the N_{t}^{+∞} are conformal for all t ∈ [−∞, +∞] with at most one exception.

Proof: This is immediate. The exceptional possibility is that the N_{t}^{+∞} may focus on a point (when the Σ_{t} are all pieces of spheres or horospheres – in this latter case the focal point is at infinity) or on a line (when the Σ_{t} are all pieces of geodesic cylinders). □

The following theorem is an immediate consequence of the above two lemmas and the theorem of Cartan mentioned in §5.

Theorem 6. If Σ_{+∞} is complete and the N_{t}^{+∞} are conformal the for all t we have one of the following cases:

1. Σ_{+∞} = S_{∞}^{2} and the Σ_{t} are concentric spheres;

2. Σ_{+∞} = S_{∞}^{2} \ {θ} for some θ ∈ S_{∞}^{2}, endowed with the affine structure of C, and the Σ_{t} are horospheres centered at θ;

3. Σ_{+∞} = S_{∞}^{2} \ {θ_{1}, θ_{2}} endowed with the complete flat structure of C^{∗} ≅ C/(z ↦ z+1), and the Σ_{t} are the equidistant surfaces from the geodesic connecting θ_{1} and θ_{2}; or

4. Σ_{+∞} is a disk with the hyperbolic metric bounded by a geometric circle in S_{∞}^{2}, and the Σ_{t} are the surfaces in H^{3} equidistant from the totally geodesic plane spanning C.

And finally we have

Theorem 7. Given a holomorphic quadratic differential A on either C or D, there exists a unique complete surface Σ_{+∞} of constant mean curvature ±1 at infinity of H^{3} with A as the holomorphic part of its second fundamental form. In the first case, Σ_{+∞} is flat and in the second it is hyperbolic. Thus the surfaces of constant mean curvature at ∂_{∞}H^{3} are parametrized up to isometry by holomorphic quadratic differentials on either C or D, with the exception of the sphere S_{∞}^{2} itself.

Proof: The existence of such structures has been established in §7 and 8. This amounted, in effect, to solving the Schwarzian differential equation {f, z} = A on C or D. We are left with the problem of showing that the family of equidistant surfaces in H^{3} arising therefrom is unique.

This, though, is the content of the theorem of §4 and theorem 3.

Thus our support function arises from one of the two Ahlfors–Weill constructions.

It follows that except in the cases covered by theorem 6 the complete surfaces at infinity are precisely those arising from solving the Schwarzian differential equation, and the corresponding deformation of surfaces in H^{3} is given by extension of the Ahlfors–Weill construction. In the exceptional cases we see that we have the sphere itself, the quadratic differential A = 0 on C (case 2 of theorem 6), the quadratic differential A = const ≠ 0 on C (case 3 of theorem 6, see also the comments after theorem 5 as to why all constants give the same foliation here), and finally the quadratic differential A = 0 on D (case 4 of theorem 6). This finishes the proof. □

This can be reformulated as

Theorem 8. The complete constant mean curvature surfaces at infinity of H^{3} are in a natural 1–1 correspondence with quadratic differentials on either S^{2} = Ĉ, C, or D. The map from quadratic differentials to surfaces (up to ambient isometry) is given by solving the Schwarzian differential equation, and the inverse is via the map from surfaces to the zz–part of their second fundamental forms.

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