Constant Mean Curvature Surfaces in Hyperbolic Space

Nicholas J. Korevaar^{∗} Mathematics Department

University of Utah Salt Lake City, UT 84112

Rob Kusner^{†} Mathematics Department University of Massachusetts Amherst, MA 01003

William H. Meeks, III^{‡} Mathematics Department University of Massachusetts

Amherst, MA 01003

Bruce Solomon^{§} Mathematics Department Indiana University Bloomington, IN 47405

August 27, 1990

∗Supported by the National Science Foundation grant DMS-8808002. †Supported by the National Science Foundation grant DMS-8908064. ‡The research described in this paper was supported by research grant DE-FG02-

86ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS-8900285.

§Supported by the National Science Foundation grant DMS-8800414.

Contents

1 Introduction 2

2 Preliminaries 5

3 Cylindrical boundedness and topological obstructions 13

4 Conservation laws and momenta 20

5 Positive flux lemma 29

6 Asymptotic symmetry of cylindrically bounded ends 41

1 Introduction

A basic problem in differential geometry is the global structure and classification of complete properly embedded constant mean curvature surfaces in three-manifolds of constant sectional curvature. The models for these manifolds are three-dimensional Euclidean space ^{3}, the three sphere S^{3}, and hyperbolic three-space ^{3}.

In 1986, Meeks [16] proved that each annular end of a properly embedded surface Σ of nonzero constant mean curvature in ^{3} remains a bounded distance from a ray, i.e. each annular end of Σ is cylindrically bounded. This theorem, together with a simple application of the Alexandrov reflection principle, showed that Σ cannot be homeomorphic to a closed surface with a single point removed. He also used this cylindrical boundedness theorem to show that when Σ is topologically a closed surface minus two points, the entire surface stays a bounded distance from a line.

In the same year, many new examples of complete properly embedded surfaces in ^{3} of finite topology and nonzero constant mean curvature were constructed by Kapouleas [10]. The ends of his examples converge exponentially to the constant mean curvature surfaces of revolution known as Delaunay surfaces.

Korevaar, Kusner and Solomon [11] then developed a theory to deal with the global structure and asymptotic geometry of complete, properly embedded constant mean curvature surfaces. In particular, they proved that each annular end of any properly embedded surface Σ in ^{3} with nonzero constant mean curvature converges exponentially to a Delaunay surface. The first step in their proof was to show that if Σ stays within bounded distance of a line, it is Delaunay. Together with Meeks’ result, this established that every properly embedded constant mean curvature surface homeomorphic to a closed surface minus two points is Delaunay.

Lately there has been renewed interest in the geometry of surfaces in hyperbolic manifolds. (See for example, [1, 4, 5, 7, 15, 22].) Let _{c}^{n+1} denote the complete simply-connected spaceform of constant negative sectional curvature c. As in Euclidean space, we will call constant mean curvature surfaces of revolution in _{c}^{3} Delaunay surfaces. We shall prove the following theorems on the geometry of constant mean curvature surfaces in _{c}^{3}.

Theorem 1.1 Suppose Σ ⊂ _{c}^{3} is a complete properly embedded surface with constant mean curvature greater than that of a horosphere. Then

1. Σ is not homeomorphic to a closed surface punctured in one point.

2. If Σ is homeomorphic to a closed surface punctured in two points, Σ is Delaunay. In particular, Σ is topologically a cylinder.

3. If Σ is homeomorphic to a closed surface punctured in three points, then Σ remains a bounded distance from a geodesic plane of reflective symmetry. Furthermore, each half of Σ determined by the plane of reflective symmetry is a graph over this plane with respect to the distance function to the plane.

See § 2.5 for the definition of horosphere and a brief explanation of its role here. The proof of the above theorem is similar to that of the corresponding theorem for constant mean curvature surfaces in ^{3} whose proof we outlined earlier. Namely, we first prove that each annular end of Σ is cylindrically bounded. This boundedness property has several immediate consequences when Σ has finite topology:

1. Σ has more than one end.

2. When Σ has two ends, it is cylindrically bounded.

3. When Σ has three ends, it stays a bounded distance from a hyperplane.

An application of the Alexandrov reflection argument, due to Hsiang [9], together with these three properties for Σ proves Theorem 1.1. This proof is carried out in detail in § 3.

As in the Euclidean case, the annular ends of the surface Σ described in Theorem 1.1 must converge to Delaunay surfaces. More precisely,

Theorem 1.2 If A ⊂ _{c}^{3} is a properly embedded annulus with constant mean curvature greater than that of a horosphere, then A converges exponentially to a fixed Delaunay surface. In particular, each end of a complete properly embedded surface Σ ⊂ _{c}^{3}, with finite topological type and constant mean curvature exceeding that of a horosphere, is asymptotically Delaunay. It follows that Σ is conformally a compact Riemann surface with finitely many punctures, and that the limit set of Σ ⊂ _{c}^{3} is a finite collection of points in the sphere at infinity S_{∞}.

The above theorem results directly from the cylindrical boundedness property for the annular end A and the following general theorem.

Theorem 1.3 Suppose Σ ⊂ _{c}^{n+1} is a properly embedded noncompact hypersurface with constant mean curvature and compact boundary ∂Σ. If Σ stays within bounded distance of a geodesic ray in _{c}^{n+1}, then it converges exponentially to one end of a Delaunay hypersurface of revolution. In particular, Σ has one end, which is diffeomorphic to a punctured ball.

The proof of Theorem 1.3 introduces a useful new tool we call the Positive Flux Lemma (Theorem 5.3). It also depends on some careful applications of the Alexandrov reflection method. We remark that the Positive Flux Lemma holds in Euclidean space too, and combined with a more refined Alexandrov reflection argument may help prove an n-dimensional Euclidean version of Theorem 1.3.

For simplicity we will prove all of our theorems in _{−1}^{n+1} = ^{n+1}.

2 Preliminaries

Here we introduce some background material and notation.

We will use three models of hyperbolic space: the upper halfspace, the ball, and the Minkowski-space models.

(2.1) Upper halfspace model. Introduce the metric

ds^{2} = \frac{dx_{1}^{2} + dx_{2}^{2} + ⋯ + dx_{n}^{2} + dy^{2}}{y^{2}}

on the upper halfspace {(x, y) ∈ ^{n+1} | x ∈ ^{n}, 0 < y ∈ }. Familiar calculations show this is a complete metric with constant sectional curvature −1, hence isometric to ^{n+1}.

Horizontal (Euclidean) translations and rotations (i.e., elements of the Euclidean group) are clearly isometries of this model. There are two other families of transformations associated with each x_{0} ∈ ^{n}: the vertical hyperbolic translations

(x, y) ⟼ (e^{t}(x − x_{0}), e^{t}y) , (t ∈ ) ,

(i.e., Euclidean homotheties about (x_{0}, 0) with factor e^{t}), and the hyperbolic reflections

(x, y) ⟼ (x_{0}, 0) + \frac{t^{2}(x − x_{0}, y)}{|x − x_{0}|^{2} + y^{2}} , (t ∈ ) ,

which are simply Euclidean inversions of the upper halfspace in the radius t sphere centered at (x_{0}, 0). (Reflection in a vertical hyperplane should also be regarded as a hyperbolic reflection, with x_{0} = ∞, t = ∞.) As the fixed-point set of its reflection, any such sphere is immediately seen to be totally geodesic. Taking intersections, one then deduces: Hemispheres of any dimension and radius meeting the y ≡ 0 “hyperplane at infinity” orthogonally are totally geodesic. In particular, circles meeting the y ≡ 0 hyperplane at right angles and vertical lines are the geodesics of the upper halfspace model.

The hyperbolic translations will prove useful later. The ones described above associate to each x_{0} ∈ ^{n} a one-parameter group of isometries obtained by translating along the vertical geodesic through (x_{0}, 0) with unit speed. Since every geodesic can be mapped to a vertical geodesic by some hyperbolic reflection (reflect in a sphere of appropriately larger radius having a great circle tangent to the given geodesic at y = 0 in the 2-plane it determines), one sees: Every constant speed geodesic γ ∈ ^{n+1} uniquely determines a oneparameter group of hyperbolic translations, whose Killing field extends the velocity vector field of γ.

(2.2) Ball model. Introduce a complete, rotationally invariant conformal metric

ds^{2} = \frac{4 (dx_{1}^{2} + dx_{2}^{2} + ⋯ + dx_{n+1}^{2})}{(1 − r^{2})^{2}}

on the open ball B^{n+1} with radius 1 centered at the origin 0 in ^{n+1} (here r^{2}: = x_{1}^{2}+x_{2}^{2}+⋯+x_{n+1}^{2}). Completeness of this metric and constant curvature −1 are again easy to check. Euclidean rotations (i.e., elements of the group SO(n + 1) ) and reflections across hyperplanes through 0 are obvious isometries, and it is straightforward to verify that the following maps from ball to the upper halfspace and back, (treating both as subsets of (x, y)-space) are mutually inverse isometries:

(x, y) ⟼ \frac{(2x, −|x|^{2} − y^{2} + 1)}{(1 − y)^{2} + |x|^{2}}

and

(x, y) ⟼ \frac{(2x, |x|^{2} + y^{2} − 1)}{(1 + y)^{2} + |x|^{2}}

Since both models are conformally flat, these are conformal maps of ^{n+1}. Thus, in the ball model, hyperbolic reflections are again (Euclidean) hyperplane reflections and inversions in spheres meeting the (r ≡ 1) sphere at infinity S_{∞} = ∂B^{n+1} orthogonally. Any point q outside B^{n+1} centers a unique such reflecting sphere, whose Euclidean radius R, satisfies R^{2} = |q|^{2} − 1. Each radial ray emanating from B^{n+1} thus determines a one-parameter family of such reflections and these are the families we shall use to implement the Alexandrov reflection method in hyperbolic space. (Recall in Euclidean space, this method involves successive reflection across each of a family of parallel hyperplanes; see 6.5). The above also shows that k-dimensional spherical caps and disks meeting S_{∞} orthogonally (k ≤ n + 1) are totally geodesic in the ball model; Euclidean circular arcs and line segments meeting S_{∞} orthogonally are hyperbolic geodesics.

Explicit expressions for hyperbolic translations in the ball model are not particularly easy (or illuminating) to write down, but their behavior is clear from the upper halfspace model: given any two points on S_{∞}, there is a one-parameter group of hyperbolic translations moving with unit speed along their unique “connecting” geodesic.

(2.3) Minkowski-space model. Here we write ^{1,n+1} for ^{n+2} with the Lorentz metric

Q = −dx_{0}^{2} + dx_{1}^{2} + dx_{2}^{2} + ⋯ + dx_{n+1}^{2} .

The following spacelike hypersurface S ⊂ ^{1,n+1} with induced Riemannian metric then provides the Minkowski space model for ^{n+1}:

S: = {x = (x_{0}, x) ∈ ^{n+2} | x ∈ ^{n+1} , 0 < x_{0} ∈ , Q(x) = |x|^{2} − x_{0}^{2} = −1}

In this way, we regard ^{n+1} as the “unit sphere” in (n + 2)-dimensional Minkowski space. By definition, the group O^{+}(1, n+1) preserves this metric, and acts transitively on S. To see the latter, consider that any (x_{0}, x) ∈ S can be sent to (1, 0) by an element of the group. For the matrices

cosh(t) sinh(t) 0 … 0
sinh(t) cosh(t) 0 … 0
0  0  1 … 0
.  . .  .
.  .  . .
.  .  . .
0  0  0 … 1

form a one-parameter subgroup of O^{+}(1, n + 1), and any (x_{0}, x) ∈ S can be mapped to (x_{0}, |x|e_{1}) by a “horizontal” rotation (i.e., element of O(n + 1); we regard the x_{0}-axis as “vertical”). Belonging to S, the latter vector can be written (cosh(s) , sinh(s)e_{1}) for some s ∈ , and we obtain an isometry mapping it to (1, 0) by applying the matrix above with t = −s. In fact, since the isotropy group of S at (1, 0) contains O(n + 1), S must have constant curvature. Computing at (1, 0), one finds it to be −1. Completeness is also easy to check, as noted below.

Observe further that the matrices above act transitively on the set of all hyperplanes in ^{n+2} containing the subspace x_{1} = 0. We therefore obtain involutive isometries of S by conjugating reflection across the hyperplane x_{1} = 0, (which induces an obvious isometry) with any of the above matrices. These involutions are the hyperbolic reflections encountered previously in the ball and upper halfspace models.

Moreover, conjugating these reflections with horizontal rotations and taking intersections, we see that the totally geodesic subspaces of S are precisely the intersections of S with vector subspaces of ^{1,n+1}; all geodesics of S are intersections of S with Euclidean planes through (0, 0) ∈ ^{1,n+1}. In particular, the curve cosh(t)e_{0} +sinh(t)e_{1} in S is a geodesic, and the one-parameter group of matrices above translates it along itself with unit speed. This makes the completeness of S obvious, as well as the following: the one-parameter hyperbolic translation subgroups on S are all conjugate to the matrix subgroup of O^{+}(1, n + 1) displayed above.

Note also that in this model, the sphere at infinity S_{∞} naturally identifies with the (projectivized) null-cone {x = (x_{0}, x) ∈ ^{1,n+1} | Q(x) = |x|^{2} − x_{0}^{2} = 0}.

(2.4) Hypersurfaces of constant mean curvature. We define the mean curvature vector h of a hypersurface Σ ∈ ^{n+1} as the trace of the second fundamental form A on Σ:

h : = trace(A) = (∑_{i=1}^{n} ∇_{τ_{i}}τ_{i})^{⊥} .

Here τ_{1}, τ_{2}, …, τ_{n} is any local orthonormal frame field on Σ, “∇” is the covariant derivative operator on ^{n+1}, and “⊥” denotes projection on the normal bundle of Σ. We say Σ has mean curvature H if each of its points has a neighborhood with a continuous unit normal vector field ν such that h = −Hν.

It is well-known that h can also be characterized in variational terms, according to the first variation formula for area (§ 9 of [20]):

\frac{d}{dt} |_{t=0} |Σ(t) ∩ K| = − ∫_{Σ} h · X .

Here “| · |” signifies n-area on Σ , and the integral is taken with respect to same. Σ(t) is any one-parameter family of hypersurfaces having Σ(0) = Σ , with initial velocity vector field X supported in a compact set K disjoint from ∂Σ. This formula simply says that the vector field h represents the “gradient” of the n-area functional on hypersurfaces of ^{n+1}.

(2.5) Some examples. Any hypersurface of ^{n+1} on which the isometries of ^{n+1} act transitively must have constant mean curvature: for example, any sphere centered at 0 in the ball model, and any “horizontal” hyperplane y = y_{0} in the upper halfspace model. The latter are known as horospheres; in the ball model they appear as Euclidean spheres tangent to ∂B^{n+1}.

In the ball model, a sphere of hyperbolic radius r centered at 0 is quickly seen to have Euclidean radius tanh(r/2), hence (multiply by the conformal factor and raise to the n^{th} power), n-area proportional to sinh^{n}(r). The first variation of area formula above then gives it constant mean curvature H ≡ n coth(r). In the upper halfspace model, the horospheres are similarly found to have constant mean curvature H ≡ n.

The fact that spheres in ^{n+1} have mean curvature bounded away from zero as their radii tend to infinity — the lower bound is exactly n, the mean curvature of the horospheres — illustrates a key qualitative difference between the theories of constant mean curvature hypersurfaces of ^{n+1}, and of ^{n+1}, where a sphere of radius r has mean curvature n/r. This difference shows up again when we consider Delaunay hypersurfaces in ^{n+1} (Lemma 6.3). Indeed, cylinders of radius r have mean curvature H ≡ (n − 1) coth(r) + tanh(r) > n, and we will see that cylindrically bounded Delaunay hypersurfaces in ^{n+1} exist only when H > n. This is why the latter inequality appears as an important hypothesis in all our main results (e.g., Theorems 1.1 and 1.2). The theory of constant mean curvature hypersurfaces with H ≤ n is more like the theory of minimal (H = 0) hypersurfaces in Euclidean space than the constant mean curvature theory of [11] and [16] which guides the present work.

(2.6) Two elliptic equations. In the Minkowski space model, a constant mean curvature hypersurface Σ^{n} in ^{n+1} may be regarded as a spacelike codimension-two submanifold of ^{1,n+1}. Two elliptic PDE’s associated with this situation will help us get height estimates for constant mean curvature graphs in Lemma 3.3 and § 5.4. We sketch their derivations here.

Let ν be a unit vector field normal to Σ (and tangent to S). Then the position vector field X on Σ forms, with ν, an orthonormal pair spanning the normal bundle to Σ in ^{1,n+1}. These complete any local orthonormal tangent frame field τ_{1}, τ_{2} …, τ_{n} on Σ to an orthonormal frame for the pullback to Σ of the ^{1,n+1} tangent bundle. It turns out the connection Laplacian on this bundle induces a simple endomorphism of the normal bundle:

Δ_{Σ}X = nX − Hν

(2.7)

Δ_{Σ} ν = −HX + |A|^{2}ν .

Here H and A are respectively the (constant) mean curvature and second fundamental form of Σ. The Laplacian Δ_{Σ} can be written as

Δ_{Σ} : = D_{i}D_{i} − D_{(D_{i}τ_{i})^{⊤}} ,

where D_{v} is the standard derivative along v ∈ ^{n+2} (D_{i} abbreviates the case v = τ_{i}), and “⊤” signifies projection onto T Σ.

To obtain equations (2.7), compute at a single point, choosing a “normal coordinate” frame {τ_{i}} so that (D_{i}τ_{j})^{⊤} vanishes there for each i, j = 1, …, n. Write

h_{ij}: = −ν · D_{i}τ_{j} = τ_{j} · D_{i}ν = τ_{i} · D_{j}ν

for the components of A, (hence H = Σh_{ii}) and observe that D_{i}X = τ_{i}. Using the summation convention one then has

Δ_{Σ}X = D_{i}τ_{i} = (ν · D_{i}τ_{i})ν − (X · D_{i}τ_{i})X = −h_{ii}ν + (τ_{i} · D_{i}X)X ,

and the first equation in (2.7) follows immediately. (Geometrically, this equation establishes that the mean curvature vector Δ_{Σ}X of Σ , as a codimensiontwo submanifold of Lorentz space, is the sum of its mean curvature vector as a hypersurface in S, and that of S itself in ^{1,n+1}.)

For the second equation, first check that the normal components of D_{i}ν automatically vanish on Σ, so that

D_{i}D_{i}ν = D_{i} ((τ_{j} · D_{i}ν)τ_{j}) = D_{i}(h_{ij}τ_{j}) .

But recall that the Codazzi equations for hypersurfaces of space forms imply D_{i}h_{jk} = D_{j}h_{ik} for all i, j, k = 1, …, n. So,

D_{i}D_{i}ν = (D_{j}h_{ii})τ_{j} + h_{ij}D_{i}τ_{j}
= h_{ij} ((ν · D_{i}τ_{j})ν − (X · D_{i}τ_{j})X)
= −h_{ij}^{2}ν + h_{ii}X ,

which establishes the desired fact.

3 Cylindrical boundedness and topological obstructions

In this section we generalize Meeks’ cylindrical boundedness theorem [16] to constant mean curvature surfaces in ^{3}. Our method of proof, which also applies in ^{3}, further simplifies that given in [11]. Throughout this section A will denote a properly embedded annulus (≈ S^{1}×[0, ∞)) in ^{3} with boundary ∂A a simple closed curve.

Lemma 3.1 Let γ_{R} be a circle of radius R in a geodesic plane in ^{3}. Let N_{ε} denote the closed ε-tubular neighborhood of γ_{R} for some fixed ε < R. Suppose Σ is a compact connected oriented surface of positive mean curvature in N_{ε} with ∂Σ ⊂ ∂N_{ε}. If Σ does not separate N_{ε}, then its mean curvature is somewhere less than or equal to that of a sphere S_{ε} ⊂ ^{3} with radius ε.

Proof. We first show that the map i_{∗}: π_{1}(Σ) → π_{1}(N_{ε}) induced by the inclusion map is the trivial map. Since Σ does not separate N_{ε}, there is a loop α in N_{ε} intersecting Σ transversely in a single point, and homotopic to some nonzero multiple of γ_{R}. Thus, we may assume, after orienting γ_{R}, that Σ has positive intersection number with γ_{R}. If i_{∗}: π_{1}(Σ) → π_{1}(N_{ε}) is non-trivial, there is a simple oriented loop β on Σ homotopic to a nonzero multiple of γ_{R}. The homology class of β in H_{1}(N_{ε}) must therefore have nonzero intersection number with Σ. However, since the normal bundle to Σ is trivial, we can push the curve β off Σ to obtain a homotopic curve β̃ that is disjoint from Σ. Since β̃ is homologous to β, this is a contradiction. Hence i_{∗}: π_{1}(Σ) → π_{1}(N_{ε}) is trivial.

Let Π: Ñ_{ε} → N_{ε} denote the universal Riemannian covering space of N. Since i_{∗}: π_{1}(Σ) → π_{1}(N_{ε}) is trivial, Σ lifts to a surface Σ̃ ⊂ Ñ_{ε}. Since Ñ_{ε} is simply connected and ∂Σ̃ ⊂ ∂Ñ_{ε}, the surface Σ̃ separates Ñ_{ε} into two components. If the closure of either component were compact, then Σ would be the boundary of a compact region of N_{ε} that is the image of this component under projection by Π. This contradicts our hypothesis that Σ does not separate N_{ε} and so Σ̃ separates the two ends of Ñ_{ε}.

Let γ̃_{R}: → Ñ_{ε} be a parametization of the proper arc covering γ_{R}. For each t let S(t) denote the sphere of radius ε in Ñ_{ε} with center γ̃_{R}(t). Let t_{0} = min{t | S(t) ∩ Σ̃ ≠ ∅}, t_{1} = max{t | S(t) ∩ Σ̃ ≠ ∅} and let p_{i} ∈ S(t_{i}) ∩ Σ̃ for i = 1, 2, respectively. A simple geometric comparison of principal curvatures then shows that the mean curvature of Σ̃ at one of the points, p_{1} or p_{2}, is less than the mean curvature of the sphere S_{ε}. (In this comparison we use the fact that if S(t_{1}) is contained in one of the closures of a component of Ñ_{ε} − Σ̃, then S(t_{2}) is contained in the closure of the other component.) □

Lemma 3.2 Let A ⊂ ^{3} be a properly embedded annulus with mean curvature function greater than a constant H > 2. Let ε > arctanh(\frac{2}{H}), the radius of a sphere in ^{3} with constant mean curvature H. Let P_{+} and P_{−} be planes of distance ε from a geodesic plane P in ^{3} and let Π_{+} and Π_{−} denote the two closed halfspaces determined by these planes that do not contain P. Then Π_{+} ∩ A or Π_{−} ∩ A consists entirely of compact components.

Proof. Without loss of generality, assume P_{−} and P_{+} are transverse to A. We will argue by contradiction. If Π_{+} ∩ A and Π_{−} ∩ A each contain noncompact components, then there are proper arcs α_{+} ⊂ P_{+} ∩ Int(A) and α_{−} ⊂ P_{−} ∩ Int(A) parametized by a half-open interval. Consider α_{+} and α_{−} to be proper disjoint arcs on A and choose an embedded path β joining the end point of α_{+} to the end point of α_{−}. The arc δ = α_{+} ∪ α_{−} ∪ β is a proper embedded arc in Int(A) that separates A into two components, where one component is topologically an annulus that contains ∂A, and the other component is simply connected. Let D denote the closure of the simply connected component of A − δ (see Figure 1).

[figure: A; ∂A; i; α_{−}; β; D; α_{+}; P_{+}; P; P_{−}; Π_{+}; Π_{−}; β; p; D; α_{+}; α_{−}; Figure 1.]

Choose a ball B centered at some point p ∈ P and of radius sufficiently large so that the arc β is contained in the interior of B. Since δ intersects P in a compact set contained in the disk B ∩ P, we can choose a circle γ_{R} ⊂ P centered at p, whose radius R equals the sum of the radius of B and the positive number ε . Clearly γ_{R} has linking number 1 with the proper arc δ.

Without loss of generality, assume now that γ_{R} intersects the simply connected surface D transversely, and that the boundary of the ε-tubular neighborhood N_{ε} of γ_{R} is also transverse to D. Since γ_{R} has linking number 1 with δ, γ_{R} has odd intersection with D. So one component Σ of D ∩N_{ε} has odd intersection with γ_{R} and Σ does not separate N_{ε}. But then, Lemma 3.1 implies Σ has some point with mean curvature less than or equal to the mean curvature of a sphere of radius ε, namely, H. This contradiction completes our proof. □

The following estimate is true for constant mean curvature hypersurfaces of ^{n+1}, and is sharp on geodesic spheres.

Lemma 3.3 Let Σ ⊂ ^{n+1} be an embedded compact hypersurface with constant mean curvature H > n, and boundary ∂Σ contained in a totally geodesic hyperplane P. Then for any x ∈ Σ, dist(x, P) ≤ K(H) ≡ 2 arctanh(\frac{n}{H}).

Proof. Since Σ is compact we can choose a point p ∈ Σ of maximal distance d from the hyperplane P. Let γ be the unit speed geodesic with γ(0) ∈ P and γ(d) = p. Note that γ is orthogonal to P at γ(0). Let Y be the Killing field that generates the unit speed family of hyperbolic translations h_{t} along γ with h_{0} the identity (see Section 2.1). Let R_{t}: ^{n+1} → ^{n+1} be hyperbolic reflection in the geodesic hyperplane P_{t} = h_{t}(P).

Let Σ_{t} = ⋃_{t′≥t}(P_{t′} ∩ Σ). The Alexandrov reflection principle implies that for t ≥ \frac{d}{2}, R_{t}(Σ_{t}) ∩ Σ = ∂Σ_{t}, and that this intersection is transverse. It follows that the surface S = h_{− \frac{d}{2}} (Σ_{\frac{d}{2}} ) is a graph with respect to the Killing coordinate t over a domain Ω ⊂ P.

We claim that S is also a graph over Ω with respect to the distance function s from P. In fact, it suffices to show that S is a graph with respect to the Killing coordinate associated to any unit speed geodesic γ that meets Ω ⊂ P orthogonally, since along γ the Killing coordinate t coincides with s, and in general, t ≤ s. But this latter claim follows from the Alexandrov reflection argument used previously.

It remains to show the distance function s from P satisfies s ≤ arctanh(\frac{n}{H}) on the graph S. To do this, we model ^{n+1} by the unit sphere S in Minkowski space ^{1, n+1} (see discussion in Section 2.3).

We may assume P is the geodesic hyperplane {x_{1} ≡ 0}. In these coordinates, the distance from P is simply s = arcsinh(x_{1}). By the above, S is a graph with respect to x_{1}, and x_{1} = 0 on the boundary. Also, the first component of the unit normal ν on a graph satisfies ν_{1} ≥ 0, and in general ν_{1} ≤ \sqrt{1 + x_{1}^{2}} = \sqrt{1 + sinh^{2} s} = cosh s.

Now by the Cauchy-Schwartz inequality we have n|A|^{2} ≥ H^{2}, and so by equations 2.7,

Δ(Hx_{1} − nν_{1}) = (n|A|^{2} − H^{2})ν_{1} ≥ 0.

The maximum principle then implies

Hx_{1} − nν_{1} ≤ (Hx_{1} − nν_{1}) |_{∂S} ≤ 0 ;

that is

sinh s = x_{1} ≤ \frac{n}{H} ν_{1} ≤ \frac{n}{H} cosh s ,

which immediately gives the required estimate. □

Corollary 3.4 Suppose Σ ⊂ ^{n+1} is an embedded compact hypersurface of constant mean curvature H > n. Then the distance from Σ to the convex hull H(∂Σ) of the boundary of Σ is less than K(H) ≡ 2 arctanh(\frac{n}{H}).

Proof. Suppose p ∈ (Σ−H(∂Σ)) and d = dist(p, H(∂Σ)). Let q ∈ ∂(H(∂Σ)) be a closest point to p, let γ be the geodesic line segment joining p and q, and denote by P_{q} the geodesic plane orthogonal to γ at q. Let Π denote the halfspace with boundary P_{q} and that contains the p. Then Π ∩ Σ is a compact hypersurface with boundary contained in P_{q}. Since dist(p, P_{q}) = d, Lemma 3.3 shows that d ≤ K(H). □

Remark 3.5 The corresponding result holds in Euclidean space, replacing the constant K(H) by 2n/H. (This generalizes a result of Serrin [19].)

We now prove the main result of this section.

Theorem 3.6 (Cylindrical Boundedness) Suppose A is a properly embedded annulus in ^{3} of constant mean curvature H > 2. Then for any point p ∈ ∂A there exists a unique geodesic ray γ_{p} through p such that A stays a bounded distance C from γ_{p}. The number C depends only on the diameter of ∂A and on H.

Proof. Choose a point p ∈ ∂A and a divergent sequence {p_{1}, …, p_{n}, …} ⊂ A. Let γ_{i} denote the unit speed geodesic joining p to p_{i}. Since the unit sphere in the tangent space to ^{3} at p is compact and the geodesics γ_{i} are determined by their initial velocities, a subsequence of {γ_{i}} converges to a unit speed geodesic ray γ. We now show A stays a bounded distance from γ and that this distance can be estimated from above by a constant depending only on its mean curvature H and the diameter of ∂A. Clearly any geodesic γ satisfying these hypotheses is unique.

Let C(γ, d) denote the d-tubular neighborhood of γ where d is the diameter of ∂A. For this choice of d, we have ∂A ⊂ C(γ, d). Suppose P is a totally geodesic plane tangent to C(γ, d). Then P intersects ∂C(γ, d) at a unique point. Let P_{+} and P_{−} be the planes of constant distance ε from P where ε > arctanh(\frac{2}{H}). Let Π_{+} and Π_{−} be halfspaces defined as in the similar situation described in Lemma 3.2. Assume that P_{+}, P_{−} are indexed so that ( ^{3} − Π_{+}) contains γ.

[figure: p; p; γ; Figure 2a:; P; P_{+}; P_{−}; Π_{−}; Π_{+}; q; P; P_{+}; Π_{+}; γ; Figure 2b:]

We first claim Π_{+} ∩ A consists entirely of compact components. For, if to the contrary Π_{+} ∩ A contains a noncompact component, Lemma 3.2 makes every component of Π_{−} ∩ A compact. Since ∂(Π_{−} ∩ A) ⊂ ∂Π_{−} = P_{−}, and P_{−} is convex toward P (the interior convex hull of P_{−} is equal to ^{3} − Π_{−}), Corollary 3.1 shows Π_{−} ∩ A is within distance K(H) of P_{−}. So the limit set of A is contained in the limit set S_{+} of Π_{+}. (Here, we view the limit set of a subset of ^{3} as a subset of S_{∞} in the ball model.) On the other hand, the sequence {p_{i}} ⊂ A has the limit point p_{∞} of γ in its limit set. But γ and P diverge in ^{3} and hence p_{∞} is not contained in S_{+}. This contradiction proves every component of Π_{+} ⊂ A is compact.

Let q ∈ A be a point in the complement of C(γ, d) and let α be the geodesic from q to γ with length dist(q, γ). For this q we pick a particular geodesic plane P to apply the above estimates. Specifically, choose P to be the geodesic plane that is tangent to C(γ, d) at α ∩ ∂C(γ, d). Since P_{+} is convex toward P, Corollary 3.1 again implies dist(q, P) ≤ K(H). Therefore,

dist(q, γ) ≤ dist(q, P_{+}) + dist(P_{+}, P) + dist(P, γ) ≤ K(H) + ε + d ,

and this estimate proves the theorem. □

We can now prove Theorem 1.1 as stated in the Introduction, assuming without loss of generality that the ambient curvature c equals −1. Proof of Theorem 1.1. Suppose Σ ⊂ ^{3} is homeomorphic to a closed surface punctured in one point. By Theorem 3.1, the annular end of Σ stays a bounded distance C from some geodesic ray γ with γ(0) ∈ Σ. Hence the entire surface stays a bounded distance from γ.

Let P be a geodesic plane orthogonal to γ at γ(d) where d is greater than the constant K(H) of Lemma 3.3. Let Π be the halfspace determined by P and containing the point γ(0). Then Π ∩ Σ is a compact surface with boundary in P, and dist(γ(0), P) > K(H) contradicting Lemma 3.3. This proves part 1 of the theorem.

For the remainder of the proof we recall a theorem of Hsiang [9], which is proved using the Alexandrov reflection principle (cf. also § 5).

Hsiang’s Theorem [9]. Suppose that Σ ⊂ ^{n+1} is a complete properly embedded hypersurface of constant mean curvature that is a bounded distance from a geodesic hyperplane. Then Σ is invariant under geodesic reflection in the hyperplane. If Σ is a bounded distance from a geodesic, then Σ is a Delaunay hypersurface of revolution.

Suppose now that Σ is homeomorphic to a closed surface with two points removed. Theorem 3.1 says the annular ends A_{1} and A_{2} of Σ stay a bounded distance from geodesic rays γ_{1} and γ_{2}, respectively. Given two geodesic rays in ^{3} there is a unique geodesic γ to which γ_{1} and γ_{2} converge at infinity. It follows that Σ stays a bounded distance from γ. Hsiang’s theorem implies that Σ is a surface of revolution, which proves part 2 of the theorem.

Finally, assume Σ is homeomorphic to a closed surface with three points removed. Let A_{1}, A_{2}, A_{3} be the annular ends of Σ and let γ_{1}, γ_{2}, γ_{3} be the geodesic rays given in Theorem 3.1 such that A_{i} is a bounded distance from γ_{i} for i = 1, 2, 3. Note that the limit points at infinity of these geodesic rays must be distinct. Otherwise, the rays stay a bounded distance from some fixed geodesic, and Hsiang’s Theorem would make Σ a surface of revolution with only two ends. So there exists a unique geodesic plane P in ^{3} to which the geodesic rays γ_{1}, γ_{2}, γ_{3} are asymptotic, and Σ stays a bounded distance from P. Hsiang’s Theorem now makes Σ invariant under reflection in the plane P. Furthermore, the argument in the proof of Lemma 3.3 shows, if s is the distance function from P, then the halves of Σ above and below P are graphs over P relative to s. This completes the proof of Theorem 1.1. □

4 Conservation laws and momenta

Using the first variation formula and the symmetry group O^{+}(1, n + 1) of hyperbolic space, we derive conservation laws for a constant mean curvature hypersurface Σ ⊂ ^{n+1}. These laws permit, for example, the assignment to each end of Σ a momentum in the Lie algebra o(1, n + 1), and force the sum of these momenta (over all ends) to vanish. We also interpret these momenta geometrically, showing, for instance, how they assign geodesic “worldlines” to certain cycles on Σ (cf. [14] for a discussion of the Euclidean case).

Given a hypersurface Σ ⊂ ^{n+1} with constant mean curvature H, consider an (n−1)-cycle Γ ⊂ Σ and any “cap” n-chain K ⊂ ^{n+1} with boundary ∂K = Γ. Let η and ν denote the “outer” unit conormal and normal for Γ and K, respectively. In other words, if ν_{Σ} is the outer normal to Σ (oriented oppositely to the mean curvature vector), then the orientations (Σ , ν_{Σ}), (K , ν) and (Γ , η , ν_{Σ}) all agree with the righthanded orientation on ^{n+1}.

Each Lie algebra element Y ∈ o(1, n + 1) generates a one-parameter subgroup exp(tY) in O^{+}(1, n + 1). The velocity of the corresponding isometric flow on ^{n+1} is a Killing vector field Y. This process inverts, yielding an isomorphism between o(1, n + 1) and the Lie algebra of Killing vector fields on ^{n+1}.

Theorem 4.1 (Conservation Laws) On any hypersurface Σ ⊂ ^{n+1} with constant mean curvature H, there is a natural cohomology momentum class with coefficients in the dual Lie algebra,

μ ∈ H^{n−1}(Σ) ⊗ o(1, n + 1)^{∗}

defined by the area −H· volume flux induced by Y through (Γ, K):

⟨μ([Γ]), Y ⟩ = ∫_{Γ} η · Y − H ∫_{K} ν · Y .

Furthermore, as Σ is translated in ^{n+1} by the action of g ∈ O^{+}(1, n + 1), the momentum class transforms via the co-Adjoint representation

g^{∗}μ = Ad^{∗}(g) · μ .

Proof. To prove μ is well-defined in H^{n−1}(Σ)⊗o(1, n+1)^{∗}, it suffices to show, for fixed Y, that the flux integral depends solely upon the homology class [Γ] ∈ H_{n−1}(Σ). Since the flux integral is linear in its domains of integration, we only need to show it vanishes when (Γ, K) is null homologous, i.e. when there exists a relative (n, n + 1)-chain (S, U) ⊂ (Σ, ^{n+1}) such that

Γ = ∂S and K = ∂U − S.

In this case the area − H · volume flux induced by Y can be rewritten

⟨μ([Γ]), Y ⟩ = ∫_{∂S} η · Y − H ∫_{K} ν · Y
= ∫_{∂S} η · Y + H ∫_{S} ν_{Σ} · Y − H (∫_{S} ν_{Σ} · Y + ∫_{K} ν · Y)

But, combining the first variation formulas (cf. [20]) as in [11] for area

δ_{Y}|S| = ∫_{S} div Y = ∫_{∂S} η · Y + H ∫_{S} ν_{Σ} · Y

and for volume

δ_{Y}|U| = ∫_{U} DIV Y = ∫_{S} ν_{Σ} · Y + ∫_{K} ν · Y ,

we have

⟨μ([Γ]), Y ⟩ = ∫_{S} div Y − H ∫_{U} DIV Y
= δ_{Y}(|S| − H|U|)

as the pair (S, U) is deformed along Y; but the latter vanishes, since Y is assumed to be a Killing vector field, preserving area and volume.

The transformation property follows from a change of variables in the flux integral

⟨μ(g_{#}[Γ]), Ad(g) · Y ⟩ = ∫_{g(Γ)} g_{∗}η · g_{∗}Y − H ∫_{g(K)} g_{∗}ν · g_{∗}Y
= ∫_{Γ} η · Y − H ∫_{K} ν · Y
= ⟨μ([Γ]), Y ⟩

using the fact that the Lie algebra element Ad(g) · Y extends to the Killing vector field g_{∗}Y on ^{n+1}, which is the push-forward of Y by the derivative of g. □

As a direct consequence, we observe that the ends of a constant mean curvature hypersurface Σ ⊂ ^{n+1} must “balance”.

Corollary 4.2 On a complete, properly embedded hypersurface Σ ⊂ ^{n+1} with constant mean curvature and finitely many ends, any end E has a welldefined momentum μ_{E} ∈ o(1, n + 1)^{∗}, and the sum of the momenta corresponding to the ends is zero.

Proof. Define μ_{E} = μ([Γ]), where Γ is an appropriately oriented cycle separating E from the “compact part” of Σ. To prove the second statement, consider cycles Γ_{i} of this type, where i indexes the ends of Σ. Defining μ_{i} = μ([Γ_{i}]), we now simply observe that

μ_{1} + ⋯ + μ_{k} = μ (∑ [Γ_{i}]) = 0

since the latter sum bounds a compact domain in Σ. □

It is customary to identify the Lie algebra o(1, n + 1) and its dual o(1, n + 1)^{∗} with the Lie algebra ∧^{2} ^{1, n+1} of antisymmetric (with respect to the Lorentz form Q = −dx_{0}^{2} + dx_{1}^{2} + ⋯ + dx_{n+1}^{2}) linear maps on ^{1, n+1}

o(1, n + 1)^{∗} ≅ o(1, n + 1) ≅ ∧^{2} ^{1,n+1} ≅ {A ∈ M_{n+2}( ) | A^{t}Q + QA = 0}.

The dual pairing is then given by the Killing form ⟨A, Z⟩ = \frac{1}{2}T race(AZ). With these identifications, a little computation shows

μ([Γ]) = ∫_{Γ} η ∧ X − H ∫_{K} ν ∧ X

where X is the position vector on ^{1, n+1}, and η and ν are regarded as vectors in ^{1, n+1} via the “Minkowski-space model” embedding ^{n+1} ⊂ ^{1,n+1} discussed in § 2.3. Furthermore, the co-Adjoint action can be identified with the usual action of

O^{+}(1, n + 1) ≅ {U ∈ M_{n+2}( ) | U^{t}BUB = I}

on o(1, n + 1) by conjugation. Thus the geometric invariants of μ([Γ]) are simply its conjugacy invariants, namely its (generalized) eigenspaces and eigenvalues.

Before we work these out explicitly (we do so later for constant mean curvature surfaces in ^{3} ), recall the decomposition of O^{+}(1, n + 1) and its Lie algebra o(1, n + 1) into elliptic, parabolic, and hyperbolic elements.

Proposition 4.3 Consider a nonzero element μ ∈ o(1, n + 1) . We say μ is elliptic (E), parabolic (P), or hyperbolic (H), respectively, if one of the following four correspondingly lettered equivalent conditions holds:

1. exp(μ) , viewed as a conformal automorphism of \overline{B^{n+1}}, has:

E^{(1)} : a fixed point on the interior,

P^{(1)} : a unique fixed point on the boundary sphere S_{∞}, H^{(1)} : two fixed points on S_{∞}.

2. μ , extended to a vector field on \overline{B^{n+1}}, has:

E^{(2)} : a zero on the interior, P^{(2)} : a unique zero on S_{∞}, H^{(2)} : two zeros (a source and a sink) on S_{∞}.

3. μ, viewed as an antisymmetric endomorphism of ^{1,n+1}, has:

E^{(3)} : a ray of eigenvectors in the interior {v | Q(v) < 0} of the null

cone, P^{(3)} : one ray of eigenvectors on the null cone {v | Q(v) = 0}, H^{(3)} : two linearly independent rays of null eigenvectors (with real

eigenvalues \underline{+}m ≠ 0).

4. μ, as an (n + 2) × (n + 2) matrix has the property that:

E^{(4)} : each column (μ_{∗j}) (j ≥ 1) satisfies (μ_{0j})^{2} < ∑_{i=1}^{n+1} (μ_{ij})^{2}, P^{(4)} : neither E^{(4)} nor H^{(4)}, H^{(4)} : there is a column (μ_{∗j}) (j ≥ 1) such that (μ_{0j})^{2} > ∑_{i=1}^{n+1} (μ_{ij})^{2} .

Proof. The equivalence of (1) and (2) is clear since a zero of μ is a fixed point of exp(μ); (1) gives the usual notion of ellipticity, etc. for group elements. Items (1) and (2) are equivalent to (3) by the usual identification of \overline{B^{n+1}} = S_{∞} ∪ B^{n+1} with the projectivization of the future cone {v | Q(v) ≤ 0}. These are then equivalent to (4) by the following type of argument (we do case H^{(4)}).

Without loss of generality suppose

(μ_{0,n+1})^{2} > ∑_{i=1}^{n+1} (μ_{i,n+1})^{2} .

Now consider the hyperplane (e_{n+1})^{⊥} ⊂ ^{1,n+1} , which defines a totally geodesic \overline{^{n}} ⊂ \overline{^{n+1}} . Using the relationship between the Minkowski space and ball models (§ 2.2), one readily finds this inequality equivalent to the fact that the vector field

∑_{i,j=0}^{n+1} μ_{ij}x_{j}e_{i}

has a source and a sink on S_{∞} (one on each side of (e_{n+1})^{⊥}) corresponding to the \underline{+}m null eigenvectors of μ.

We leave the other cases to the reader. □

Remark 4.4 If μ ∈ o(1, n + 1) is hyperbolic, it defines a unique (oriented) geodesic L ⊂ ^{n+1} joining the source and sink mentioned above. L is the intersection of the oriented \underline{+}m eigenvector plane with the Lorentzian unit sphere S ⊂ ^{1,n+1} (see § 2.3). In fact, μ can be put into canonical form:

[ \begin{matrix}0 & m & 0 & . & . & . & & 0 & 0 \\ m & 0 & & & & & & & 0 \\ 0 & & 0 & s_{1} \\ . & & −s_{1} & 0 & & & & & . \\ . & & & & 0 & s_{2} & & & . \\ . & & & & −s_{2} & 0 & & & . \\ & & & & & & . \\ 0 & & & & & & & . & 0 \\ & & & & & & & 0 & s_{[n/2]} \\ 0 & 0 & 0 & . & . & . & 0 & −s_{[n/2]} & 0 \end{matrix} ]

with respect to an “orthonormal” basis {u_{0}, u_{1}, …, u_{n+1}} for ^{1,n+1}, i.e., Q(u_{0}, u_{0}) = −1, Q(u_{i}, u_{i}) = 1 (i > 0), Q(u_{α}, u_{β}) = 0 , (α ≠ β).

We call L the worldline, m the mass, and the invariants s_{1}, s_{2}, …, s_{[n/2]} the spins of μ.

To justify this remark, note that from H^{(3)} of Proposition 4.3 we have

[ \begin{matrix}m & 0 & 0 \\ 0 & −m & 0 \\ 0 & 0 & R \end{matrix} ]

where R ∈ o(n) is some antisymmetric n × n matrix. The existence of {u_{2}, u_{3}, …, u_{n}} putting R into a diagonal matrix of 2 × 2 antisymmetric blocks

[ \begin{matrix}0 & s_{i} \\ −s_{i} & 0 \end{matrix} ]

is then standard linear algebra. Now choose null eigenvectors v_{+} and v_{−} corresponding to \underline{+}m such that the remaining vectors u_{0} , u_{1} defined by u_{0} + u_{1} = v_{+} and u_{0} − u_{1} = v_{−} are orthonormal.

Remark 4.5 In the case n = 2 (corresponding to Σ ⊂ ^{3}), we have simply

μ ≅ [ \begin{matrix}0 & m & 0 & 0 \\ m & 0 & 0 & 0 \\ 0 & 0 & 0 & s \\ 0 & 0 & −s & 0 \end{matrix} ] ,

which can be viewed as the initial velocity of a screw-motion along the geodesic axis L with translation and rotation “speeds” m and s respectively.

To be completely explicit with regard to this last remark, if

μ = [ \begin{matrix}0 & w_{1} & w_{2} & w_{3} \\ w_{1} & 0 & r_{3} & −r_{2} \\ w_{2} & −r_{3} & 0 & r_{1} \\ w_{3} & r_{2} & −r_{1} & 0 \end{matrix} ] ,

is hyperbolic, then

(4.1)  m^{2} = \frac{1}{2} (B + \sqrt{B^{2} + 4C}) ,
(4.2)  s^{2} = −\frac{1}{2} (B − \sqrt{B^{2} + 4C}) ,

where, regarding w = (w_{1}, w_{2}, w_{3}) and R = (r_{1}, r_{2}, r_{3}) as vectors in ^{3} with the standard dot product, we have

B = w · w − R · R = ⟨μ, μ⟩, C = (w_{1}^{2}, w_{2}^{2}, w_{3}^{2}) · (r_{1}^{2}, r_{2}^{2}, r_{3}^{2}) .

We let the reader work out the analogous formulae when n > 2.

Remark 4.6 On the most important examples, the Delaunay surfaces of revolution, where [Γ] is represented by a cross-section, one finds that the spins all vanish, the world line L is the axis of revolution, and the mass m is computed by the area − H · volume flux generated by a unit speed dilation along the axis (cf. 6.3).

Finally, we determine which (n − 1)-cycles Γ ⊂ Σ ⊂ ^{n+1} give rise to hyperbolic momenta μ([Γ]) . (It seems natural to state the following result here, although the heart of its proof is really Theorem 5.3. For the definitions of “planar”, “interior” and “exterior” homology classes, the reader should refer to § 5.1.)

Theorem 4.7 (Geometric Positive Flux Lemma) Let [Γ] be a planar homology class on a complete constant mean curvature Σ ⊂ ^{n+1}. Then μ ([Γ]) is either hyperbolic or zero. In the former case, μ assigns [Γ] a geodesic worldline L and a nonzero mass m (positive if [Γ] is an interior class, negative if [Γ] is an exterior class). If μ([Γ]) = 0, then [Γ] is an interior class, and bounds a subdomain of Σ that is a graph over the corresponding plane domain.

Proof. We will use the “scalar” Positive Flux Lemma (5.3) to establish the inequality H^{(4)} from Proposition 4.2:

(μ_{0j})^{2} > ∑_{i=1}^{n+1} (μ_{ij})^{2} , (some j ≥ 1) .

Actually, for purposes of proof, we may assume j = n+1 by relabelling. Thus, let Π = (e_{n+1})^{⊥} ∩ S in the Minkowski space model and rotate coordinates in Π so that μ_{i,n+1} = 0 for each i > 2 . We must show

(μ_{0,n+1})^{2} > (μ_{1,n+1})^{2} .

Consider the following two parabolic elements of o(1, n + 1)

P\underline{+} = [ \begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 0 & . & & . & \underline{+}1 \\ . & . & & . & 0 \\ 0 & . & & . & 0 \\ 1 & \overline{+}1 & ⋯ & 0 & 0 \end{matrix} ] .

These extend to Killing vector fields P_{±} on ^{n+1} which are perpendicular to Π and satisy P_{±} · ν_{Π} > 0. Thus, the pairings

⟨μ , P_{+}⟩ = μ_{0,n+1} + μ_{1,n+1}
⟨μ , P_{−}⟩ = μ_{0,n+1} − μ_{1,n+1}

are both positive from Theorem 5.3 . So their product (μ_{0,n+1})^{2} − (μ_{1,n+1})^{2} is positive, as needed for H^{(4)}. □

5 Positive flux lemma

Here we prove our “Positive Flux Lemma” (Theorem 5.3) for a properly embedded constant mean curvature surface Σ with compact boundary ∂Σ.

Roughly, it says that “planar” homology classes (defined in § 5.1) which are far enough from ∂Σ must either be trivial, or have non-zero area−H ·volume flux relative to certain Killing vector fields. We first establish the analogous fact in Euclidean space. We do so because the Euclidean case is interesting on its own, and because it displays the main ideas of the hyperbolic case without as many technical details.

In § 6, we will apply the Positive Flux Lemma to cylindrically bounded Σ to deduce rotational symmetry “at infinity.” We hope this lemma will also prove useful in studying the “compact part” of constant mean curvature surfaces. As already observed (see Theorem 4.7, and [14] for the Euclidean case), the fact that a homology class has non-zero mass gives it a well-defined “worldline” which for a Delaunay end is just the axis of revolution. It should be true more generally that constant mean curvature surfaces stay close to their worldlines. For instance, one might hope to prove (at least for the case n = 2) that every properly embedded constant mean curvature surface stays near some “balanced diagram” like the examples in ^{3} constructed by N. Kapouleas [10].

(5.1) Configuration. We assume Σ comprises part of the boundary of some open set, into which the mean curvature vector of Σ points. Specifically, we assume that there exists an open, connected domain Ω with ∂Ω = Σ ∪ Σ′ and ∂Σ = Σ ∩ Σ′, where Σ′ is a smooth “cap”. (We assume nothing about the geometry of Σ′.)

The homology class [Γ] of Σ is called planar if the cycle Γ is a compact, transverse intersection between part of Σ and some (totally geodesic) hyperplane Π, and Γ is the (possibly disconnected) boundary of a component K of either Ω ∩ Π or of ( ^{n+1} − Ω) ∩ Π. In the first case, or more generally whenever Γ is trivial in Ω, we will call [Γ] an interior homology class, and in the second case (or whenever Γ is trivial in ( ^{n+1} −Ω)) we call [Γ] an exterior class. (Note that the Mayer-Vietoris exact sequence implies that interior and exterior classes span the homology of Σ ∪ Σ′, but there is no guarantee that planar classes do so.)

(5.2) Orientations. We retain the convention that on Σ the normal ν_{Σ} is the exterior normal to Ω, and require that on a planar cap K, the conormal η along Γ has positive inner product with the plane normal ν. The flux formulas (Theorem 4.1) for interior and exterior homology classes then differ by a sign in the cycle term:

⟨μ([Γ]), Y ⟩ = ∫_{Γ} η · Y − H ∫_{K} ν · Y [Γ] interior planar
⟨μ([Γ]), Y ⟩ = −∫_{Γ} η · Y − H ∫_{K} ν · Y [Γ] exterior planar .

Our main objective for the remainder of this section is now to prove:

Theorem 5.3 (Positive Flux Lemma) Let [Γ] be a planar homology class. Let Y be an Killing vector field perpendicular to Π , with Y·ν_{Π} strictly positive on Π. If [Γ] is an exterior planar homology class, then the flux ⟨μ([Γ]), Y⟩ is strictly negative, and thus Γ is nontrivial on Σ.

If [Γ] is an interior planar homology class, and if dist(K, Σ′) is sufficiently large (depending on H and the diameter of Γ), then either Γ is trivial on Σ or the flux ⟨μ([Γ]), Y⟩ is strictly positive. In the trivial case, Γ actually bounds a subset of Σ that is a graph (with respect to distance) above K.

In particular, only the planar homology classes near ∂Σ and the trivial class can have momentum zero. If ∂Σ is empty, every non-trivial planar class has hyperbolic momentum and nonzero mass (cf. Theorem 4.7).

Proof. For exterior planar classes, the definition of flux immediately yields the result, since η · Y and ν · Y are non-negative and positive, respectively, on Γ and K.

The main content of our theorem therefore concerns interior planar classes. Our plan is to “blow a bubble” K_{H} inside Ω with mean curvature H which is bounded by Γ, and then to compare the flux through Γ relative to Σ with the flux relative to K_{H} (which must be zero since Γ bounds K_{H}).

Specifically, we will construct a one-parameter family of graphs K_{λ} over K with constant mean curvature λ ∈ [0, H] and with ∂K_{λ} = Γ. Let R denote geodesic reflection across Π. Since the K_{λ} form a continuous family of constant mean curvature surfaces, the strong maximum principle implies that:

1. For λ < H, the graph K_{λ} intersects Σ and its reflected image Σ̃ = R(Σ) transversely along Γ and K_{λ} ∩ (Σ ∪ Σ̃) = Γ.

2. Either K_{H} ⊂ Σ ∪ Σ̃, or K_{H} ∩ (Σ ∪ Σ̃) = Γ and K_{H} is transverse to Σ and Σ̃ along Γ.

In case 2 above when K_{H} ⊂ Σ̃, the reflected image R(K_{H}) is contained in Σ. Hence when K_{H} ⊂ Σ ∪ Σ̃, Γ is the boundary of a compact graph in Σ. On the other hand, if Γ does not bound a graph in Σ, the angle of Σ with the plane Π along Γ exceeds the angle of K_{H} (see Figure 3). Since Y ⊥ Π, this means η_{K_{H}} · Y < η_{Σ} · Y along Γ. (Here subscripts distinguish between the conormals to Σ and K_{H}.) Hence

⟨μ([Γ]), Y ⟩ = ∫_{Γ} η_{Σ} · Y − H ∫_{K_{H}} ν · Y (5.1)

> ∫_{Γ} η_{K_{H}} · Y − H ∫_{K_{H}} ν · Y = ⟨μ_{K_{H}}([Γ]), Y ⟩ = 0.

The inequality here is precisely what we sought; it remains to construct the graphs K_{λ} to complete the proof of the Positive Flux Lemma.

K_{λ} K_{H}  Σ̃ Σ  Π Figure 3.

We first carry out the construction in Euclidean space. In this setting, Y generates a translation perpendicular to the plane Π. We may assume Π is the {x_{1} = 0} plane and that its normal ν_{Π} = −e_{1} = Y. The family of graphs over K will be K_{λ} = {x_{1} = u_{λ}(x_{2}, …, x_{n+1})}, were the non-negative function u_{λ} solves the nonparametric constant mean curvature λ equation:

\{ \begin{matrix}div (\frac{Du_{λ}(y)}{\sqrt{1+|Du_{λ}(y)|^{2}}} ) = −λ & y ∈ K \\ u_{λ}(y) = 0 & y ∈ ∂K = Γ . \end{matrix} (5.2)

for 0 ≤ λ ≤ H.

Note that u_{0} ≡ 0, so that K_{0} = K. It follows from the implicit function theorem for elliptic PDE’s that we can solve (5.2) for small λ > 0. The following sequence of estimates will enable us to continue this solution-family all the way through λ = H.

(5.4) Height estimates. Choose an origin in Π so that Γ lies in the ball of radius R = diam(Γ). Consider the spherical graph x_{1} = \sqrt{R^{2} − |y|^{2}}, with non-parametric mean curvature −n/R, and non-negative boundary values on Γ. For λ ≤ n/R, it provides a barrier above the graph K_{λ} of u_{λ}. That is,

u_{λ}(y) ≤ R, 0 ≤ λ ≤ n/R. (5.3)

Recall the estimate λx_{1} ≤ nν_{1} , (5.4)

which holds for a graph x_{1} = u(y) having mean curvature λ and upper normal ν. (This estimate is proven in [11] by the same method used in § 2.6 above for the analogous hyperbolic estimate: Δ(λx_{1}−nν_{1}) = (−λ^{2}+n|A|^{2})ν_{1} ≥ 0 holds for Euclidean graphs as well.) Applying (5.4) to estimate x_{1}, we conclude that u_{λ}(y) ≤ n/λ ≤ R, n/R ≤ λ ≤ H . (5.5)

(5.5) Gradient estimates and barriers at the boundary. Note that the u_{λ}(y) increase monotonically in λ: if λ_{1} < λ_{2} then u_{λ_{1}}(y) ≤ u_{λ_{2}}(y), with strict (interior) inequality, since the maximum principle applies to equation (5.2). Estimate (5.4) combines with this monotonicity in λ to bound ν_{1}(y) below by a positive number depending on y in (the interior of) K, but not on λ. The gradient |Du_{λ}(y)| is therefore bounded uniformly in λ on compact subsets of K, as long as the graphs K_{λ} exist.

Observe next that for λ ≤ H, our original surface Σ ∩ {x_{1} ≥ 0} , and the “positive” half of its reflection Σ̃ through {x_{1} = 0}, namely Σ̃ ∩ {x_{1} ≥ 0}, are barriers “above” the graphs of u_{λ}, provided that Σ′ is more than R units away from K. (This is why Theorem 5.3 requires the distance between Σ′ and K to be sufficiently large: we can thereby guarantee that any first point of tangential contact between ∂Ω (or the reflection ∂Ω̃) and u_{λ} occurs on Σ (or the reflection Σ̃). The maximum principle then controls contact between Σ (and Σ̃) and the graph K_{λ} of u_{λ}.)

At the boundary of K, since Σ ∩ {x_{1} ≥ 0} and Σ̃ ∩ {x_{1} ≥ 0} are barriers, we immediately get gradient estimates for the u_{λ}, independent of λ, wherever the conormal η of Σ is not perpendicular to Π along Γ. Of course, η may indeed be perpendicular to Π at many points. The following curvature estimates cover this contingency.

(5.6) Curvature estimates. One could apply geometric measure theory arguments to produce curvature estimates independently of gradient estimates: constant mean curvature graphs minimize (parametric) elliptic variational problems, hence have Holder continuous normals with norm depending only on the smoothness of ∂K = Γ [3]. Such an estimate implies that one has functions parameterizing uniformly-sized neighborhoods U(p) of points p ∈ K_{λ} as Lipschitz graphs above the tangent planes to K_{λ} at p. Elliptic regularity theory then bounds the curvatures.

However, a certain amount of machinery is required to obtain Holder continuity of the normal, so we use direct PDE techniques below to find tilted planes near Γ, above which the K_{λ} are (locally) uniform Lipschitz graphs. (These techniques relate to methods used in [13]; more technical versions of “tilting” apply to a variety of Dirichlet and contact angle problems [12].) We then combine the resulting curvature estimates near Γ with the interior estimates of § 5.5 to derive global curvature (and higher derivative) estimates independent of λ and |Du|.

To begin, let n be the exterior normal vector field in a tubular neighborhood of Γ ⊂ Π. By a mild abuse of notation, we also use n for the normal vector field (0, n) near the cylinder × Γ lying “above” Γ, and then extend n smoothly (again without changing notation) to the entire cylinder × K.

Now consider the functional

F = θe^{kx_{1}} − ν · n − Mν_{1} (5.6)

on K_{λ}. It suffices to show that we can find positive constants θ, k and M so that F ≤ 0 on K_{λ}. For, in a neighborhood of p ∈ K_{λ}, F ≤ 0 can be rewritten as

ν · (Me_{1} + n(p)) ≥ θe^{kx_{1}} + ν · (n(p) − n) ,

and one has a uniformly Lipschitz graph above a plane perpendicular to Me_{1} +n(p) as long as the quantity on the right is uniformly positive. Elliptic regularity theory for solutions to uniformly elliptic equations having locally smooth boundary data (i.e., the projections of Γ onto such planes when p is near × Γ) then produces uniform estimates on the curvature (and all its derivatives) on K_{λ}.

To find θ, k, M, recall first (paragraph after equation (5.2)), that for λ near 0, say 0 ≤ λ ≤ λ_{0}, the implicit function theorem gives us solutions u_{λ} with uniformly bounded gradient. For this range of λ, and for given θ, k, we can therefore ensure F ≤ 0 by making M large enough. It then suffices to find θ, k, M for the range λ ≥ λ_{0}.

For this purpose, we again use the strong maximum principle, which implies u_{λ_{0}} has strictly positive normal derivative, i.e., ν · n > θ along Γ for some positive number θ. Since the u_{λ} increase monotonically in λ, this normal derivative estimate continues to hold for all λ ≥ λ_{0}. Using this θ in our definition of F we see that F < 0 on ∂K_{λ} = Γ, for all λ ≥ λ_{0}.

Now, if for a given k, we already have F ≤ 0 with M = 0, we are done. If not, increase M from 0 until F ≤ 0 on K_{λ} and there exists an interior point p ∈ K_{λ} where F = 0. (This is possible because F < 0 on Γ and ν_{1} > 0 on the interior of K_{λ}.) Then at p, ∇F = 0 and ΔF ≤ 0, where ∇ and Δ are the surface gradient and Laplacian, respectively.) Using the identities Δx_{1} = −λν_{1}, Δν = −|A|^{2}ν, we compute ΔF (p) explicitly, and obtain an estimate at p:

\begin{matrix}θe^{kx_{1}}(−kλν_{1} + k^{2}|∇x_{1}|^{2}) \\ + |A|^{2}(ν · n + Mν_{1}) − C|A| − C & ≤ 0, \end{matrix} (5.7)

The error terms −C|A| − C here come from cross derivatives of Δ(ν · n). The constants C depend on the domain K, but not on λ. Since F (p) = 0 we can substitute ν · n + Mν_{1} = θe^{kx_{1}} into equation (5.7), obtaining

θe^{kx_{1}}(kλν_{1} + k^{2}|∇x_{1}|^{2} + |A|^{2}) − C|A| − C ≤ 0. (5.8)

But |∇x_{1}|^{2} = 1 − (ν_{1})^{2} > \frac{1}{2} if (ν_{1})^{2} < \frac{1}{2}, the latter holding for any M ≥ M_{0}(θ, k) because x_{1} is bounded from above (equation (5.4)), and F (p) = 0 implies

ν_{1} = \frac{θe^{kx_{1}} − ν · n}{M} . (5.9)

For M ≥ M_{0} equation (5.7) therefore implies at p:

θe^{kx_{1}} (\frac{k^{2}}{2} + |A|^{2} − Ck − C|A| − C) ≤ 0. (5.10)

But the arithmetic-geometric mean inequality contradicts this for k sufficiently large. Take one such as our value of k. It follows that our value of M, gotten by increasing from zero until the inequality F ≤ 0 is first obtained, must satisfy M ≤ M_{0}(θ, k). This gives the uniform curvature (and higher regularity) estimates we seek in the Euclidean case: we have shown that F ≤ 0 on K_{λ} for certain values of θ, k, and M = M_{0}(θ, k).

(5.7) Final step. Using the estimates from § 5.4, § 5.5, § 5.6 we deduce that the set of λ ≤ H, for which u_{σ} exists for all σ ≤ λ, is closed. For the limit of u_{σ} as σ → λ from below must exist, and must be a smooth graph by our apriori estimates. We now claim that this closed interval must actually be [0, H]. If not, denote it by [0, λ] for λ < H. Then |Du_{λ}| must be infinite at some boundary point p ∈ Γ; otherwise the equation (5.2) for u_{λ} is uniformly elliptic and we may again use the implicit function theorem to obtain solutions with larger mean curvature. Since K_{λ} lies “beneath” both Σ∩{x_{1} ≥ 0} and the reflection Σ̃∩{x_{1} ≥ 0}, it follows that at p the conormal η to Σ is perpendicular to Π, hence K_{λ} and Σ ∩ {x_{1} ≥ 0} make one-sided tangential contact there. This is a contradiction: it violates the maximum principle for λ < H. Hence the path of graphs includes all values λ ∈ [0, H].

The same basic strategy proves the theorem in hyperbolic space. We indicate the necessary technical modifications below. We work in the Minkowski space model (§ 2.3). Assume Π = {x_{1} = 0}∩ ^{n+1}, and that along Π, ν_{Π} and the Killing vector field Y are positive (function) multiples of the projection of −e_{1} onto ^{n+1} (i.e. of −(e_{1} − (e_{1} · x)x), where x = (x_{0}, x) can be viewed as both position and normal vector to ^{n+1} ⊂ ^{1,n+1}).

Definition 5.8 Let γ(y) denote the unit speed geodesic in ^{n+1} perpendicular to Π, passing through y ∈ Π, with tangent vector at y pointing in the positive x_{1}-direction. Let γ(y, s) be its parametrization, where s is the arclength, γ(y, 0) = y. A surface S is a graph (with respect to distance) above Π if there exists a function v defined on a subset K of Π such that S = {γ(y, v(y)) | y ∈ K}.

Now we will construct a one-parameter family of graphs K_{λ} of functions u_{λ} ≥ 0 above K ⊂ Π having mean curvature λ ∈ [0, H], with K_{0} = K, as in the Euclidean case. Recall from Section 3 that the geodesic distance s from Π relates to x_{1} by sinh s = x_{1}. Consider the functions (x_{λ})_{1} ≡ v_{λ} ≡ sinh u_{λ}. If estimates corresponding to those in § 5.4, § 5.5, and § 5.6 can be established for the v_{λ}, then the equivalent estimates will hold for the “hyperbolic” u_{λ}. The equation corresponding to (5.2) is

\{ \begin{matrix}Δv_{λ} = nv_{λ} − λ(ν_{λ})_{1} & y ∈ K \\ v_{λ}(y) = 0 & y ∈ ∂K = Γ . \end{matrix} (5.11)

This equation restates the identity Δx_{1} = nx_{1} − λν_{1} from § 2.3, so the Laplacian in (5.11) is that associated to the pull-back of the metric on K_{λ} onto K ⊂ Π. It is uniformly elliptic for y ∈ K as long as u and |Du| (equivalently v and |Dv|) are uniformly bounded above. By considering the (upper) normal ν to K_{λ} one also finds this equivalent to the existence of a uniform upper bound on u and lower bound ν · Y ≥ δ > 0. But for bounded u, Y is a uniform multiple of the projection of e_{1} onto ^{n+1}, so we may replace ν · Y ≥ δ > 0 by ν_{1} = ν · e_{1} ≥ δ > 0 to guarantee uniformity.

As long as (5.11) remains uniformly elliptic, i.e., as long as one has a solution u_{λ} for which u_{λ_{−}} and |Du_{λ}| are bounded, the implicit function theorem guarantees an interval of solutions about u_{λ}. In particular, solutions exist in a neighborhood of u_{0} ≡ 0.

(5.9) Height estimates. As in Euclidean space, make a choice of Π so that the origin (0, 0, …, 0, 1) of ^{n+1} is in K, and so that a geodesic ball of radius R = diam(Γ) about (0, 0, …, 0, 1) contains Γ ∪ K.

Consider the equidistant hemisphere obtained by intersecting the plane {x_{0} = C} with ^{n+1}. As C → ∞ this hemisphere has mean curvature approaching n. Choose C = C(R) so large that the plane intersects Π outside Γ. The resulting hemisphere has mean curvature H_{0} > n hence provides a barrier above for K_{λ} when 0 ≤ λ ≤ H_{0}. But for H_{0} ≤ λ ≤ H, one can use Lemma 3.3 (a consequence of λx_{1} ≤ nν_{1}) to derive uniform estimates above for x_{1} = v_{λ}.

(5.10) Gradient estimates and barriers at the boundary. The monotonicity (strict in the interior K) of v_{λ} with respect to λ follows from the maximum principle, just as in the Euclidean case. The estimate λx_{1} ≤ nν_{1} and the monotonicity in λ yield interior gradient estimates. Again, the original surface Σ ∩ {x_{1} ≥ 0} and the reflection Σ̃ ∩ {x_{1} ≥ 0} of Σ through {x_{1} = 0} are barriers “above” the K_{λ}, provided that Σ′ is sufficiently far from Π, depending on the height estimates of § 5.9.

(5.11) Curvature estimates. Follow the reasoning used in § 5.6, considering again the function F from equation (5.6) on the surface K_{λ}. Extend the normal n as before; first from a tubular neighborhood of Γ to all of K, then from K to the solid cylinder above it by parallel transport along the geodesics γ(y). By the arguments after equation (5.11) it again suffices to find θ, k, M for which F ≤ 0 on K_{λ}. Find θ as before. For a given k, we let M vary as before and find an interior point p at which F (p) = 0, knowing that F ≤ 0 elsewhere. The analog of estimate (5.7) (computing ΔF (p), using the formulas for Δx_{1} and Δν in § 2, and noting that ΔF (p) is non-positive) is:

\begin{matrix}θe^{kx_{1}}(k(−λν_{1} + nx_{1}) + k^{2}|∇x_{1}|^{2} \\ + |A|^{2}(ν · n − λx · n + Mν_{1}) − C|A| − C & ≤ 0. \end{matrix} (5.12)

But x · n = 0 since x is the normal to ^{n+1} at x. Using F (p) = 0 we now estimate as in equation (5.8): θe^{kx_{1}}(k^{2}|∇x_{1}|^{2} + |A|^{2} − Ck − C|A| − C) ≤ 0. (5.13)

In the present setting we have ∇x_{1} = e_{1} − (x_{1}) x − ν_{1}ν, and, because x · x = −1, we have |∇x_{1}|^{2} = 1 + (x_{1})^{2} − (ν_{1})^{2}. Estimate (5.13) lets us again find an M_{0} such that M ≥ M_{0} implies |∇x_{1}|^{2} ≥ \frac{1}{2}. Therefore we can complete the argument as in § 5.6, and conclude uniform curvature and higher regularity estimates for the K_{λ}, as in the Euclidean case. □

6 Asymptotic symmetry of cylindrically bounded ends

We now show that any noncompact properly embedded constant mean curvature hypersurface Σ ⊂ ^{n+1}, which is cylindrically bounded and has compact boundary ∂Σ, must approach a Delaunay surface exponentially at infinity (Theorem 6.9). The proof is conceptually simpler than that [11] in the Euclidean case Σ^{2} ⊂ ^{3}, and holds in all dimensions. This happens mainly because Alexandrov reflection is “stronger” in hyperbolic space, since there are “more” totally geodesic hyperplanes parallel to a given hyperplane.

(6.1) Configuration. Let Σ be a properly embedded noncompact constant mean curvature hypersurface in ^{n+1}, with compact boundary ∂Σ and the property that Σ stays in a solid half cylinder C(γ, R), i.e., the geodesic radius R tubular neighborhood about a geodesic ray γ.

Let Y be the Killing vector field generating unit-speed translation along γ, and write Y = ∂_{t}, where arclength t varies from 0 to ∞ along γ = γ(t). For each fixed t, let D_{t} be the cross-section of C(γ, R) at t, i.e., the totally geodesic n-disk of radius R perpendicular to γ and containing γ(t). Use ρ to denote distance from γ i.e., ρ(p) is the distance from p ∈ D_{t} to γ(t).

We may assume, after translating along Y and discarding some compact pieces of Σ, that ∂Σ is contained in D_{0}, and that Σ bounds a possibly disconnected domain Ω ⊂ C(γ, R). We allow the possibility that Σ is now a finite union of disconnected ends, each one bounding a connected domain. Our analysis will show a posteriori that there is really only one end.

Next, perform an isometry of ^{n+1} so that in the upper half-space model (§ 2.1) γ is the y-axis, {0 < y < 1}. Then C(γ, R) is part of a solid Euclidean cone vertex at the origin; its intersection with each Euclidean n-sphere centered at 0 is a totally geodesic hyperbolic disk D_{t} of hyperbolic radius R. We then have Y = ∂_{t} = −r∂_{r}, for t = − ln r, and r = Euclidean distance to 0.

Lemma 6.2 Let Σ and Y be as in Configuration 6.1. Then the flux μ = ⟨μ([∂Σ]), Y ⟩ is positive. (Here we orient the conormal to ∂Σ to be inward on Σ, i.e. it points toward infinity).

Proof. Note that Γ = ∂(Ω ∩ D_{t}) is an interior planar cycle homologous to ∂Σ. Hence by Theorem 5.3, for t large, Γ must have positive flux with respect to Y. This proves ∂Σ has positive flux. □

Lemma 6.3 For fixed H > n there exists a unique (up to translations generated by Y = ∂_{t}) embedded axially symmetric Delaunay surface D(m) about γ with mean curvature H and mass m = ⟨μ([D(m) ∩ D_{t}]), Y ⟩. The mass is positive, is maximized for the cylinder of mean curvature H, and approaches zero as the family evolves into a chain of spheres.

Proof. Fix a Delaunay surface D and express it as a graph in cylindrical coordinates, {ρ = ρ_{D}(t)}. Let Ω be the solid region of ^{n+1} “inside” D.

Taking rotational symmetry into account, the flux through the homology class [D ∩ D_{t}] generated by Y = ∂_{t} is actually the mass:

m = ∫_{D∩D_{t}} η · ∂_{t} − H ∫_{Ω∩D_{t}} ν · ∂_{t} . (6.1)

We calculate m explicitly now in terms of ρ and ρ_{t}, using elementary hyperbolic geometry. Noting first that geodesic distance ρ is related to polar angle φ (r cos φ = y) by dρ = sec φ dφ we integrate to get

tan φ = sinh ρ. (6.2)

Letting ω_{n} be the area of the unit Euclidean n-sphere S^{n}, we next evaluate (6.1) with the aid of (6.2):

(6.3) m = ω_{n−1}(tan φ)^{n−1} sec φ\frac{1}{\sqrt{1 + φ_{t}^{2}}} − \frac{H}{n} ω_{n−1}(tan φ)^{n}.

Thus ρ_{D} satisfies a first order ODE whose derivative is the usual second order ODE for mean curvature. Inspection of this first order equation immediately reveals that ρ_{D} must be periodic in t.

Writing z = tan φ, we find that when φ_{t} = 0 , (6.3) becomes

m = ω_{n−1}z^{n−1} [\sqrt{1 + z^{2}} − \frac{H}{n} z] . (6.4)

The function m(z) in (6.4) is zero at z = 0 (corresponding to a chain of spheres), positive for small z > 0, and has a second zero at Hz = n(1+z^{2})^{1/2}. It is straightforward to check that m(z) has one positive critical point, so m is monotone increasing until that value and monotone decreasing afterwords. Thus, except at the maximum, each non-negative value is attained exactly twice. So each embedded Delaunay surface is then uniquely determined (up to translation) by its minimum (or maximum) ρ-value, since ρ_{D} solves the second order initial value problem ρ = ρ_{min}, (or ρ = ρ_{max}), ρ_{t} = 0 . But (6.4) determines these two values of ρ (and the corresponding z-values) uniquely in terms of the mass m. Since ρ_{max} = ρ_{min} on the cylinder, that surface maximizes the mass. □

Lemma 6.4 (Earp-Rosenberg [6]) Given Configuration 6.1, the mean curvature of Σ is at least as big as the mean curvature H_{1} > n of the cylinder containing it. Furthermore, there is a sequence {p_{i}} ⊂ Σ with t(p_{i}) → ∞ and with ρ(p_{i}) ≥ ρ_{0}, where ρ_{0} is the geodesic radius of a cylinder having mean curvature H.

Proof. (For the reader’s convenience, we sketch the Earp-Rosenberg argument.) Fix a T so that p ∈ ∂Σ implies t(p) < T. Let the mean curvature of C(γ, R) be H_{1}. Deform the cylinder along the one-parameter family of Delaunay surfaces with constant mean curvature H_{1} that have maximum “bulge” at t = T. The periods of these Delaunay surfaces are bounded from above by the diameter of the corresponding sphere, and since the family converges to a chain of spheres, the “necks” pinch to zero radius. The bulge radius increases, so the bulge region of these surfaces stays “outside” of Σ. But since Σ extends beyond the maximum period of the Delaunay surfaces, there must be a Delaunay surface in the family that first makes one-sided tangential contact with Σ at a point within one period of t = T. Hence the mean curvature H of Σ is at least that of C(γ, R).

To prove the second part of the lemma, assume the sequence does not exist. Repeat the above argument, using the cylinder with mean curvature H and with T large enough so that p ∈ Σ, t(p) ≥ T implies that p is inside the cylinder. At the first point of one-sided tangential contact the strong maximum principle would be violated. □

(6.5) Alexandrov Reflection. Recall that in the ball model of ^{n+1} (Section 2.2) the totally geodesic hypersurfaces are Euclidean n-spheres meeting S_{∞} = ∂B^{n+1} at right angles. “Reflection,” corresponding to inversion through these spheres, is an isometry of ^{n+1}. Alexandrov reflection applied to a constant mean curvature hypersurface Σ = ∂Ω is the process of inverting through a continuous family of these spheres {S(q)}, and studying the possibility of one-sided interior reflection contact between the part of Σ exterior to the sphere and its reflection from inside the sphere. The strong maximum principle implies reflection symmetry if they contact.

(6.6) New Configuration. We adapt Configuration 6.1 to the ball model. Let (1, 0, …, 0) be the vertex v at infinity of our cylinder C(γ, R), which we may assume is axially symmetric about the x_{1}-axis. Given t as in Configuration 6.1, make a hyperbolic translation along the x_{1}-axis so that D_{t} crosses it at the origin. We will use hyperbolic translations and a particular set of reflection families to study the asymptotic symmetry of D_{t} ∩ Σ. Denote the translated (Σ, ∂Σ) by (Σ_{t}, ∂Σ_{t}). Note that as t → ∞, ∂Σ_{t} moves uniformly towards the point (−1, 0, …, 0). In fact, the translation moves D_{0} = {x_{1} = 0} ∩ C(γ, R) to the Euclidean reflection of D_{t} through {x_{1} = 0}. Hence ∂Σ_{t} is within Euclidean distance O(e^{−t}) of (−1, 0, …, 0).

Lemma 6.5 (Weak gradient estimate) For every δ > 0 there is a T = T (δ) and an ε = ε(δ) so that for any p ∈ Σ satisfying t(p) > T and dist(p, γ) = ρ(p) > δ, we have the estimate ν(p) · ∂_{ρ} > ε. (Here ν(p) is the exterior normal to Σ at p and ∂_{ρ} is the unit vector field in D_{t} which is the derivative of geodesic distance ρ from γ(t).)

Proof. Consider the translations Σ_{t} in Configuration 6.6. There are various possibilities for Alexandrov reflection from “above” (say, after Euclidean rotation about the x_{1}-axis, from the positive x_{2} direction). The simplest one begins with a small Euclidean sphere S(q) orthogonal to S_{∞}, centered at q on the x_{2}-axis just above (0, 1, 0, …, 0), with radius chosen to satisfy the orthogonality condition. As q approaches infinity along the x_{2}-axis, the sphere S(q) approaches the {x_{2} = 0} plane.

Because ∂Σ_{t} → (−1, 0, …, 0) as t → ∞, one can approach arbitrarily close to this limiting plane as t → ∞ without attaining boundary reflection contact for Σ_{t}, nor interior contact, by the strong maximum principle.

If one only wants the reflection spheres to reach (0, δ, 0, … 0) before first reflection contact, nearby reflection families are permitted: Consider families of spheres whose centers are Euclidean distance r from the origin, 1 < r < ∞, and which lie on rays making small angle with the x_{2}-axis. The geometry of these spheres as r → ∞ implies that for t sufficiently large and angle sufficiently small (O(δ)), first reflection contact only occurs after the family passes through (0, δ, 0, …, 0). But by the definition of first reflection contact, the angle between the reflection sphere’s interior normal and the exterior normal ν to Σ_{t} is strictly less than π/2 at any point on the intersection of such a sphere and Σ_{t}. We see that the set of all such reflection families for a point p = (0, ρ(p), 0, …, 0) ∈ Σ_{t} with ρ(p) ≥ δ, includes a whole cone of directions about the positive x_{2} direction, and this cone has aperture greater than or equal to O(δ). Hence we conclude that ν(p) · ∂_{ρ} > O(δ), for t sufficiently large. This proves Lemma 6.5, with ε ≥ O(δ) as δ → 0. □

Lemma 6.6 (Weak asymptotic axial symmetry) For every δ > 0 there is a T = T (δ) so that whenever t > T and D_{t} ∩ Σ contains a point of distance ρ > δ from γ(t), then D_{t} ∩ Σ is a radial graph with respect to ρ over S^{n−1}(γ(t)), the unit sphere in D_{t} with center γ(t). Furthermore the function ρ(θ) parametrizing D_{t} ∩ Σ above S^{n−1}(γ(t)) is smooth and satisfies |ρ_{θ}| < Ce^{−t} for some C = C(δ, T).

Proof. We study the translates Σ_{t}, and reflection from above as in the beginning of Lemma 6.5 (i.e., the admissible reflection spheres are now those symmetric with respect to the {x_{1} = 0} plane and centered on a ray perpendicular to the x_{1}-axis). Because ∂Σ_{t} is within O(e^{−t}) of (−1, 0, …, 0), such a reflection family of spheres can be followed until the reflecting spheres approach to distance O(e^{−t}) of the origin, before any reflection contact occurs.

Let δ > 0 be given. Consider t for which the maximum ρ value on Σ∩D_{t} is at least δ. Pick any p ∈ Σ ∩ D_{t} with ρ(p) = \overline{ρ} > δ/2. Then after translation, the corresponding point p_{t} ∈ D_{0} also satisfies ρ(p_{t}) = \overline{ρ}. The reflecting spheres considered above map D_{0} to itself. Write ε (= O(e^{−t})) for a distance to which any such sphere may approach the origin before reflection contact, and assume t has been chosen so that ε ≪ δ.

Now study the images of p_{t} in D_{0}, under all these reflection spheres, until the reflection sphere of distance ε from the origin is attained. They all lie in the interior of Ω ∩ D_{t}, since there has been no reflection contact. Call the image set U_{p} and note that its boundary ∂U_{p} is the set of reflection images of p by the final (distance = ε) sphere in each family. We will prove:

Claim 6.7 U_{p} is a teardrop-shaped region, symmetric about the axis \overline{Op}_{t}. Furthermore, ∂U_{p} lies within O(ε) of the sphere of radius \overline{ρ}, the “vertex” of the teardrop is at p_{t}, and its tangent cone is almost flat, with aperture half-angle within O(ε) of π/2.

Assuming the claim, Lemma 6.6 follows easily: Apply the claim to the point p ∈ Σ ∩ D_{t} that has maximum ρ = ρ(p) > δ. It follows that all of Σ ∩ D_{t} lies within an O(ε) distance of ρ · S^{n−1}(γ(t)). From the weak gradient estimate in Lemma 6.5, we conclude that since ν · ∂_{ρ} is positive (for t large enough), Σ ∩ D_{t} is a transverse intersection and is given as the graph of a smooth function ρ = ρ(θ). Applying the claim to each q ∈ Σ ∩ D_{t}, the statement about aperture angle implies that |ρ_{θ}| is at most O(e^{−t}).

So it remains to verify the claim. By a rotation about the x_{1}-axis, we assume p_{t} = (0, \overline{ρ}, 0, …, 0). Because our reflection spheres are symmetric about the x_{1}-axis and preserve D_{0}, it suffices to consider the two-dimensional picture: families of reflection circles in the x_{2}x_{3}-plane. For simplicity write y = x_{2}, z = x_{3}. Then p_{t} corresponds to (0, \overline{ρ}) and all reflection families of circles in the yz-plane continue up to the reflection circle of distance ε from the origin.

It is possible to show (and geometrically clear, see Figure 4), that if one replaces the reflecting circles with reflecting lines (perpendicular to the reflection family ray and tangent to the given circle at the point on the ray nearest the origin), the corresponding image set V_{p} of p_{t} is contained in U_{p}. Hence it suffices to establish the claim for V_{p}. Let θ be the angle between the y-axis and the lines in an admissible family able to reach p_{t}. (So sin^{−1}(\overline{ρ}) < θ ≤ π/2). The image of p_{t} under this family will be a line segment extending from p_{t} to the reflection p̃_{t} of p_{t} through the final line in the family. One can calculate the coordinates of p̃_{t} explicitly using elementary geometry. Assuming the lines have positive slope we do so:

(6.5)  p̃_{t} = (0, \frac{ε}{sin θ} ) + (\overline{ρ} − \frac{ε}{sin θ} ) (sin 2θ, cos 2θ).

Taking norms gives

|p̃_{t}|^{2} = \overline{ρ}^{2} + 4ε^{2} − 4ε\overline{ρ} sin θ. (6.6)

Clearly |p̃_{t}| is within O(ε) of \overline{ρ}, and this establishes part of the claim.

γ ∩ D_{t} P_{t}  V_{p}  U_{p} ε  D_{t} Figure 4.

To estimate the vertex-angle of the teardrop, compute (p̃_{t})_{θ} from (6.5), and evaluate at the limiting θ = sin^{−1}(ε/\overline{ρ}):

(p̃_{t})_{θ} = (2(\overline{ρ} − \frac{ε^{2}}{\overline{ρ}}), 2ε\sqrt{1 − \frac{ε^{2}}{\overline{ρ}^{2}}} − \frac{ε}{\overline{ρ}^{2}} ). (6.7)

Since this tangent vector is within O(ε) of being horizontal, the angle in question is within O(ε) of π/2. □

Lemma 6.8 There exist T, C, and ρ_{min} so that {p ∈ Σ | t(p) ≥ T } is the cylindrical graph of a function ρ(t, θ) with ρ uniformly bounded. Then R > ρ > ρ_{min} > 0, with uniformly bounded gradient |(ρ_{t}, ρ_{θ})| < C, and with |ρ_{θ}| < Ce^{−t}.

Proof. Given δ > 0, choose T, ε so that Lemmas 6.5 and 6.6 hold. Use Lemma 6.5 to pick p with ρ(p) ≥ ρ_{0}, and |ρ_{θ}| ≪ δ for t ≥ t(p) and ρ > δ. Then as t increases from t(p), the cross-sections D_{t} ∩ Σ remain nearly spherical, provided their maximum radius ρ exceeds δ. But as the radius approaches δ, the flux μ = ⟨μ([∂Σ]), ∂_{t}⟩ is of order δ^{n−1}. If δ is chosen sufficiently small (depending on the positive value of μ in Lemma 6.2), this contradicts the homology-invariance of flux. So the radius never approaches δ, and ρ is uniformly bounded above and below by positive numbers. The estimates for |Dρ| now follow from Lemmas 6.5 and 6.6. □

Theorem 6.9 (Exponential convergence to Delaunay surface) Given Σ as in Configuration 6.1, there exists an axially-symmetric Delaunay surface D about γ to which Σ converges exponentially ^{1}. That is, for t sufficiently large, Σ is the graph in geodesic cylindrical coordinates of a function ρ, there is a Delaunay D surface that is the graph of a function ρ_{D}, and |ρ − ρ_{D}| < Ce^{−t} as t → ∞. (Convergence holds in any C^{k} norm.)

Proof. From Lemma 6.8 we know that for t sufficiently large, Σ is a graph in geodesic cylindrical coordinates of a function ρ(t, θ) satisfying R > ρ > ρ_{min} > 0, |(ρ_{t}, ρ_{θ})| < C and |ρ_{θ}| < Ce^{−2t}.

Since Σ has constant mean curvature, ρ satisfies a quasilinear elliptic equation in (t, θ). The equation is H = g^{ij}h_{ij} = M(ρ), where [g^{ij}] and [h_{ij}] are the inverse of the first and second fundamental forms, respectively, in the (t, θ) coordinates. The uniformity of ρ (0 ≤ ρ_{0} ≤ ρ ≤ R) and the bound on D_{ρ} imply that in the (t, θ) coordinate system the pull-back metric [g_{ij}] is uniformly positive, and hence the operator M(ρ) = g^{ij}h_{ij} is uniformly elliptic. So ρ is C^{∞} , with uniform C^{k} estimates for sufficiently large t.

Now by differentiating M(ρ) = H in any θ direction, and interchanging derivatives, we conclude that ρ_{θ} also satisfies a uniformly elliptic equation. Interior elliptic regularity and the estimate |ρ_{θ}| < Ce^{−t} then imply that |D(ρ_{θ})| (and |(Dρ)_{θ}|) are also bounded by C̃e^{−t} for sufficiently large t. That is, all second derivatives involving at least one θ differentiation are uniformly bounded by C̃e^{−t}.

1This Delaunay surface D is unique if we require its axis to be the world-line of [∂Σ], i.e. if D and Σ have the same momenta.

Because of this decay in θ derivatives of ρ, we may consider solutions of M(ρ) = H as perturbations of Delaunay ODE solutions. Define an “approximate bulge” of Σ to be a t value for which the area of the cap D_{t} ∩ Ω is maximized. At a bulge ρ_{t} cannot be of one sign, and since |ρ_{tθ}| is O(e^{−t}), |ρ_{t}(t, θ)| must be O(e^{−t}) as well. It follows from the flux formula, and the fact that the variation of ρ and |ρ_{θ}| are also O(e^{−t}), that ρ must be within O(e^{−t}) of the maximum ρ value on a Delaunay surface D having momentum equal to that of ∂Σ. By continuous dependence of parameters for ODE’s, it now follows that the time interval until the next approximate bulge (assuming D is not the cylinder) differs by O(e^{−t}) from the period of the Delaunay solution ρ = ρ_{D}(t). (Compare the PDE and ODE solutions, where the ODE has initial values ρ = max(ρ_{D}) , ρ_{t} = 0.) If D is the cylinder, our estimates immediately show |ρ − ρ_{0}| is O(e^{−t}). In any case, the total variation of the difference between PDE and ODE “periods” from t = t_{0} to t = ∞ is now seen to be O(e^{−t_{0}}). But then, ρ(t, θ) converges to a translate ρ_{D}(t + T), and the total variation of period differences is O(e^{−t_{0}}), which also bounds the rate at which ρ(t, θ) converges to ρ_{D}(t + T). Now elliptic regularity, applied to the difference of the two solutions, yields C^{k} convergence. □

Theorem 1.3 in the introduction follows immediately.

Remark 6.10 One may want to consider the asymptotic behavior of a properly embedded surface Σ ⊂ ^{n+1} with constant mean curvature H ≤ H_{horosphere}. In this case the ends of Σ are never cylindrically bounded. In fact, examples show that the asymptotic boundary behavior is essentially arbitrary when H < H_{horosphere}. However, the case H = H_{horosphere} in ^{3} is similar to the case of minimal surfaces in ^{3} [5]. Perhaps in this case the asymptotic behavior of annular ends can be classified (see [8] for partial results on this problem in the case of minimal surfaces in ^{3}).

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