Introductionw3.impa.br/~rosen/MxR-Hopf.pdf · 2003. 12. 19. · A HOPF DIFFERENTIAL FOR CONSTANT...

A HOPF DIFFERENTIALFOR CONSTANT MEAN CURVATURE SURFACES

IN S2 × R AND H

2 × R

UWE ABRESCH AND HAROLD ROSENBERG

Dedicated to Hermann Karcher on the Occasion of his 65th Birthday

Abstract. A basic tool in the theory of constant mean curvature (cmc)surfaces Σ2 in space forms is the holomorphic quadratic differential dis-covered by H. Hopf. In this paper we generalize this differential toimmersed cmc surfaces Σ2 in the product spaces S2 × R and H2 × Rand prove a corresponding result about the geometry of immersed cmcspheres S2 in these target spaces.

Introduction

In 1955 H. Hopf [13] discovered that the complexification of the tracelesspart of the second fundamental form hΣ of an immersed surface Σ2 withconstant mean curvature H in euclidean 3–space is a holomorphic quadraticdifferential Q on Σ2. This observation has been the key to his well-knowntheorem that any immersed constant mean curvature (cmc) sphere S2 # R

3

is in fact a standard distance sphere with radius 1H .

Hopf’s result has been extended to immersed cmc spheres S2 in the sphereS

3 or in hyperbolic space H3. In other words, the result has been extended toimmersed cmc spheres S2 #M3

κ in space forms with arbitrary curvature κ.Futhermore, W.-Y. Hsiang has conjectured [14, Remark (i) on p. 51] thatimmersed cmc spheres in the product space H2 × R should be embeddedrotationally-invariant vertical bi-graphs.

The holomorphic quadratic differential Q itself has other important appli-cations; it guarantees the existence of conformal curvature line coordinateson cmc tori in space forms, and thus it has played a significant role in thediscovery of the Wente tori [25] and in the subsequent development of ageneral theory of cmc tori in R3 (see [1, 21, 6, 12, 7]). For cmc surfaces Σ2

g

of genus g > 1 the holomorphic quadratic differential still provides a linkbetween the genus of the surface and the number and index of its umbilics,a link that has been very useful in investigating the geometric properties ofsuch cmc surfaces (see [8, 11, 22]).

2000 Mathematics Subject Classification. Primary 53A10; Secondary 53C42.Key words and phrases. Surfaces with constant mean curvature, Codazzi equations,

holomorphic quadratic differentials, integrable distributions, isoperimetric domains.The first named author likes to thank the Institut de Mathematiques de Jussieu

and the CNRS for their hospitality and support during a visit to Paris in fall 2003. Thebasic ideas for the results in this paper have been developed during that visit.

1

2 UWE ABRESCH AND HAROLD ROSENBERG

1. Main Results

Our main goal is to introduce a generalized quadratic differential Q forimmersed surfaces Σ2 in the product spaces S2×R and H2×R and to extendHopf’s result to cmc spheres in these target spaces; we shall prove that suchimmersed spheres are surfaces of revolution. In order to treat the two casessimultaneously, we write the target space as M2

κ × R where M2κ stands for

the complete simply-connected surface with constant curvature κ.A lot of information about the product structure of these target spaces

is encoded in the properties of the height function ξ : M2κ × R → R that

is induced by the standard coordinate function on the real axis. We notein particular that ξ has a parallel gradient field of norm 1, and that thefibers of ξ are the leaves M2

κ × {ξ0}. In addition to the function ξ weonly need the second fundamental form hΣ(X,Y ) = 〈X,AY 〉, the meancurvature H = 1

2 trA, and the induced (almost) complex structure J inorder to define the quadratic differential Q for immersed surfaces Σ2 inproduct spaces M2

κ × R . We set

q(X,Y ) := 2H · hΣ(X,Y )− κ · dξ(X) dξ(Y )(1)

and

Q(X,Y ) := 12 ·(q(X,Y )− q(JX, JY )

)−1

2 i ·(q(JX, Y ) + q(X, JY )

)(2)

Clearly, Q(JX, Y ) = Q(X, JY ) = i · Q(X,Y ). In dimension 2, tracelesssymmetric endomorphisms anticommute with the almost complex structureJ , whereas multiples of the identity commute with J . Thus the quadraticdifferential Q defined in (2) depends only on the traceless part q0 of thesymmetric bilinear form q introduced in (1) and, conversely, it is the tracelesspart q0 that can be recovered as the real part of Q.Theorem 1. Let Σ2 # M2

κ × R be an immersed cmc surface in a productspace. Then its quadratic differential Q as introduced in (2) is holomorphicw.r.t. the induced complex structure J on the surface Σ2.

The proof of this theorem is a straightforward computation based on theCodazzi equations for surfaces Σ2 # M2

κ × R , which differ from the morefamiliar Codazzi equations for surfaces in space forms M3

κ by an additionalcurvature term. The details will be given in Section 3.

Following H. Hopf, we want to determine the geometry of immersed cmcspheres in product spaces M2

κ × R using Theorem 1. In these target spacesone has the embedded cmc spheres S2

H studied by W.-Y. Hsiang [14] andR. Pedrosa [20] that are invariant under an isometric SO(2)–action, rota-tion about a vertical geodesic. From now on we shall refer to them as theembedded rotationally invariant cmc spheres S2

H ⊂M2κ × R .

Their geometry will be described in Proposition 2.5i in the next section.In contrast to cmc spheres in space forms, they are not totally-umbilicalunless κ = 0 or κ > 0 and H = 0. The traceless part q0 of the symmetricbilinear form introduced in (1), however, vanishes on all of them. For us, thisobservation has been the major clue when guessing the proper expressionfor the quadratic differential Q for cmc surfaces in such product spaces.

A HOPF DIFFERENTIAL FOR CMC SURFACES IN S2 × R AND H

2 × R 3

Theorem 2. Any immersed cmc sphere S2 # M2κ × R in a product space

is actually one of the embedded rotationally invariant cmc spheres S2H ⊂

M2κ × R.This theorem in particular establishes Hsiang’s conjecture from [14, Re-

mark (i) on p. 51] about immersed cmc spheres in H2 × R.It is a well-known fact that any holomorphic quadratic differential on the

2–sphere vanishes identically. Thus Theorem 1 implies that the quadraticdifferential Q of an immersed cmc sphere S2 # M2

κ × R vanishes. HenceTheorem 2 follows directly from the following classification result:Theorem 3. There are 4 distinct classes of complete, possibly immersed,cmc surfaces Σ2 #M2

κ × R with vanishing quadratic differential Q.Three of these classes are comprised of embedded rotationally invariant

surfaces; they are the cmc spheres S2H ⊂ M2

κ × R of Hsiang and Pedrosa,their non-compact cousins D2

H , and the surfaces C2H of catenoidal type. The

fourth class is comprised of certain orbits P 2H of 2–dimensional solvable

groups of isometries of M2κ × R.

The geometry of the rotationally invariant embedded cmc surfaces of typeS2H , D2

H , or C2H will be described in detail in Propositions 2.5 and 2.9.

The fact that they all have Q ≡ 0 will be verified in Section 2, too. Theremaining family P 2

H , where the parabolic symmetries are prevalent, willonly be introduced in Proposition 4.9 in Section 4. Explicit formulas for themeridians of all these surfaces and some pictures thereof will be provided inthe appendix.Remark 4. If κ ≥ 0, then only the spheres S2

H ⊂M2κ ×R occur, whereas for

κ < 0 all 4 cases do actually occur.The fact that there are no other cmc surfaces with Q ≡ 0 will be estab-

lished in Section 4. A systematic approach to this classification problem isto study a suitable overdetermined ODE system for the unit normal field ofthe surfaces to be classified. In doing so, the surfaces P 2

H in fact show upquite naturally.

In a sense the preceding theorems amount to saying that, when studyingsurfaces Σ2 in product spaces M2

κ × R rather than in space forms, the roleof umbilics is taken over by the points where Σ2 approximates one of theembedded cmc surfaces S2

H , D2H , C2

H , or P 2H up to second order.

Theorem 2 provides substantial information even when studying the ge-ometry of embedded cmc spheres S2 ⊂ S2 × R . Moving planes argumentsalong the lines of A. D. Alexandrov [3] only prove that a closed embeddedcmc surface Σ2 ⊂ S2 × R is a vertical bi-graph. In contrast to target spaceswith non-positive sectional curvature, moving planes arguments do not im-ply that such a surface Σ2 ⊂ S2 × R is rotationally invariant. The latterclaim would only follow, if we assumed that Σ2 were contained in the prod-uct of a hemisphere with the real axis. Such an assumption, however, is byfar too strong. It does not even hold for the embedded rotationally invariantcmc spheres S2

H ⊂ M2κ × R if 0 < 4H2 < κ. This particular aspect of their

geometry is explained in more detail in Remark 2.8 in Subsection 2.4.Furthermore — unlike in euclidean space R3 — there even exists a fam-

ily of embedded cmc tori in S2κ × R , most of them not being rotationally

invariant. They rather look approximately like “undoloids” around some


great circle in a leaf S2κ × {ξ0} ⊂ S

2κ × R and are thus not contained in

the product of a hemisphere with the real axis either. They have been con-structed by J. de Lira [17] by a bifurcation argument along the lines of thework done by R. Mazzeo and F. Pacard [18]. Presumably one could alsoobtain these surfaces using a singular perturbation argument similar to theone that N. Kapouleas’ work is based on [16].

Finally, we observe that the holomorphic quadratic differential Q of The-orem 1 is likely to open the doors for investigating the geometry of cmc toriin product spaces M2

κ × R , and — as in the case of space forms — theremight be interesting connections to the theory of integrable systems. Asomewhat more subtle question is whether all the above can be generalizedto cmc surfaces Σ2 in homogeneous 3–manifolds.

2. The rotationally invariant cmc spheres S2H ⊂M2

κ × R

In this section we shall describe the embedded rotationally invariant cmcspheres S2

H ⊂M2κ×R of Hsiang and Pedrosa. Our main goal is to compute

the symmetric bilinear form q introduced in (1) for these cmc spheres andthus verify that their quadratic differential Q vanishes. This goal will beaccomplished in Proposition 2.7.

The work of both authors, Hsiang and Pedrosa, has been somewhat moregeneral than what we need here. They have studied rotationally invariantcmc spheres SnH in product spaces including Hn × R [14] or in Sn × R [20],respectively. For our purposes we are only interested in properties of thesurfaces S2

H , and, specializing our summary accordingly, we thus work onproduct manifolds M3 := M2

κ × R that are equipped with an isometricSO(2)–action.

2.1. Isometric SO(2)–actions on the product spaces M2κ × R. Such

an action necessarily operates trivially on the second factor. In other words,the group SO(2) ⊂ Isom0(M2

κ ×R) can be considered as the isotropy groupconsisting of those rotations that preserve some fixed axis `0 := {x0}×R . Ifκ ≤ 0, the fixed point set of this SO(2)–action is precisely the axis `0 itself;however, if κ > 0, it encompasses the antipodal line ˆ

0 = {−x0}×R as well.In order to understand this action better, it is useful to pick a normal

geodesic γ : R→M2κ with γ(0) = x0. The product map γ×id : R2 →M2

κ×Ris a totally-geodesic isometric immersion. Its image intersects all SO(2)–orbits orthogonally. The existence of such a slice means that the groupaction under consideration is polar.

The subset Iκ×R ⊂ R2, where Iκ denotes the half axis (0,∞) if κ ≤ 0 andthe interval (0, π/√κ) if κ > 0, is a fundamental domain whose image underthe immersion γ×id intersects each principal SO(2)–orbit inM2

κ×R preciselyonce. Its closure Iκ × R represents the entire orbit space SO(2)\(M2

κ × R).The canonical projection p onto this orbit space lifts to a map p : M2

κ×R→Iκ×R such that the composition p ◦ (γ× id) is the identity on Iκ×R . Thislift is given by p(x) =

(r(x), ξ(x)

)where r(x) := dist(x, `0) and where ξ is

the height function introduced beforehand. The restriction of p to the unionof all principal orbits is clearly a Riemannian submersion.

A conceptually nice way of thinking about the slice im(γ× id) is to extendthe SO(2)–action to some larger subgroup of Isom(M2

κ × R). The natural


2 × R 5

candidate is the group O(2) consisting of all isometries of M2κ × R that fix

the axis `0. The cosets of SO(2) ⊂ O(2) are represented by the subgroupconsisting of the identity and of the reflection % at the slice. The extendedaction has the same orbits as the SO(2)–action that we began with. How-ever, the principal isotropy group of the extended action is Z2 = 〈%〉 ratherthan the trivial group.

The slice im(γ × id) can be recovered as the fixed point set of the re-flection %. Moreover, % preserves the height function ξ and thus also itsgradient. In particular, grad ξ is tangential to the slice everywhere.

2.2. Connected SO(2)–invariant surfaces Σ2 #M2κ×R. As explained

above, the given action extends to an isometric O(2)–action on M2κ×R with

precisely the same orbits. Hence any rotationally invariant cmc surfaceΣ2 # M2

κ × R is also invariant under the full O(2)–action. The additionalsymmetry tells a lot about the geometry of the surfaces in question:Proposition 2.1. Any rotationally invariant surface Σ2 # M2

κ × R inter-sects the slice im(γ × id) orthogonally in a set of regular curves c that aresimultaneously geodesics and curvature lines on Σ2.

If the surface Σ2 is connected, then so is its image under the projectionp from M2

κ ×R onto the orbit space. Therefore this image can be describedby a single regular curve c in Iκ × R that may start and/or end at theboundary and that can meet the boundary only orthogonally. The surfaceΣ2 # M2

κ × R can be recovered from c by letting SO(2) act on its imagec := (γ × id) ◦ c in M2

κ ×R . Moreover, any regular curve s 7→ c(s) ∈ Iκ×Rwith the proper behavior at the boundary of Iκ ×R is the generating curvefor some connected, smooth, rotationally invariant surface Σ2 #M2

κ × R .The unit normal field ν of the surface is best described in terms of its

angle function s 7→ θ(s) along the generating curve s 7→ c(s) = (r(s), ξ(s)),a function that is defined by the equation

ν|((γ×id)◦c)(s) = d(γ × id)|c(s) ·(cos θ(s), sin θ(s)

)We assume that the generating curve c is parametrized by arc length andthat its tangent vector is c′(s) = (− sin θ(s), cos θ(s)). Our convention is towork with “exterior” unit normal vectors and write the Weingarten equationwith a “+”–sign, i.e. to write A = D ν. In particular, the principal curvatureof Σ2 in the direction of the meridian c = (γ × id) ◦ c is given by ∂

∂s θ(s).Using the formula of Meusnier to determine the other principal curvature,

it is straightforward to compute the second fundamental form hΣ and thetensor field dξ2|Σ along the meridian c on Σ2 with respect to the orthonormalbasis consisting of the vectors c′(s) and Jc′(s) :

hΣ =(

∂∂s θ 00 cos(θ) · ctκ(r)

)and dξ2|Σ =

(cos2(θ) 0

0 0

)(3)

Here ctκ stands for the generalized cotangent function1; the value ctκ(r) isthe curvature of the circle of radius r > 0 in the surface M2

κ of constantcurvature κ.

1Recall that the generalized cotangent function ctκ is by definition the logarithmicderivative of the generalized sine function snκ which in turn is defined to be the solutionof the differential equation y′′ + κ y = 0 with initial data y(0) = 0 and y′(0) = 1.


2.3. Differential equations for SO(2)–invariant cmc surfaces Σ2.The mean curvature H of a rotationally invariant surface Σ2 # M2

κ × R isgiven by the identity 2H = trA = trg(hΣ). Thus the preceding consider-ations yield the following system of ordinary differential equations for thegenerating curve s 7→ c(s) = (r(s), ξ(s)):

∂∂s r = − sin(θ)∂∂s ξ = cos(θ)∂∂s θ = 2H − cos(θ) · ctκ(r)

(4)

The key to solving this ODE system is the observation that its r.h.s. dependsneither explicitly on s nor on ξ. The latter property is a direct consequenceof the fact that the euclidean factor in the product spaces M2

κ × R is 1–dimensional. It follows that the ODE system (4) has a first integral . Infact, W.-Y. Hsiang already found that

L := cos(θ) · snκ(r)− 2H ·∫ r

0snκ(t) dt

= cos(θ) · snκ(r)− 4H · snκ(12r)

2

(5)

is constant along any solution of (4).Remark 2.2. The value of L can be used to characterize the special solutionsof (4) that correspond to particularly simple cmc surfaces in M2

κ × R :(i) If H = 0, there are the totally-geodesic leaves M2

κ ×{ξ0}. These leavescorrespond to solutions with ξ = ξ0 and θ = ±1

2π, and thus they canbe characterized by the condition L = 0.

(ii) If H = 12 ctκ(r0) for some r0 ∈ Iκ, there is the cylinder of radius r0

around the axis `0. This surface corresponds to solutions with r = r0

and θ = 0, and thus it can be characterized by the condition L =ctκ(1

2r0)−1 = (2H +√

4H2 + κ)−1.If κ is positive, the vertical cylinders with radius π/(2

√κ ) have mean

curvature H ≡ 0, too. Moreover, there exist vertical undoloids with H ≡ 0that oscillate around these cylinders. These undoloids are discussed fromdifferent points of view in [20] and in [23, Sections 1 and 3]. Together, allthese examples give a good indication of how rich the class of rotationallyinvariant minimal surfaces in M2

κ × R is.Furthermore, there is a 2–parameter family of minimal annuli whose level

curves are geodesic circles which are not rotationally invariant. We expectthe quadratic differential Q to play a key role in studying these examples.Proposition 2.3. A connected, rotationally invariant, immersed cmc sur-face Σ2 # M2

κ × R is diffeomorphic to a torus or an annulus, unlessthe first integral L for its generating curve c vanishes, or, unless κ > 0and L = −4H/κ.

Proof. The orbit structure described in § 2.1 implies that any connected,SO(2)–invariant surface Σ2 # M2

κ × R is topologically the product of itsgenerating curve c in Iκ×R and a circle, provided that the generating curvestays in the interior of Iκ × R . If c has an end point on the boundary ofIκ × R , a disk must be glued to the corresponding boundary component ofthe remaining part of Σ2.


2 × R 7

0 rπ√κ

ξ

1√κ

−1√κ

H=0.033√κ

H=0.121√κ

H=0.375√κ

H=1.075√κ

Figure 1. The meridians of the spheres S2H for κ > 0.

Inspecting the expression in (5), it is evident that L must vanish if ctouches the component of the boundary of Iκ×R where r ≡ 0. If κ ≤ 0 thiscomponent is already the entire boundary of the fundamental domain Iκ×R .However, if κ > 0, the boundary of Iκ×R has one further component wherer ≡ π/√κ. Clearly, c can only reach the latter component if L = −4H/κ. �

Remark 2.4. The components of the boundary of Iκ × R correspond tothe components of the fixed point set of the SO(2)–action. In particular,interchanging the axes `0 and ˆ

0 induces a symmetry of (4) that interchangesthe cases L = 0 and L = −4H/κ as well.

2.4. SO(2)–invariant cmc surfaces Σ2 #M2κ×R with L = 0. In this

subsection we shall see that a rotationally-invariant cmc surface with L = 0does indeed intersect the axis `0. Thus, by the argument used in the proofof Proposition 2.3 the surface Σ2 must be either a sphere S2

H or a disk D2H .

Since snκ(12r) > 0 for any r ∈ Iκ, it is possible to rewrite the condition

that the first integral L introduced in (5) vanishes as follows:

cos(θ) · ctκ(12r) = 2H(6)

This equation has a number of important consequences.It shows in particular that cos(θ) converges to 0 as r → 0. This property

essentially reflects the regular singular nature of the ODE system in (4); itimplies that the generating curve c can only meet the boundary of Iκ × Rorthogonally. Hence the cmc surface Σ2 is automatically smooth at all thepoints where it intersects the axis `0.Proposition 2.5. Let Σ2 #M2

κ × R be a connected, SO(2)–invariant cmcsurfaces with mean curvature H 6= 0 that intersects the axis `0. Then either

(i) 4H2 + κ > 0, and Σ2 is an embedded sphere S2H ⊂M2

κ × R , or(ii) 4H2 + κ ≤ 0, and Σ2 is a convex rotationally invariant graph D2

H

over the horizontal leaves M2κ × {ξ0}, which is asymptotically conical

whenever the inequality for 4H2+κ is strict. There are two possibilitiesfor the range of the normal angle θ; the image of sin(θ) is one ofthe two half open intervals, [−1 ,− sin θ0) or (sin θ0 , 1], where θ0 :=arcsin(

√1 + 4H2/κ ).


0 r3√−κ

ξ

2√−κ

H=1.15√−κ

H=0.87√−κ

H=0.66√−κ

H=0.5√−κ

H=0.33√−κ

H=0.12√−κ

S 2H

D 2H

Figure 2. The meridians of the spheres S2H and the

disk-like surfaces D2H for κ < 0.

The surfaces described in this proposition are the embedded rotationallyinvariant cmc spheres S2

H ⊂M2κ × R of Hsiang and Pedrosa and their non-

compact cousins D2H . In the limit H → 0, these cmc surfaces converge on

compact subsets of M2κ × R towards the SO(2)–invariant minimal surfaces

described in Remark 2.2i. Thus it is sometimes convenient to denote theleaves M2

κ × {ξ0} as S20 or D2

0, respectively.

Proof. The idea is to evaluate the condition L = 0 at all points of thegenerating curve s 7→ c(s) = (r(s), ξ(s)) that lie in the interior of Iκ × R .Using the monotonicity properties of the functions r 7→ ctκ(1

2r) defined onthe open intervals Iκ, it is easy to determine their range, too. Thus onefinds that the pair of inequalities H cos(θ) > 0 and 4H2 + κ cos2(θ) > 0 area necessary and sufficient condition in order to solve equation (6) for r and,moreover, that this solution is always unique.

In particular, identity (6) can be used to eliminate the factor ctκ(r) fromthe last equation in (4); one finds that

∂∂s θ = 1

4H

(4H2 + κ cos2(θ)

)(7)

The r.h.s. of this differential equation is uniformly bounded, and thus its so-lutions are defined on the entire real axis. As explained above, H cos θ(s) >0 for all points c(s) on the generating curve that lie in the interior of Iκ×R.By equation (6), the curve c(s) approaches the boundary component of Iκ×Rcorresponding to the axis `0 whenever cos θ(s) converges to 0, and the pointson c that lie in the interior of Iκ × R correspond to a maximal interval(s1, s2) ⊂ R such that H cos θ(s) > 0 for all s ∈ (s1, s2).

The discussion gets much more intuitive when observing that the anglefunction θ can be used as a regular parameter on c. In order to see this,we recall that by our analysis of equation (6) we have 4H2 + κ cos2(θ) > 0.Thus by equation (7) the principal curvature ∂

∂s θ has always the same signas the mean curvature H. In particular, θ is a strictly monotonic functionof s. The same holds for the function sin(θ). It remains to analyze whether


2 × R 9

the image of (s1, s2) under sin(θ) is the entire interval (−1, 1) or just somesubinterval thereof.

If 4H2 +κ > 0, equation (7) implies that ∂∂s θ is bounded away from zero.

Hence the range of sin(θ) is the entire interval [−1, 1], and the generatingcurve c is an embedded arc that begins and ends at the boundary componentof Iκ × R corresponding to the axis `0. Thus by the argument in the proofof Proposition 2.3 the surface must be a sphere.

If 4H2 + κ ≤ 0 , the condition 4H2 + κ cos2(θ) > 0 implies that sin(θ)does not vanish anywhere along c. In fact, sin(θ) /∈ [− sin(θ0), sin(θ0)] whereθ0 := arcsin(

√1 + 4H2/κ ), and the function θ(s) is asymptotic to one of

the stationary solutions θ0 or −θ0 of equation (7). This explains the claimabout the range of the normal angle θ, and again, reasoning as in the proofof Proposition 2.3, the surface must be homeomorphic to a disk.

Moreover, if 4H2 + κ < 0, the asymptotic normal angle θ0 is differentfrom 0. In other words, sin θ(s) is bounded away from 0, and thus theradius function r is a regular parameter for the generating curve c as well;the range of this parameter is the entire half axis Iκ = [0,∞). Finally,equation (7) implies that

∂∂s

(θ(s)± θ0

)= κ

4H

(sin θ0 − sin θ(s)

)·(sin θ0 + sin θ(s)

)Hence θ(s) converges exponentially to θ0 or −θ0, respecticely. Thus thegenerating curve c has real asymptotes rather than merely some asymptoticslope and the generated surface is asymptotically conical as claimed.

On the other hand, if 4H2 + κ = 0, we find that θ0 = 0, and θ(s) decaysonly like O(1

s ). In particular, the integral∫∞s sin θ(σ) dσ does not converge;

hence in this case there do not exist any asymptotes. Yet, the radius functionr is still a regular parameter along c, and its range is still the entire halfaxis Iκ = [0,∞). �

Remark 2.6. W.-Y. Hsiang has already computed both, the area and theenclosed volume of the embedded cmc spheres S2

H ⊂ H2×R. In [20] Pedrosahas extended these computations to embedded cmc spheres S2

H ⊂ S2×R andhas then used this information to determine candidates for the isoperimetricprofiles of the product spaces M2

κ × R .

Proposition 2.7. Let Σ2 #M2κ ×R be a connected, rotationally invariant

cmc surface that intersects the axis `0. Then the tensor field q introducedin (1) is traceless, and thus the quadratic differential Q of Σ2 vanishesidentically. More precisely,

2H · hΣ − κ · dξ2|Σ =(2H2 − 1

2κ cos2(θ))· g(8)

As pointed out in the introduction, it is this simple proposition that hasbeen the key to finding the proper expression for the quadratic differentialof cmc surfaces in the product spaces M2

κ × R .

Proof. If H = 0, we consider some point c(s) on the generating curve wherethe surface intersects the axis `0. At such a point snκ(r(s)) = 0, and thusL = 0. It follows from (6) that cos θ(s) ≡ 0, and therefore Σ2 must be oneof the totally-geodesic leaves M2

κ×{ξ0} described in Remark 2.2i. For thesesurfaces equation (8) holds, as both its sides evidently vanish identically.


If H 6= 0, we may simplify the expression for the second fundamentalform hΣ from (3) using equations (4) and (7):

hΣ =(H + κ

4H · cos2(θ) 00 H − κ

4H · cos2(θ)

)(9)

Combining this formula with the expression for dξ2|Σ from (3), we againarrive at equation (8). �

Remark 2.8. The rotationally invariant cmc spheres S2H ⊂M2

κ×R of Pedrosaare not convex if 0 < 4H2 < κ. In fact, the principal curvatures computedin (9) have opposite signs at all points on S2

H ⊂ M2κ × R that are closer to

the antipodal axis ˆ0 than to `0 itself.

2.5. SO(2)–invariant cmc surfaces Σ2 #M2κ×R of catenoidal type.

In this subsection we are going to describe another family of rotationallyinvariant cmc surfaces with vanishing quadratic differential Q. For thispurpose we study the solutions of (4) with first integral L = −4H/κ.

In case κ > 0, the surfaces corresponding to solutions with L = −4H/κare precisely the surfaces that intersect the antipodal axis ˆ

0 rather than`0. As explained in Remark 2.4, they are congruent to the spheres thatcorrespond to the solutions with L = 0 and that have been studied byPedrosa.

In case κ < 0, however, the surfaces obtained from the solutions of theODE system (4) with L = −4H/κ are obviously not congruent to the cmcspheres of Hsiang, anymore. Yet, it is conceivable that they still have Q ≡ 0.We shall establish this property in Proposition 2.10. Using the expressionfor the first integral from (5), the condition L = −4H/κ reads

0 = κ cos(θ) · snκ(r) + 4H ·[1− κ snκ(1

2r)2]

In particular, we find that the radius r is bounded away from 0, and hencewe may rewrite this equation in a form similar to (6) :

−κ · cos(θ) = 2H · ctκ(12r)(6cat)

In fact, there is an extremely close relationship between the equations (6)and (6cat). For instance2, either one of them implies that

ctκ(r) = 12

(ctκ(1

2r)− κ ctκ(12r)−1)

= H cos−1(θ)− κ4H cos(θ)

Inserting this expression into the third equation in (4), we obtain of coursethe same differential equation for θ as in the proof of Proposition 2.5 :

∂∂s θ = 1

4H

(4H2 + κ cos2(θ)

)(7cat)

Thus it should not be surprising that most arguments in the proof of Propo-sition 2.5 can be carried over to this situation. As explained above, we areonly interested in studying the case κ < 0.

2Another consequence of either one of equations (6) and (6cat) is that the Pfaffian(4H2 + κ cos2(θ)

)· dr + 4H sin(θ) · dθ

vanishes along the corresponding solutions of (4). This in effect explains how to recover(6) and (6cat) and thus eventually the ODE system (4) from the differential equations (23)when doing the classification in Section 4.


2 × R 11

0 r3√−κ

ξ

1√−κ

−1√−κ

H=0.033√−κ

H=0.121√−κ

H=0.275√−κ

Figure 3. The meridians of the surfaces C2H of catenoidal type.

Proposition 2.9. Let κ < 0, and let Σ2 #M2κ ×R be a connected, SO(2)–

invariant cmc surface whose generating curve can be described as a solutionof (4) with first integral L = −4H/κ. Then the surface is an embeddedannulus C2

H with two asymptotically conical ends. It is generated by rotatinga strictly concave curve with asymptotic slopes dr

dξ = ∓ tan(θ0) where θ0 :=arccos(2H/√−κ). Moreover, the range of θ is the interval (−θ0, θ0).

The name C2H has been picked for this family of surfaces, since their

shapes resemble the shape of a catenoid or rather the shapes of the analogoussurfaces in hyperbolic 3–space. We shall refer to them as the rotationallyinvariant cmc surfaces of catenoidal type. When H approaches zero, theyconverge to the double covering of some leaf M2

κ×{ξ0} < 0 with a singularityat the origin. This collapse is indicated in Figure 3; it is pretty similar tothe collapse of the minimal catenoids described in [23, Section 6] or [19].

Proof. Since r ∈ Iκ by construction and since κ < 0 by hypothesis, equa-tion (6cat) implies that H cos(θ) > 0 and 4H2 + κ cos2(θ) < 0. Hence themaximal existence interval for the solutions of the differential equation (7cat)is the entire real axis. The range of the function s 7→ θ(s) is as claimed, and— because of the first two equations in (4) — all the other assertions arestraightforward consequences hereof. �

Proposition 2.10. Let κ < 0, and let Σ2 #M2κ×R be a connected, SO(2)–

invariant cmc surface whose generating curve can be described as a solutionof (4) with first integral L = −4H/κ. Then the tensor field q introducedin (1) is traceless, and thus the quadratic differential Q of Σ2 vanishesidentically. More precisely,

2H · hΣ − κ · dξ2|Σ =(2H2 − 1

2κ cos2(θ))· g

Proof. The proof of Proposition 2.7 only uses equations (3), (4), and (7).The first two of these sets of formulas are clearly valid in the present contextas well. Moreover, the differential equation for θ obtained in (7cat) happensto coincide with (7), as has been observed beforehand. Thus the proof ofProposition 2.7 carries over verbatim. �


3. ∂–operators, quadratic differentials, and Codazzi equations

The purpose of this section is to prove Theorem 1. In other words, wewant to compute the ∂–derivative of the quadratic differential Q defined informula (2) in Section 1 and verify that ∂ Q indeed vanishes identically. Thekey ingredients for this computation are, firstly, a formula that expressesthe ∂–operator in terms of covariant derivatives and, secondly, the Codazziequations for surfaces Σ2 in the product spaces M2

κ×R. These are somewhatmore complicated than the Codazzi equations for surfaces in space forms.Yet, all this is still essentially standard material, and so we shall summarizethe necessary details in Lemmas 3.1 and 3.4 as we go along.

3.1. Geometry and Complex Analysis on Riemann surfaces. In thissubsection we recall the basic facts about the complex analytic structure onthe surface (Σ2, g). In particular, we explain how the ∂–operator that comeswith this complex structure is related to covariant differentiation. Passingto a 2–fold covering if necessary, we may assume that Σ2 is oriented. Thusthere exists a unique almost complex structure J ∈ C∞(EndTΣ2) that iscompatible with the Riemannian metric g and the given orientation.

A celebrated theorem first stated by C.-F. Gauß [10] guarantees the ex-istence of isothermal coordinates. In such a coordinate chart ψ : U ⊂ Σ2 →C = R

2 the almost complex structure J is given as multiplication by i, andthe metric g is of the form e2λ g0 for some C∞–function λ : U → R ; here wehave used g0 to denote the standard euclidean metric on R2. The transitionfunctions between such charts are clearly orientation-preserving conformalmaps and are thus holomorphic. In other words, the existence of isothermalcoordinates turns the surface (Σ2, J) into a complex 1–dimensional manifoldΣ. Its complex tangent bundle TΣ, its cotangent bundle K := T ∗Σ, and allits tensor powers K⊗m are thus holomorphic line bundles. In other words,these are complex vector bundles that come with a natural ∂–operator.

By construction, the quadratic differential Q of an immersed cmc surfaceΣ2 # M2

κ × R is a section in a subbundle of the bundle of complex-valuedsymmetric bilinear 2–forms on the real tangent bundle of Σ2, a subbun-dle that can be canonically identified with K⊗2. As mentioned above, wewant to express the ∂–operator in terms of the Levi-Civita connection ∇associated to the Riemannian metric g. The basic link between the realand the complex picture is the fact that on Riemann surfaces ∇ J vanishes3

identically.Lemma 3.1. Let η ∈ C∞(K⊗m), m ∈ Z, be a section in the m-th powerof the canonical bundle on a Riemann surface (Σ2, g, J). Then ∂ η can becomputed in terms of Riemannian covariant differentiation as follows:(

∂ η)(X) = 1

2

(∇X η + i · ∇JX η

), ∀X ∈ C∞(TΣ2).(10)

The lemma is an immediate consequence of a couple of basic facts. Firstly,whenever a holomorphic vector bundle E over some complex manifold isequipped with a hermitian inner product g, there exists a unique metrical

3In higher dimensional case this fact does not hold anymore for an arbitrary complexmanifold with a hermitian inner product on the tangent bundle. In fact, the condition∇ J = 0 is one way of saying that (Mn, g, J) is Kahlerian.


2 × R 13

connection ∇ such that(∂ η)(X) = 1

2

(∇Xη + i ∇JXη

). This connection is

known as the hermitian connection of (E, g). On the tangent bundle of aKahler manifold the hermitian connection ∇ coincides with the Levi–Civitaconnection (see [24]). The extension of this identity to arbitrary tensorpowers is then done using the standard product rules for ∇ and ∇.

By definition the ∂–operator depends only on the almost complex structureJ , i.e. on the conformal structure and the orientation, whereas ∇ dependsalso on the choice of the metric g representing the conformal class. Clearly,this dependence must cancel when taking the particular linear combinationof covariant derivatives appearing on the r.h.s. of (10). This observation canbe used to verify the lemma directly.

Elementary Proof. By the very definition of the ∂–operator, formula (10)holds in isothermal coordinates provided that covariant differentiation is re-placed by euclidean differentiation d with respect to that particular isother-mal chart. In such a coordinate system the metric is by definition conformalto the euclidean metric g0, i.e. g = e2λ ·g0, and the Christoffel formula spe-cializes to

∇X Y − dX Y = dX λ · Y + dY λ ·X − g0(X,Y ) · gradg0λ

Because of the product rules for ∂ and ∇ it is sufficient to handle the casem = 1. For this purpose we consider a C–valued 1–form η of type (1, 0).This means that η(J Y ) = i · η(Y ), i.e., that η represents a section of thecomplex cotangent bundle K = T ∗Σ. Thus a straightforward computationbased on the Christoffel formula shows that:

2(∂η)(X;Y )− (∇X η · Y + i ∇JX η · Y )

= (dX η −∇X η) · Y + i (dJX η −∇JX η) · Y= η(∇X Y − dX Y ) + i η(∇JX Y − dJX Y )

= (dX λ+ i dJX λ) · η(Y ) + dY λ ·(η(X) + i η(JX)

)−(g0(X,Y ) + i g0(JX, Y )

)· η(gradg0

λ)

= g0(X, gradg0λ) · η(Y ) + g0(JX, gradg0

λ) · η(JY )

− g0(X,Y ) · η(gradg0λ)− g0(JX, Y ) · η(J gradg0

λ)

In order to explain the third equality sign, we simply observe that the co-efficient of the factor dY λ vanishes, as η is of type (1, 0). Picking an or-thonormal basis e1, e2 = Je1 such that Y is a multiple of e1, we find that

g0(X,Y ) · η(gradg0λ) = g0(X, e1) · g0(e1, gradg0

λ) · η(Y )

+ g0(X, e1) · g0(e2, gradg0λ) · η(JY )

g0(JX, Y ) · η(J gradg0λ) = g0(JX, e1) · g0(e1, gradg0

λ) · η(JY )

− g0(JX, e1) · g0(e2, gradg0λ) · η(Y )

These identities reveal that the last line in the preceding display does indeedvanish. �

3.2. Surface theory in the product spaces M2κ × R. There are two

aspects in regard to which the surface theory in the product spaces M2κ ×R

differs significantly from the surface theory in space forms. Firstly, there is


the global parallel vector field grad ξ, and secondly the Codazzi equations aremore involved. Except for some inevitable changes caused by these differ-ences, the computation of the ∂–derivative ∂ Q of the quadratic differentialintroduced in formulas (1) and (2) follows the same basic pattern as thecorresponding computation in the case of space forms. Clearly,

Q(Y1, Y2) = H ·(〈Y1, AY2〉 − 〈JY1, A JY2〉

)− iH ·

(〈JY1, AY2〉+ 〈Y1, A JY2〉

)− 1

2κ ·(dξ(Y1) dξ(Y2)− dξ(JY1) dξ(JY2)

)+ 1

2 iκ ·(dξ(JY1) dξ(Y2) + dξ(Y1) dξ(JY2)

)(11)

Remark 3.2. Since the quadratic differential is a field of type (2, 0), its ∂–derivative ∂ Q is a field of type (2, 1), i.e., ∂ Q is a section of the bundleK⊗2 ⊗ K. This means in particular that

∂ Q(X;JY1, Y2) = ∂ Q(X;Y1, JY2) = i · ∂ Q(X;Y1, Y2)

∂ Q(JX;Y1, Y2) = −i · ∂ Q(X;Y1, Y2)

In other words, ∂ Q is invariant under some T3–action, and we are going toarrange the terms in the subsequent computations accordingly.

The differential dξ of the height function is clearly a parallel 1–form withrespect to the Levi–Civita connection D of the target manifold M2

κ × R .However, its restriction is not parallel w.r.t. the induced Levi–Civita con-nection ∇ on the surface; one rather hasLemma 3.3. The covariant derivative of the restriction of dξ to any sur-face Σ2 # M2

κ × R can be expressed in terms of the gradient of the heightfunction ξ, the unit normal field ν of the surface, and its Weingarten mapA = D ν as follows4:

∇X(dξ) · Y = −〈ν, grad ξ〉 · 〈X,AY 〉 .

For intuition, we think of the term dξ(ν) = 〈ν, grad ξ〉 as the sine of someangle θ.

Proof. Extending Y to a tangential vector field, a straightforward compu-tation shows that

∇X(dξ) · Y = dX(dξ · Y

)− dξ

(∇X Y

)= DX(dξ) · Y + dξ

(DX Y −∇X Y

)= DX(dξ) · Y + dξ(ν) · 〈ν,DX Y 〉= DX(dξ) · Y − 〈ν, grad ξ〉 · 〈X,AY 〉

The lemma follows when taking into account that the vector field grad ξ isglobally parallel, i.e. that DX(grad ξ) and DX(dξ) vanish. �

Differentiating formula (11) using Lemmas 3.1 and 3.3, we find that

∂ Q(X;Y1, Y2)

= H · T1(X;Y1, Y2) + κ 〈ν, grad ξ〉 · T2(X;Y1, Y2)(12)

4The symmetric endomorphism A = D ν is sometimes also called the shape operator,and the associated bilinear form is the second fundamental form hΣ.


2 × R 15

where

T1(X;Y1, Y2) := 12 ·[〈Y1,∇X A · Y2〉 − 〈JY1,∇X A · JY2〉

+〈JY1,∇JX A · Y2〉+ 〈Y1,∇JX A · JY2〉]

− 12 i ·[〈JY1,∇X A · Y2〉+ 〈Y1,∇X A · JY2〉−〈Y1,∇JX A · Y2〉+ 〈JY1,∇JX A · JY2〉

],

T2(X;Y1, Y2) := 14 ·[〈Y1, AX〉 dξ(Y2) + dξ(Y1) 〈AX,Y2〉−〈JY1, AX〉 dξ(JY2)− dξ(JY1) 〈AX, JY2〉+〈JY1, A JX〉 dξ(Y2) + dξ(JY1) 〈AJX, Y2〉+〈Y1, A JX〉 dξ(JY2) + dξ(Y1) 〈AJX, JY2〉

]− 1

4 i ·[〈JY1, AX〉 dξ(Y2) + dξ(JY1) 〈AX,Y2〉

+〈Y1, AX〉 dξ(JY2) + dξ(Y1) 〈AX, JY2〉−〈Y1, A JX〉 dξ(Y2)− dξ(Y1) 〈AJX, Y2〉+〈JY1, A JX〉 dξ(JY2) + dξ(JY1) 〈AJX, JY2〉

]Clearly, T1 is just four times the (2, 1)–part of ∇A. It is the same expressionthat appears when proving that the standard Hopf differential for cmc sur-faces in space forms M3

κ is holomorphic. It will be evaluated in Lemma 3.5using the Codazzi equations.

The other term, however, is new. Since in dimension 2 the tracelesspart A0 of the Weingarten map anticommutes with the almost complexstructure J , it follows that the tensor T2 depends only on the mean curva-ture H but not on the traceless part A0 := A − H · id of the Weingartenmap A. Thus

T2(X;Y1, Y2) = H · T red2 (X;Y1, Y2)(13)

where

T red2 (X;Y1, Y2) = 1

2 ·[〈Y1, X〉 dξ(Y2) + dξ(Y1) 〈X,Y2〉−〈JY1, X〉 dξ(JY2)− dξ(JY1) 〈X, JY2〉

]− 1

2 i ·[〈JY1, X〉 dξ(Y2) + dξ(JY1) 〈X,Y2〉

+〈Y1, X〉 dξ(JY2) + dξ(Y1) 〈X, JY2〉](14)

The expression on the r.h.s. in this formula is actually the simplest tensorof type (2, 1) that can be constructed from 〈., .〉dξ and that is symmetricw.r.t. interchanging Y1 and Y2.Lemma 3.4. The Codazzi equations for surfaces Σ2 #M2

κ × R are:

〈∇XA · Y −∇Y A ·X,Z〉 = 〈RD(X,Y )ν, Z〉= κ 〈ν, grad ξ〉 ·

(〈Y, Z〉 dξ(X)− 〈X,Z〉 dξ(Y )

)Here RD denotes the Riemannian curvature tensor of the product space

M2κ×R, and, with our sign conventions, the Weingarten equation is A = D ν,

and the sectional curvature K of a 2–dimensional subspace in T (M2κ ×R) is

given by K(span{X,Y }) = ‖X ∧ Y ‖−2 · 〈RD(X,Y )Y,X〉.


Proof. In principle the first half of the claimed formula is well-known. How-ever, in order to make sure that the sign of the curvature term is correct, itis easiest to include the short computation. Working locally, we may assumethat Σ2 is embedded, and thus we may extend X, Y , and ν as smooth vectorfields onto a neighborhood of the surface:

〈∇X A · Y −∇Y A ·X,Z〉 = 〈∇X(AY )−∇Y (AX)−A · [X,Y ], Z〉= 〈DX(AY )−DY (AX)−A · [X,Y ], Z〉= 〈DX(DY ν)−DY (DX ν)−D[X,Y ] ν, Z〉= 〈RD(X,Y )ν, Z〉

The second equality sign in the claimed formula is obtained upon com-puting the curvature tensor RD for the product manifolds M2

κ ×R . Indeed,the bilinear form g − dξ2 represents the metric on the leaves M2

κ × {ξ0};hence

RD = κ · (g − dξ2)©∧ (g − dξ2) = κ · (g©∧ g − 2 g©∧ dξ2)

Like in the case of space forms, the tensor g©∧ g vanishes on the relevantcombinations of arguments, and therefore

〈RD(X,Y )ν, Z〉 = −2κ · g©∧ dξ2(X,Y ; ν, Z)

= −κ ·[〈X,Z〉 dξ(Y ) dξ(ν)− 〈Y, Z〉 dξ(X) dξ(ν)

−〈X, ν〉 dξ(Y ) dξ(Z) + 〈Y, ν〉 dξ(X) dξ(Z)]

= κ 〈ν, grad ξ〉 ·(〈Y, Z〉 dξ(X)− 〈X,Z〉 dξ(Y )

)�

Lemma 3.5. For cmc surfaces Σ2 #M2κ×R the tensor field T1 introduced

in the context of formula (12) can be evaluated as follows:

T1(X;Y1, Y2) = −κ 〈ν, grad ξ〉 · T red2 (X;Y1, Y2)

where T red2 is the field introduced in (14).

Proof. On cmc surfaces it is evident that the tensor ∇w A is traceless andtherefore anticommutes with the almost complex structure J for any vec-tor w ∈ TΣ2. We conclude that the real part of T1 can be rewritten as

<T1(X;Y1, Y2) = 〈Y1,∇X A · Y2〉+ 〈JY1,∇JX A · Y2〉 ,

and that the sum 〈Y1,∇Y2 A ·X〉+ 〈JY1,∇Y2 A ·JX〉 vanishes for all vectorsX, Y1, and Y2. Subtracting this sum from the expression for <T1, we mayuse the Codazzi equations from the Lemma 3.4 in order to evaluate thedifferences 〈Y1,∇X A · Y2 −∇Y2 A ·X〉 and 〈JY1,∇JX A · Y2 −∇Y2 A · JX〉.Hence we find that

<T1(X;Y1, Y2) = κ 〈ν, grad ξ〉 ·[〈Y1, Y2〉 dξ(X)

+ 〈JY1, Y2〉 dξ(JX)− 2〈X,Y1〉 dξ(Y2)]

By construction, T1 is a tensor field of type (2, 1). Firstly, this implies thatT1(X;Y1, Y2) = −T1(X;JY1, JY2), and thus

<T1(X;Y1, Y2) = 12

[<T1(X;Y1, Y2)−<T1(X;JY1, JY2)

]= κ 〈ν, grad ξ〉 ·

[〈X, JY1〉 dξ(JY2)− 〈X,Y1〉 dξ(Y2)

]


2 × R 17

Moreover, the field T1 is clearly symmetrical w.r.t. the permutation of Y1

and Y2, and thus we may symmetrize the r.h.s. accordingly. We concludethat <T1 = −κ 〈ν, grad ξ〉 · <T red

2 , as claimed. Referring to the invarianceproperties of type (2, 1)–tensors one more time, we can deduce that theimaginary parts are equal, too. �

Proof of Theorem 1. Combining equations (12) and (13), one obtains

∂ Q = H ·[T1 + κ 〈ν, grad ξ〉 · T red

2

],

and by Lemma 3.5 the expression on the r.h.s. of this equation vanishes. �

Remark 3.6. The proof of Theorem 1 is much more robust than it mightappear at first: The tensors T1, T2, and T red

2 are the (2, 1)–parts of basicgeometric objects like ∇A or the tri-linear form 〈., .〉dξ. When applying theCodazzi equations to T1, we evidently obtain a tensor of type (2, 1) that is asum of terms of the form κ〈ν, grad ξ〉 · 〈., .〉dξ. The upshot of such structuralconsiderations is that ∂ Q must be a multiple of κH 〈ν, grad ξ〉 · T red

2 withsome factor that is a universal constant.

There is also an independent argument that this constant factor mustindeed be zero. One simply considers the rotationally invariant cmc spheresS2H ⊂ M2

κ × R of Hsiang and Pedrosa and observes that their quadraticdifferential Q vanishes identically. Therefore ∂ Q ≡ 0, too. On the otherhand, however, neither the function κH 〈ν, grad ξ〉 nor the field T red

2 vanishon any set of positive measure in S2

H ⊂M2κ × R.

4. Cmc surfaces with vanishing quadratic differential Q

Our goal is to classify the cmc surfaces Σ2 # M2κ × R with vanishing

quadratic differential Q, unless H and κ vanish simultaneously. By the verydefinition of the quadratic differential Q in equations (1) and (2) we shall ineffect classify complete surfaces in M2

κ × R such that

2H · hΣ − κ · dξ ⊗ dξ|Σ =(2H2 − 1

2κ ‖dξ|Σ‖2)· g(15)

At this point it gets very clear why we have to exclude the case that Hand κ vanish simultaneously; the preceding condition holds trivially despitethe fact that there is an ample supply of interesting minimal surfaces ineuclidean 3–space.

The remainder of the minimal surface case on the other hand is easy: ifH = 0 and κ 6= 0, it follows directly from equation (15) that dξ ⊗ dξ|Σ = 0,and thus dξ|Σ vanishes identically. Therefore Σ2 must be a totally-geodesicleaf M2

κ × {ξ0}. These considerations tie in nicely with the fact that bygeneral theory the height function ξ has to be harmonic, provided that Σ2

is compact.From now on we shall concentrate on the case H 6= 0. In this case equa-

tion (15) can be solved for the second fundamental form hΣ. Clearly, for anysurface Σ2 # M2

κ × R, the vector fields (grad ξ)tan := grad ξ − 〈ν, grad ξ〉 νand J · (grad ξ)tan are principal directions. This implies that 〈ν, grad ξ〉 and∥∥(grad ξ)tan

∥∥2 = 1 − 〈ν, grad ξ〉2 are constant along horizontal sections andthat, moreover, these level curves have constant curvature.

The proper way to formalize the treatment of equation (15) is to prolongthe system once and interpret (15) as a problem about integral surfaces of a


suitable distribution EH in the unit tangent bundle of M2κ×R. This prolon-

gation will be defined in § 4.1. In § 4.2 we reap the easy consequences that areimplied by the still fairly large isometry groups of the product spaces M2

κ×Rin the presence of the rotationally invariant examples described in Section 2.The properties of the distribution EH will then be analyzed in full detailin § 4.3 based on the simple observations described in the preceding para-graph. The argument culminates in the proof of Theorem 3 on page 25.

4.1. The Gauss section of cmc surfaces with Q ≡ 0. In order tohandle equation (15) in the non-minimal case in a conceptually nice way,we find it better not to work with the immersion F : Σ2 # M2

κ × R itself,but to think of the unit normal field ν as the primary unknown. It willbe easiest to consider the Gauss section ν simply as an immersion of Σ2

into the total space N5 := T1M3 of the unit tangent bundle of the manifold

M3 := M2κ × R. Clearly, F = π ◦ ν where π : N5 = T1M

3 → M3 denotesthe standard projection map. Thus the immersion F can be recovered, onceits lift ν : Σ2 # N5 is known.

It is well-known that any affine connection on a manifold M can be un-derstood as a vector bundle isomorphism TTM → π∗(TM ⊕ TM), whichidentifies the bi-tangent bundle TTM with a more familiar vector bundleover TM . In particular, the Levi–Civita connection D on M3 = M2

κ × Rinduces an injective bundle map

ΦD : TN5 → π∗(TM3 ⊕ TM3)(16)

such that ΦD

(∂∂sν(s)

)=(∂∂s(π ◦ ν(s), D

∂sν(s))

for any smooth curve s 7→ν(s) ∈ N5. The image of this map is the 5–dimensional subbundle given by

ΦD

(TvN

5)

={

(w1, w2) ∈ Tπ(v)M3 ⊕ Tπ(v)M

3∣∣ 〈v, w2〉 = 0

}With these preparations it is now possible to translate the classification

problem for cmc surfaces in M3 := M2κ ×R with mean curvature H 6= 0 and

vanishing quadratic differential Q into the classification problem for integralsurfaces of some 2–dimensional distribution EH ⊂ TN5.Proposition 4.1. Let Σ2 # M3 = M2

κ × R be a cmc surface with H 6= 0and vanishing quadratic differential Q. Then its Gauss section ν : Σ2 →N5 := T1M

3 is necessarily an integral surface of the 2–dimensional distri-bution EH ⊂ TN5 given by

ΦD

((EH)v

)={

(w,Av · w)∣∣ w ∈ v⊥}

where

Av · w := κ2H 〈w, (grad ξ)tan〉 · (grad ξ)tan

+[H − κ

4H

(1− 〈v, grad ξ〉2

)]·(w − 〈v, w〉 v

)for all w ∈ Tπ(v)M

3 and where (grad ξ)tan := grad ξ − 〈v, grad ξ〉 v .Conversely, for any number H 6= 0 an integral surface of the distribution

EH necessarily projects onto a surface Σ2 # M2κ × R with constant mean

curvature equal to H and with vanishing quadratic differential Q.In this proposition we do not claim yet that any of the distributions EH

with H 6= 0 are integrable.


2 × R 19

Proof. Each unit vector v clearly lies in the kernel of the correspondingsymmetric endomorphism Av introduced in the proposition. Moreover, therestriction of Av to v⊥ is precisely the tensor that one obtains when solvingequation (15) for the Weingarten map of the surface Σ2 # M2

κ × R, pro-vided that v is chosen to be the unit normal vector ν at the point underconsideration.

Thus the proposition immediately follows when reading this expressionwith the Weingarten equation A = D ν and with the definition of the iso-morphism ΦD in mind. �

4.2. Symmetries of the distributions EH . In many cases the integralsurfaces of the distribution EH ⊂ TN5 can be determined quite easily usingthe invariance properties of EH , and the symmetry properties themselvescan be established without much effort, too.

In fact, there is an induced action G×N5 → N5 of the 4–dimensional Liegroup G := Isom0(M2

κ ×R) on the unit tangent bundle N5 := T1(M2κ ×R).

Since the vector field grad ξ is even invariant under all isometries of M2κ ×R

that preserve the orientation of the second factor, it is clear that the function

Θ: N5 →[−1

2π,12π]

v 7→ arcsin(〈v, grad ξ〉

)(17)

is invariant under the action of G on N5. Since the isotropy group Gxat any point x ∈ M2

κ × R acts on TxM2κ × R as the group of rotations

preserving (grad ξ)|x, it follows that the function Θ separates the G–orbitsin N5. The fibers over −1

2π and 12π correspond to 3–dimensional singular

orbits with isotropy group isomorphic to SO(2), whereas all other fibers ofΘ correspond to regular orbits; the principal isotropy group of the G–actionon N5 is trivial.Lemma 4.2. For any H 6= 0 the distribution EH ⊂ TN5 introduced inProposition 4.1 is invariant under the action of G = Isom0(M2

κ × R).

Proof. We may think of v as a G–invariant section of π∗ TM3 where πdenotes the canonical projection N5 = T1M

3 → M3 = M2κ × R. From

this point of view, the fields (grad ξ)tan and Av are G–invariant sections ofπ∗TM3 and π∗ End(TM3), respectively. �

Proposition 4.3. Suppose that 4H2 + κ > 0 and H 6= 0. Then the dis-tribution EH ⊂ TN5 introduced in Proposition 4.1 is integrable, and all itsintegral surfaces are congruent to Gauss sections of the embedded rotation-ally invariant cmc spheres S2

H ⊂M2κ × R.

This proposition actually proves the part of the Theorem 3 about theclassification in the case that 4H2 + κ > 0.

Proof. Let v0 ∈ N5 be arbitrary. The monotonicity properties establishedin the proof of Proposition 2.5i show that there exists some point c(s0) onthe generating curve of the sphere S2

H ⊂M2κ ×R such that Θ(v0) = θ(s0) =

Θ(ν(s0)). Thus there exists an isometry ψ ∈ G = Isom0(M2κ ×R) that maps

the point c(s0) to the foot point π(v0) and ν(s0) to v0 itself.By Proposition 2.7 the quadratic differential Q of the sphere S2

H vanishes.Thus Proposition 4.1 implies that the Gauss sections of S2

H and of all its


isometric images are integral surfaces of the distribution EH , and the Gausssection of ψ(S2

H) is an integral surface through the given point v0 ∈ N5. �

Proposition 4.4. Suppose that 4H2 +κ ≤ 0 and H 6= 0. Then the distribu-tion EH ⊂ TN5 introduced in Proposition 4.1 has complete non-compact in-tegral surfaces through any point v0 ∈ N5 such that 4H2+κ cos2(Θ(v0)) 6= 0.These integral surfaces are congruent to the Gauss sections of

(i) one of the non-compact cousins D2H ⊂ M2

κ × R of the rotationallyinvariant cmc spheres S2

H ⊂M2κ × R, or,

(ii) one of the embedded, rotationally invariant cmc surfaces C2H ⊂M2

κ×Rof catenoidal type.

The first case occurs iff 4H2 + κ cos2(Θ(v0)) > 0, whereas the second caseoccurs iff 4H2 + κ cos2(Θ(v0)) < 0.

Proof. The argument is essentially the same as in the proof of the precedingproposition. Note that by hypothesis 4H2 + κ cos2(Θ(v0)) is non-zero.

Looking at the ranges of the angle function s 7→ θ(s) that have beendetermined in Propositions 2.5ii and 2.9, we see that there exists a pointc(s0) on either the generating curve of D2

H or the generating curve of C2H

such that Θ(v0) = θ(s0) = Θ(ν(s0)). Thus there exists an isometry ψ ∈ G =Isom0(M2

κ ×R) that maps the point c(s0) to the foot point π(v0) and ν(s0)to v0 itself.

By Propositions 2.7 and 2.10 the quadratic differentials of both kinds ofsurfaces vanish identically, and so we have identified the integral surfacethrough v0 as the Gauss section of D2

H or C2H , respectively. �

The preceding proposition, however, does not finish the proof of Theo-rem 3. This is so despite the fact that we have identified the integral surfacesof the distribution EH through an open dense set of points v ∈ N5.

It remains to study the distribution EH on the hypersurface in N5 givenby the equation 4H2 +κ cos2 Θ = 0. As we shall see in the next subsection,the integral surfaces of EH that are contained in this hypersurface are notcongruent to the Gauss section of any rotationally symmetrical cmc surfacein M2

κ × R.

4.3. Integral surfaces of EH . In this subsection we shall determine allintegral surfaces of the 2–dimensional distribution EH in the unit tangentbundles N5 of the product spaces M3 = M2

κ × R in a systematic way.The idea is to compute the integral curves of some distinguished vertical

and horizontal vector fields e1 and e2. The properties of these integral curveswill then be used in Proposition 4.8 to verify that EH is integrable and thateach of its integral surfaces is invariant under some 1–parameter group ofisometries. This will give us a unified approach to the classification resultstated in Theorem 3.

The union of the regular orbits under the action of G = Isom0(M2κ ×R) is

the open dense subset N50 := π−1

((−1

2π,12π)). The vector fields v and grad ξ

are linearly independent at all points in N50 . Applying the Gram-Schmidt

process, we thus obtain two adapted orthonormal bases of π∗TM3|N50 that

are compatible with the given orientation of M3 = M2κ×R and that depend

smoothly on the foot point v. The first one is of the form grad ξ, e01, e0

2 =


2 × R 21

J0 e01 where J0 denotes the (almost) complex structure of the leaf M2

κ ⊂M2κ × R and where

e01 := 1

cos Θ

(v − 〈v, grad ξ〉 grad ξ

)(18)

denotes the horizontal unit vector in the plane spanned by v and grad ξ thathas positive inner product with v. The other adapted frame is of the formv, e1, e2; it is given by:

e1 := 1cos Θ (grad ξ)tan ≡ 1

cos Θ

(grad ξ − 〈v, grad ξ〉 v

)e2 := v × e1 .

(19)

It follows directly from the definition of the distribution EH in Proposi-tion 4.1 that the vector fields e1 and e2 defined by

ΦD(e1) =(e1, Av · e1

)=(e1, (H + κ

4H cos2 Θ) · e1

),

ΦD(e2) =(e2, Av · e2

)=(e2, (H − κ

4H cos2 Θ) · e2

)(20)

are a smooth basis for EH |N50 .

Our plan is to compute the integral curves of these two vector fields andthen use this information to recover the integral surfaces of EH . For thesecomputations it will be useful to note that

v = sin(Θ) · grad ξ + cos(Θ) · e01 ,

e1 = cos(Θ) · grad ξ − sin(Θ) · e01 ,

e2 = −e02 .

(21)

Lemma 4.5 (The meridians). Let H 6= 0, and let v0 be a point in theregular set N5

0 . Moreover, let r 7→ γ0(r) denote the unit speed geodesic inthe horizontal leaf M2

κ such that π(v0) = (γ0(0), ξ0) for some ξ0 ∈ R andsuch that γ0

′(0) = e01|v0. Then the integral curve s 7→ ν1(s) of the field e1

through v0 and its projection π ◦ ν1 onto M3 = M2κ × R satisfy

π ◦ ν1(s) =(γ0(r(s)) , ξ(s)

)ν1(s) =

(cos θ(s) · γ0

′(r(s)) , sin θ(s))(22)

where the triple of functions(r(s), ξ(s), θ(s)

)is obtained as the solution of

the ODE system∂∂s r = − sin θ∂∂s ξ = cos θ∂∂s θ = H + κ

4H cos2 θ

(23)

with initial data(r(0), ξ(0), θ(0)

)=(0, ξ0,Θ(v0)

).

Here the domain for the independent variable s is not the full existenceinterval for the differential equation. It is restricted by the condition thatcos(θ(s)) must remain strictly positive, as the integral curve must not leavethe regular set N5

0 . In short, what matters are the solutions of the thirdequation in (23).

Moreover, we observe that the lemma is consistent with Propositions 4.3and 4.4. The third equation in (23) is actually identical to the differentialequations for θ obtained in (7) and (7cat), respectively.


Remark 4.6. If 4H2 + κ ≤ 0, the ODE system in (23) has one class ofparticularly simple solutions that do not correspond to any of the surfacesS2H , D2

H , and C2H discussed in Section 2. They are given by

r(s) := r0 − s sin(θ0) , ξ(s) := ξ0 + s cos(θ0) , θ(s) := θ0

where θ0 := ± arcsin(√

1 + 4H2/κ), and where r0 and ξ0 are arbitrary con-stants of integration. In fact, all the other solutions of (23) can be de-termined explicitly, too. The corresponding formulas will be listed in theappendix.

Proof of the Lemma. Using equations (21)–(23), it is easy to verify that∂∂s

(π ◦ ν1(s)

)=(− sin θ(s) · γ0

′(r(s)) , cos θ(s))

= e1|ν1(s) ,

∇∂s ν1(s) = ∂

∂sθ(s) ·(− sin θ(s) · γ0

′(r(s)) , cos θ(s))

=(H + κ

4H cos2 θ(s))· e1|ν1(s) .

By the first line in (20) the preceding equations can be identified as the twocomponents of ∂

∂s ν1(s) = e1|ν1(s). Thus we have verified that equations (22)and (23) in the lemma indeed define an integral curve of e1. �

Next to the meridians, the circles of latitude constitute a second distin-guished family of curves on a surface of revolution. The latter curves areall strictly horizontal, and so the idea is to recover them as projections ofintegral curves of e2.Lemma 4.7 (The horizontal curves). Let v0 be a point in the regular set N5

0 .Then the integral curve t 7→ ν2(t) of the vector field e2 through v0 projectsto a curve of constant curvature

k(v0) := H · cos−1 Θ(v0)− κ4H · cos Θ(v0)

in the leaf M2κ × {ξ0} through the foot point π(v0).

Proof. Since by construction the gradient of the height function ξ is alwaysperpendicular to the field e2, it is clear that the projection π ◦ ν2 remainsinside the totally-geodesic leaf M2

κ × {ξ0} ⊂M3 through π(v0). A straight-forward computation based on the second equation in (20) shows that

∂∂t Θ

(ν2(t)

)= ∂

∂t 〈ν2(t), grad ξ〉 = 〈 D∂t ν2(t), grad ξ〉

=[H − κ

4H cos2 Θ(ν2(t)

)]· 〈e2|ν2(t), grad ξ〉 = 0 .

In other words, Θ is constant along any integral curve of e2. By constructione0

1 is the (exterior) normal along the curve t 7→ π ◦ ν2(t) ∈M2κ ×{ξ0}. With

the help of the first equation in (21) and the second equation in (20), wethus find that

D∂t

(e0

1

)= 1

cos ΘD∂t

(ν2(t)− sin(Θ) · grad ξ

)= 1

cos Θ

(H − κ

4H cos2(Θ))· e2|ν2(t) �

Proposition 4.8. Let H be some constant 6= 0. Then, for any regularpoint v0 ∈ N5

0 , the 2–dimensional distribution EH is integrable in someneighborhood of v0. Moreover, all these local integral surfaces are invariantunder the action of some 1–parameter subgroup t 7→ φt×id ∈ Isom0(M2

κ×R).


2 × R 23

Proof. An abstract verification that EH is integrable is not of much usewhen we want to determine the invariance properties of the local integralsurfaces.

Thus it seems better to construct some explicit candidates for these in-tegral surfaces from the very beginning. We determine the integral curves 7→ ν1(s) of the vector field e1 through the given point v0 ∈ N5

0 as ex-plained in Lemma 4.5 consider the orbit of this curve under the flow ofe2. In other words, we consider the map (s, t) 7→ ν(s, t) ∈ N5

0 such that∂∂t ν(s, t) = e2|ν(s,t) and ν(s, 0) = ν1(s) .

Firstly, we claim that the maps t 7→ π ◦ ν(s, t) yield a family of parallelcurves of constant curvature in the surface M2

κ . In fact, by constructioneach of these maps describes a curve in the first factor of M3 = M2

κ × Rthat emanates perpendicularly from the unit speed geodesic γ0 at the pointγ0(r(s)), and by Lemma 4.5 this curve has constant geodesic curvature

kg(s) := H · cos−1 θ(s)− κ4H · cos θ(s) .(24)

It is well-known that the geodesic curvature k of a family of parallel curves ofconstant curvature emanating perpendicularly from γ0 satisfies the Riccatiequation

∂∂r k + k2 + κ = 0 .(25)

The converse is not hard to prove either: if the geodesic curvature k of afamily of curves of constant curvature that emanate perpendicularly fromγ0 satisfies this differential equation, then the family is in fact a family ofparallel curves.

Thus we only need to show that the geodesic curvatures kg(s) of thecurves t 7→ ν(s, t) can be written as kg(s) = k(r(s)) for some function ksatisfying (25). Using equations (23) and (24), we find that

∂∂s kg(s) = sin θ(s) ·

(H cos−2 θ(s) + κ

4H

)· ∂∂s θ(s)

= sin θ(s) ·(kg(s)2 + κ

).

(26)

Since ∂∂s r(s) = − sin θ(s), the preceding differential equation indeed asserts

that the function s 7→ kg(s) factors over some function r 7→ k(r) solv-ing (25), thereby establishing the claim from the beginning of the precedingparagraph.

Parallel curves of constant curvature in the 2–dimensional space formsM2κ are known to be the orbits of a suitable 1–parameter group t 7→ φt of

isometries. This in turn means that

π ◦ ν(s, a(s) t

)=(φt × id

)(π ◦ ν1(s))

where a(s) :=∥∥ ∂∂t |t=0 φt ◦ γ0(r(s))

∥∥. By construction, the tangent vectorsto the s– and t–parameter lines are linearly independent as long as ν(s, t) iscontained in the regular set N5

0 . Passing to the 1–jet, we therefore obtain

ν(s, a(s) t

)=(φt × id

)∗(ν1(s)) .(27)

Since the vector field e1 on N50 is by construction invariant under isometries

of M3 = M2κ × R, we conclude that the image of ν1 under each of the

maps (φt × id)∗ is again an integral curve of e1. Hence the map ν is indeed


D 2H

P 2H

C 2H

Figure 4. The meridian of a parabolic surface P 2H comes as

a limit of the meridians of the corresponding disk-like surfacesD2H and surfaces C2

H of catenoidal type, as the axis is movedfurther and further out.

an integral surface of EH . By construction it passes through the givenpoint v0, and by formula (26) it is invariant under the induced action of the1–parameter group t 7→ φt × id ∈ Isom0(M2

κ × R) of isometries. �

Proposition 4.9. Suppose that 4H2 + κ ≤ 0, and let Σ2 # M2κ × R be

one of the cmc surfaces with Q ≡ 0 that corresponds to the special solutionsof (23) from Remark 4.6. Then Σ2 is embedded; it is an orbit under some2–dimensional solvable subgroup An N ⊂ Isom0(M2

κ × R).If 4H2 + κ < 0, the solvable group A n N in the proposition is closely

related to the Iwasawa decomposition Isom0(M2κ) = SO+(2, 1) = KAN.

However, if 4H2 + κ = 0, the group N is still the nilpotent factor from theIwasawa decomposition, whereas A degenerates into the group of verticaltranslations, and the semi-direct product turns into a direct product.

We consider the parabolic nature of the elements in the subgroup N asthe characteristic property, and so we call this family of embedded cmcsurfaces P 2

H . The surfaces P 2H can be viewed as pointed limits of sequences

of disk-like cmc surfaces D2H or of sequences of cmc surfaces C2

H of catenoidaltype. In either case, the axis of the rotational symmetry disappears toinfinity in the limit as indicated in Figure 4.

Proof. The formulas in Remark 4.6 directly imply that the expression

kg(s)2 + κ ≡[H cos−1 θ(s) + κ

4H cos θ(s)]2

vanishes identically. Thus the horizontal curves t 7→ ν(s, t) yield a fam-ily of parallel horocycles in M2

κ . This means that the isometries φt con-structed in the proof of the preceding proposition must be parabolic ele-ments in Isom0(M2

κ). In other words, the image of the homomorphismR→ Isom0(M2

κ), t 7→ φt, constructed in the proof of the preceding proposi-tion is a nilpotent subgroup N ⊂ Isom0(M2

κ).The meridians s 7→ ν(s, t) project to geodesics in M2

κ×R. Each of them isthe orbit under some 1–parameter subgroup At ⊂ Isom0(M2

κ×R) consistingof transvections. The action of At clearly maps the horizontal horocyclesthat intersect the given geodesic s 7→ π◦ν(s, t) perpendicularly to horocycles


2 × R 25

of the same kind. Hence the group At maps the surface Σ2 = im(π ◦ ν) intoitself. The various groups At associated to distinct meridians are mutuallyconjugate, and thus the semi-direct product At n N does not depend on t.It acts isometrically and simply transitively on Σ2 ⊂M2

κ × R. �

Proof of Theorem 3. It is a property of the Riccati equation (26) for thegeodesic curvatures kg(s) of the horizontal curves t 7→ π◦ν(s, t) that the signof the expression kg(s)2 + κ is independent of s. In fact, it follows directlyfrom formula (24) that

kg(s)2 + κ =[H cos−1 θ(s) + κ

4H cos θ(s)]2 ≥ 0 .

Hence there are just two possibilities; the expression kg(s)2 + κ may eitherbe strictly positive for all s, or it may vanish identically. In the latter casethe term in the square brackets itself must vanish, and so we are in the caseanalyzed in the preceding proposition.

In the other case, k2g +κ > 0 everywhere, and so the horizontal curves are

geodesic circles, and the image of the homomorphism R→ Isom0(M2κ × R) ,

t 7→ φt × id , is isomorphic to SO(2). Hence the integral surfaces must becongruent to the Gauss sections of the rotationally invariant cmc surfacesΣ2 # M2

κ × R described in Section 2. By Propositions 4.3 and 4.4 weknow that we are only seeing the surfaces S2

H , D2H , and C2

H described inPropositions 2.5 and 2.9. �

Appendix A. Explicit Formulas for the Meridians

It is not hard to see that the system of differential equations (23) inLemma 4.5 is actually integrable. Its flow vector field lies in the kernel ofthe following set of Pfaffian 1–forms:

η0 := cos(θ) · dr + sin(θ) · dξη1 :=

(4H2 + κ cos2(θ)

)· dr + 4H sin(θ) · dθ

η2 :=(4H2 + κ cos2(θ)

)· dξ − 4H cos(θ) · dθ

(28)

In order to compute explicit first integrals from η1 and η2, it is necessaryto integrate Riccati equations for u1 := cos(θ) and u2 := sin(θ), respectively.Substituting u := tan(θ) into the third equation in (23), we find that therelation between the angle θ and the arc length parameter s is given by adifferential equation of Riccati type, too.

Proposition A.1. The meridian curves of the cmc surfaces Σ2 #M2κ ×R

with vanishing quadratic differential Q and non-zero mean curvature H canbe described as zero sets of suitable elementary functions:

(i) If 4H2 +κ > 0, then Σ2 is necessarily one of the rotationally-symme-trical spheres S2

H of Hsiang and Pedrosa, and up to vertical transla-tions the meridian is the smooth variety

1 = 4H2 · sn2−κ(

12ξ√

1 + κ4H2

)+(4H2 + κ

)· sn2

κ

(12r).(29S)

(ii) If 4H2 + κ = 0, then Σ2 must be either one of the disk-like surfacesD2H , or Σ2 is a cylinder over a horocycle, which is a borderline case


of a surface of type P 2H . Up to vertical translations, their meridians

are the smooth varieties

H ξ = ± cosh(H r) , and(30D)

r = r0 ,(30P )

respectively.

(iii) If 4H2 + κ < 0, then — depending on the slope of the meridian —the surface Σ2 must be either a disk-like surface D2

H , or a parabolicsurface P 2

H , or a surface C2H of catenoidal type. Up to vertical trans-

lations, their meridians are the smooth varieties

sinh2(

12ξ√κ (1 + κ

4H2 ))

= −(1 + κ

4H2

)· cosh2

(12r√−κ)

,(31D) √−(4H2 + κ) · ξ = ± 2H r ,(31P )

cosh2(

12ξ√κ (1 + κ

4H2 ))

= −(1 + κ

4H2

)· sinh2

(12r√−κ)

,(31C)

respectively. The asymptotes to the meridians of D2H and C2

H intersectthe axis at a point with ξ ·

√κ (1 + κ

4H2 ) = ± ln(−1− κ

4H2

).

Some pictures of these meridians have been provided in Figures 1–3 onpages 7, 8, and 11 in Section 2. In fact, the graphs shown there have beenplotted using the explicit formulas from this proposition.Remark A.2. If κ = 0 equation (29S) is just a somewhat involved way towrite the standard equation 1 = H2 · ξ2 +H2 · r2 for a circle with radius 1

Hin euclidean 3–space. If κ is non-zero, (29S) turns out to be a shorthandfor one of the following two formulas

κ = 4H2 · sinh2(

12ξ√κ (1 + κ

4H2 ))

+(4H2 + κ

)· sin2

(12r√κ),

−κ = 4H2 · sin2(

12ξ√−κ (1 + κ

4H2 ))

+(4H2 + κ

)· sinh2

(12r√−κ).

Remark A.3. In order to make it very clear that the surfaces C2H really

resemble catenoids, it is best to consider the following scaling limits: wepick some constant λ > 0 and a sequence (κj)j of negative numbers thatconverges to 0. Moreover, we set Hj := 1

4λ κj . Thus 4H2j + κj < 0 for j

sufficiently large, and so we are indeed dealing with a sequence of surfacesof catenoidal type. Passing to the limit in equation (31C), we find that themeridians of these surfaces converge to the zero set of the equation

λr = ± cosh(λξ) ,

which is obviously the equation for the meridian of a standard catenoid ineuclidean 3–space.

Proof of the Proposition. As suggested above the proof starts out integrat-ing the Riccati equations corresponding to the Pfaffians η1 and η2. It followsdirectly from Remark 4.6 and Proposition 4.9 that neither cos(θ) can be in-dependent of r, nor can sin(θ) be independent of ξ, unless Σ2 is a horizontalleaf or a surface of the parabolic type P 2

H . An explicit equation describingthe meridians of the latter surfaces has already been given in Proposition 4.9.

It remains to compute the equations for the meridians of the surfacesS2H , D2

H , and C2H . From the discussion of the qualitative properties of these


2 × R 27

surfaces we already know that cos(θ) must be an odd function of r. For thesurfaces of type S2

H and D2H this function must vanish at r = 0, whereas the

radius function r must be bounded away from 0 for the surfaces of catenoidaltype. In short, when integrating η1, we recover equations (6) and (6cat).

Clearly, when integrating the Pfaffian η2 the solution only matters upto translation in the ξ–variable. We can normalize the solutions such thatsin(θ) is an odd function of ξ. For the surfaces S2

H and C2H we thus want

sin(θ) to vanish at ξ = 0, whereas we want sin(θ) to be bounded away from0 for the disk-like surfaces D2

H . Hence the first integrals obtained from η2

are:

4H2 + κ = 2H · sin(θ) · ct−κ (1+κ/4H2)

(12 ξ),(32)

κ · sin(θ) = 2H · ct−κ (1+κ/4H2)

(12 ξ).(32disk)

We merely need to insert these expressions for cos(θ) and sin(θ) into thestandard identity cos2 + sin2 = 1 in order to obtain the claimed equationsfor the meridians. �

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193–243.

Ruhr-Universitat Bochum, Fakultat fur Mathematik, Universitatsstr. 150,

D–44780 Bochum, Germany

E-mail address: [email protected]

Centre de Mathematiques de Jussieu, Universite Paris VII — Denis Diderot,

2 place Jussieu, F–75251 Paris Cedex 05, France

E-mail address: [email protected]

Introductionw3.impa.br/~rosen/MxR-Hopf.pdf · 2003. 12. 19. · A HOPF DIFFERENTIAL FOR CONSTANT...

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Transcript of Introductionw3.impa.br/~rosen/MxR-Hopf.pdf · 2003. 12. 19. · A HOPF DIFFERENTIAL FOR CONSTANT...