A HOPF DIFFERENTIAL FOR CONSTANT MEAN CURVATURE SURFACES IN S 2 R AND H 2 R

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1 A HOPF DIFFERENTIAL FOR CONSTANT MEAN CURVATURE SURFACES IN S 2 R AND H 2 R UWE ABRESCH AND HAROLD ROSENBERG Dedicated to Hermann Karcher on the Occasion of his 65 th Birthday Abstract. A basic tool in the theory of constant mean curvature (cmc) surfaces Σ 2 in space forms is the holomorphic quadratic differential discovered by H. Hopf. In this paper we generalize this differential to immersed cmc surfaces Σ 2 in the product spaces S 2 R and H 2 R and prove a corresponding result about the geometry of immersed cmc spheres S 2 in these target spaces. Introduction In 1955 H. Hopf [13] discovered that the complexification of the traceless part of the second fundamental form h Σ of an immersed surface Σ 2 with constant mean curvature H in euclidean 3 space is a holomorphic quadratic differential Q on Σ 2. This observation has been the key to his well-known theorem that any immersed constant mean curvature (cmc) sphere S 2 R 3 is in fact a standard distance sphere with radius 1 H. Hopf s result has been extended to immersed cmc spheres S 2 in the sphere S 3 or in hyperbolic space H 3. In other words, the result has been extended to immersed cmc spheres S 2 M 3 κ in space forms with arbitrary curvature κ. Futhermore, W.-Y. Hsiang has conjectured [14, Remark (i) on p. 51] that immersed cmc spheres in the product space H 2 R should be embedded rotationally-invariant vertical bi-graphs. The holomorphic quadratic differential Q itself has other important applications; it guarantees the existence of conformal curvature line coordinates on cmc tori in space forms, and thus it has played a significant role in the discovery of the Wente tori [25] and in the subsequent development of a general theory of cmc tori in R 3 (see [1, 21, 6, 12, 7]). For cmc surfaces Σ 2 g of genus g > 1 the holomorphic quadratic differential still provides a link between 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 of such cmc surfaces (see [8, 11, 22]) 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 Mathématiques de Jussieu and the CNRS for their hospitality and support during a visit to Paris in fall The basic ideas for the results in this paper have been developed during that visit. 1

2 2 UWE ABRESCH AND HAROLD ROSENBERG 1. Main Results Our main goal is to introduce a generalized quadratic differential Q for immersed surfaces Σ 2 in the product spaces S 2 R and H 2 R and to extend Hopf s result to cmc spheres in these target spaces; we shall prove that such immersed spheres are surfaces of revolution. In order to treat the two cases simultaneously, we write the target space as Mκ 2 R where Mκ 2 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 : Mκ 2 R R that is induced by the standard coordinate function on the real axis. We note in particular that ξ has a parallel gradient field of norm 1, and that the fibers of ξ are the leaves Mκ 2 {ξ 0 }. In addition to the function ξ we only need the second fundamental form h Σ (X, Y ) = X, A Y, the mean curvature H = 1 2 tr A, and the induced (almost) complex structure J in order to define the quadratic differential Q for immersed surfaces Σ 2 in product spaces Mκ 2 R. We set (1) and (2) q(x, Y ) := 2H h Σ (X, Y ) κ dξ(x) dξ(y ) Q(X, Y ) := 1 2 (q(x, Y ) q(jx, JY ) ) 1 2 i (q(jx, Y ) + q(x, JY ) ) Clearly, Q(JX, Y ) = Q(X, JY ) = i Q(X, Y ). In dimension 2, traceless symmetric endomorphisms anticommute with the almost complex structure J, whereas multiples of the identity commute with J. Thus the quadratic differential Q defined in (2) depends only on the traceless part q 0 of the symmetric bilinear form q introduced in (1) and, conversely, it is the traceless part q 0 that can be recovered as the real part of Q. Theorem 1. Let Σ 2 M 2 κ R be an immersed cmc surface in a product space. Then its quadratic differential Q as introduced in (2) is holomorphic w.r.t. the induced complex structure J on the surface Σ 2. The proof of this theorem is a straightforward computation based on the Codazzi equations for surfaces Σ 2 Mκ 2 R, which differ from the more familiar Codazzi equations for surfaces in space forms Mκ 3 by an additional curvature term. The details will be given in Section 3. Following H. Hopf, we want to determine the geometry of immersed cmc spheres in product spaces Mκ 2 R using Theorem 1. In these target spaces one has the embedded cmc spheres SH 2 studied by W.-Y. Hsiang [14] and R. Pedrosa and M. Ritoré [20] that are invariant under an isometric SO(2) action, rotation about a vertical geodesic. From now on we shall refer to them as the embedded rotationally invariant cmc spheres SH 2 M κ 2 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-umbilical unless κ = 0 or κ > 0 and H = 0. The traceless part q 0 of the symmetric bilinear form introduced in (1), however, vanishes on all of them. For us, this observation has been the major clue when guessing the proper expression for the quadratic differential Q for cmc surfaces in such product spaces.

3 A HOPF DIFFERENTIAL FOR CMC SURFACES IN S 2 R AND H 2 R 3 Theorem 2. Any immersed cmc sphere S 2 Mκ 2 R in a product space is actually one of the embedded rotationally invariant cmc spheres SH 2 Mκ 2 R. This theorem in particular establishes Hsiang s conjecture from [14, Remark (i) on p. 51] about immersed cmc spheres in H 2 R. It is a well-known fact that any holomorphic quadratic differential on the 2 sphere vanishes identically. Thus Theorem 1 implies that the quadratic differential Q of an immersed cmc sphere S 2 Mκ 2 R vanishes. Hence Theorem 2 follows directly from the following classification result: Theorem 3. There are 4 distinct classes of complete, possibly immersed, cmc surfaces Σ 2 Mκ 2 R with vanishing quadratic differential Q. Three of these classes are comprised of embedded rotationally invariant surfaces; they are the cmc spheres SH 2 M κ 2 R of Hsiang and Pedrosa, their non-compact cousins DH 2, and the surfaces C2 H of catenoidal type. The fourth class is comprised of certain orbits PH 2 of 2 dimensional solvable groups of isometries of Mκ 2 R. The geometry of the rotationally invariant embedded cmc surfaces of type SH 2, D2 H, or C2 H 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. The remaining family PH 2, where the parabolic symmetries are prevalent, will only be introduced in Proposition 4.9 in Section 4. Explicit formulas for the meridians of all these surfaces and some pictures thereof will be provided in the appendix. Remark 4. If κ 0, then only the spheres SH 2 M κ 2 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 established in Section 4. A systematic approach to this classification problem is to study a suitable overdetermined ODE system for the unit normal field of the surfaces to be classified. In doing so, the surfaces PH 2 in fact show up quite naturally. In a sense the preceding theorems amount to saying that, when studying surfaces Σ 2 in product spaces Mκ 2 R rather than in space forms, the role of umbilics is taken over by the points where Σ 2 approximates one of the embedded cmc surfaces SH 2, D2 H, C2 H, or P H 2 up to second order. Theorem 2 provides substantial information even when studying the geometry of embedded cmc spheres S 2 S 2 R. Moving planes arguments along the lines of A. D. Alexandrov [3] only prove that a closed embedded cmc surface Σ 2 S 2 R is a vertical bi-graph. In contrast to target spaces with non-positive sectional curvature, moving planes arguments do not imply that such a surface Σ 2 S 2 R is rotationally invariant. The latter claim would only follow, if we assumed that Σ 2 were contained in the product of a hemisphere with the real axis. Such an assumption, however, is by far too strong. It does not even hold for the embedded rotationally invariant cmc spheres SH 2 M κ 2 R if 0 < 4H 2 < 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 R 3 there even exists a family of embedded cmc tori in S 2 κ R, most of them not being rotationally invariant. They rather look approximately like undoloids around some

4 4 UWE ABRESCH AND HAROLD ROSENBERG great circle in a leaf S 2 κ {ξ 0 } S 2 κ R and are thus not contained in the product of a hemisphere with the real axis either. They have been constructed by J. de Lira [17] by a bifurcation argument along the lines of the work done by R. Mazzeo and F. Pacard [18]. Presumably one could also obtain these surfaces using a singular perturbation argument similar to the one that N. Kapouleas work is based on [16]. Finally, we observe that the holomorphic quadratic differential Q of Theorem 1 is likely to open the doors for investigating the geometry of cmc tori in product spaces Mκ 2 R, and as in the case of space forms there might be interesting connections to the theory of integrable systems. A somewhat more subtle question is whether all the above can be generalized to cmc surfaces Σ 2 in homogeneous 3 manifolds. 2. The rotationally invariant cmc spheres S 2 H M 2 κ R In this section we shall describe the embedded rotationally invariant cmc spheres SH 2 M κ 2 R of Hsiang and Pedrosa. Our main goal is to compute the symmetric bilinear form q introduced in (1) for these cmc spheres and thus verify that their quadratic differential Q vanishes. This goal will be accomplished in Proposition 2.7. The work of both authors, Hsiang and Pedrosa, has been somewhat more general than what we need here. They have studied rotationally invariant cmc spheres SH n in product spaces including Hn R [14] or in S n R [20], respectively. For our purposes we are only interested in properties of the surfaces SH 2, and, specializing our summary accordingly, we thus work on product manifolds M 3 := Mκ 2 R that are equipped with an isometric SO(2) action Isometric SO(2) actions on the product spaces Mκ 2 R. Such an action necessarily operates trivially on the second factor. In other words, the group SO(2) Isom 0 (Mκ 2 R) can be considered as the isotropy group consisting of those rotations that preserve some fixed axis l 0 := {ˆx 0 } R. If κ 0, the fixed point set of this SO(2) action is precisely the axis l 0 itself; however, if κ > 0, it encompasses the antipodal line ˆl 0 = { ˆx 0 } R as well. In order to understand this action better, it is useful to pick a normal geodesic γ : R Mκ 2 with γ(0) = ˆx 0. The product map γ id: R 2 Mκ 2 R is a totally-geodesic isometric immersion. Its image intersects all SO(2) orbits orthogonally. The existence of such a slice means that the group action under consideration is polar. The subset I κ R R 2, where I κ denotes the half axis (0, ) if κ 0 and the interval (0, π/ κ) if κ > 0, is a fundamental domain whose image under the immersion γ id intersects each principal SO(2) orbit in Mκ R 2 precisely once. Its closure Īκ R represents the entire orbit space SO(2)\(Mκ 2 R). The canonical projection p onto this orbit space lifts to a map ˆp: Mκ 2 R Ī κ R such that the composition ˆp (γ id) is the identity on Īκ R. This lift is given by ˆp(x) = ( r(x), ξ(x) ) where r(x) := dist(x, l 0 ) and where ξ is the height function introduced beforehand. The restriction of ˆp to the union of all principal orbits is clearly a Riemannian submersion. A conceptually nice way of thinking about the slice im(γ id) is to extend the SO(2) action to some larger subgroup of Isom(Mκ 2 R). The natural

5 A HOPF DIFFERENTIAL FOR CMC SURFACES IN S 2 R AND H 2 R 5 candidate is the group O(2) consisting of all isometries of Mκ 2 R that fix the axis l 0. The cosets of SO(2) O(2) are represented by the subgroup consisting of the identity and of the reflection ϱ at the slice. The extended action has the same orbits as the SO(2) action that we began with. However, the principal isotropy group of the extended action is Z 2 = ϱ rather than the trivial group. The slice im(γ id) can be recovered as the fixed point set of the reflection ϱ. Moreover, ϱ preserves the height function ξ and thus also its gradient. In particular, grad ξ is tangential to the slice everywhere Connected SO(2) invariant surfaces Σ 2 Mκ 2 R. As explained above, the given action extends to an isometric O(2) action on Mκ 2 R with precisely the same orbits. Hence any rotationally invariant cmc surface Σ 2 Mκ 2 R is also invariant under the full O(2) action. The additional symmetry tells a lot about the geometry of the surfaces in question: Proposition 2.1. Any rotationally invariant surface Σ 2 Mκ 2 R intersects the slice im(γ id) orthogonally in a set of regular curves c that are simultaneously geodesics and curvature lines on Σ 2. If the surface Σ 2 is connected, then so is its image under the projection p from Mκ 2 R onto the orbit space. Therefore this image can be described by a single regular curve c in Īκ R that may start and/or end at the boundary and that can meet the boundary only orthogonally. The surface Σ 2 Mκ 2 R can be recovered from c by letting SO(2) act on its image c := (γ id) c in Mκ 2 R. Moreover, any regular curve s c(s) Īκ R with the proper behavior at the boundary of Īκ R is the generating curve for some connected, smooth, rotationally invariant surface Σ 2 Mκ 2 R. The unit normal field ν of the surface is best described in terms of its angle function s θ(s) along the generating curve s 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 and that its tangent vector is c (s) = ( sin θ(s), cos θ(s)). Our convention is to work with exterior unit normal vectors and write the Weingarten equation with a + sign, i.e. to write A = D ν. In particular, the principal curvature of Σ 2 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 h Σ and the tensor field dξ 2 Σ along the meridian c on Σ 2 with respect to the orthonormal basis consisting of the vectors c (s) and Jc (s) : (3) h Σ = ( s θ 0 0 cos(θ) ct κ (r) ) and dξ 2 Σ = ( cos 2 ) (θ) Here ct κ stands for the generalized cotangent function 1 ; the value ct κ (r) is the curvature of the circle of radius r > 0 in the surface M 2 κ of constant curvature κ. 1 Recall that the generalized cotangent function ctκ is by definition the logarithmic derivative of the generalized sine function sn κ which in turn is defined to be the solution of the differential equation y + κ y = 0 with initial data y(0) = 0 and y (0) = 1.

6 6 UWE ABRESCH AND HAROLD ROSENBERG 2.3. Differential equations for SO(2) invariant cmc surfaces Σ 2. The mean curvature H of a rotationally invariant surface Σ 2 M 2 κ R is given by the identity 2H = tr A = tr g (h Σ ). Thus the preceding considerations yield the following system of ordinary differential equations for the generating curve s c(s) = (r(s), ξ(s)): s r = sin(θ) (4) s ξ = cos(θ) s θ = 2H cos(θ) ct κ(r) The key to solving this ODE system is the observation that its r.h.s. depends neither explicitly on s nor on ξ. The latter property is a direct consequence of the fact that the euclidean factor in the product spaces Mκ 2 R is 1 dimensional. It follows that the ODE system (4) has a first integral. In fact, W.-Y. Hsiang already found that (5) L := cos(θ) sn κ (r) 2H r 0 sn κ (t) dt = cos(θ) sn κ (r) 4H sn κ ( 1 2 r)2 is constant along any solution of (4). Remark 2.2. The value of L can be used to characterize the special solutions of (4) that correspond to particularly simple cmc surfaces in Mκ 2 R : (i) If H = 0, there are the totally-geodesic leaves Mκ 2 {ξ 0 }. These leaves correspond to solutions with ξ = ξ 0 and θ = ± 1 2π, and thus they can be characterized by the condition L = 0. (ii) If H = 1 2 ct κ(r 0 ) for some r 0 I κ, there is the cylinder of radius r 0 around the axis l 0. This surface corresponds to solutions with r = r 0 and θ = 0, and thus it can be characterized by the condition L = ct κ ( 1 2 r 0) 1 = (2H + 4H 2 + 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 0 that oscillate around these cylinders. These undoloids are discussed from different points of view in [20] and in [23, Sections 1 and 3]. Together, all these examples give a good indication of how rich the class of rotationally invariant minimal surfaces in Mκ 2 R is. Furthermore, there is a 2 parameter family of minimal annuli whose level curves are geodesic circles which are not rotationally invariant. We expect the quadratic differential Q to play a key role in studying these examples. Proposition 2.3. A connected, rotationally invariant, immersed cmc surface Σ 2 Mκ 2 R is diffeomorphic to a torus or an annulus, unless the first integral L for its generating curve c vanishes, or, unless κ > 0 and L = 4H/κ. Proof. The orbit structure described in 2.1 implies that any connected, SO(2) invariant surface Σ 2 M 2 κ R is topologically the product of its generating curve c in Īκ R and a circle, provided that the generating curve stays in the interior of I κ R. If c has an end point on the boundary of I κ R, a disk must be glued to the corresponding boundary component of the remaining part of Σ 2.

7 A HOPF DIFFERENTIAL FOR CMC SURFACES IN S 2 R AND H 2 R 7 ξ 1 κ H=0.121 κ H=1.075 κ H=0.033 κ 0 r π κ 1 κ H=0.375 κ Figure 1. The meridians of the spheres S 2 H for κ > 0. Inspecting the expression in (5), it is evident that L must vanish if c touches the component of the boundary of Īκ R where r 0. If κ 0 this component is already the entire boundary of the fundamental domain I κ R. However, if κ > 0, the boundary of I κ R has one further component where r π/ κ. Clearly, c can only reach the latter component if L = 4H/κ. Remark 2.4. The components of the boundary of Ī κ R correspond to the components of the fixed point set of the SO(2) action. In particular, interchanging the axes l 0 and ˆl 0 induces a symmetry of (4) that interchanges the cases L = 0 and L = 4H/κ as well SO(2) invariant cmc surfaces Σ 2 Mκ 2 R with L = 0. In this subsection we shall see that a rotationally-invariant cmc surface with L = 0 does indeed intersect the axis l 0. Thus, by the argument used in the proof of Proposition 2.3 the surface Σ 2 must be either a sphere SH 2 or a disk D2 H. Since sn κ ( 1 2 r) > 0 for any r I κ, it is possible to rewrite the condition that the first integral L introduced in (5) vanishes as follows: (6) cos(θ) ct κ ( 1 2r) = 2H 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); it implies that the generating curve c can only meet the boundary of Īκ R orthogonally. Hence the cmc surface Σ 2 is automatically smooth at all the points where it intersects the axis l 0. Proposition 2.5. Let Σ 2 M 2 κ R be a connected, SO(2) invariant cmc surfaces with mean curvature H 0 that intersects the axis l 0. Then either (i) 4H 2 + κ > 0, and Σ 2 is an embedded sphere S 2 H M 2 κ R, or (ii) 4H 2 + κ 0, and Σ 2 is a convex rotationally invariant graph D 2 H over the horizontal leaves M 2 κ {ξ 0 }, which is asymptotically conical whenever the inequality for 4H 2 +κ is strict. There are two possibilities for the range of the normal angle θ; the image of sin(θ) is one of the two half open intervals, [ 1, sin θ 0 ) or (sin θ 0, 1], where θ 0 := arcsin( 1 + 4H 2 /κ ).

8 8 UWE ABRESCH AND HAROLD ROSENBERG ξ 2 κ H=0.87 κ H=1.15 κ H=0.66 κ S 2 H H=0.5 κ D 2 H H=0.33 κ H=0.12 κ 0 3 r κ Figure 2. The meridians of the spheres SH 2 and the disk-like surfaces DH 2 for κ < 0. The surfaces described in this proposition are the embedded rotationally invariant cmc spheres SH 2 M κ 2 R of Hsiang and Pedrosa and their non-compact cousins DH 2. The meridians generating these cmc surfaces are shown in Figures 1 and 2 for positive and negative values of κ, respectively. As indicated in these figures, the surfaces converge on compact subsets of Mκ 2 R towards the SO(2) invariant minimal surfaces described in Remark 2.2i, provided their mean curvature H approaches 0. Thus it is sometimes convenient to denote the leaves Mκ 2 {ξ 0 } as S0 2 or D2 0, respectively. Proof. The idea is to evaluate the condition L = 0 at all points of the generating curve s c(s) = (r(s), ξ(s)) that lie in the interior of Īκ R. Using the monotonicity properties of the functions r ct κ ( 1 2r) defined on the open intervals I κ, it is easy to determine their range, too. Thus one finds that the pair of inequalities H cos(θ) > 0 and 4H 2 + κ cos 2 (θ) > 0 are a 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 ct κ (r) from the last equation in (4); one finds that s θ = 1 ( (7) 4H 4H 2 + κ cos 2 (θ) ) The r.h.s. of this differential equation is uniformly bounded, and thus its solutions 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 I κ R. By equation (6), the curve c(s) approaches the boundary component of Īκ R corresponding to the axis l 0 whenever cos θ(s) converges to 0, and the points on c that lie in the interior of I κ R correspond to a maximal interval (s 1, s 2 ) R such that H cos θ(s) > 0 for all s (s 1, s 2 ). The discussion gets much more intuitive when observing that the angle function 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 4H 2 + κ cos 2 (θ) > 0.

9 A HOPF DIFFERENTIAL FOR CMC SURFACES IN S 2 R AND H 2 R 9 Thus by equation (7) the principal curvature s θ has always the same sign as the mean curvature H. In particular, θ is a strictly monotonic function of s. The same holds for the function sin(θ). It remains to analyze whether the image of (s 1, s 2 ) under sin(θ) is the entire interval ( 1, 1) or just some subinterval thereof. If 4H 2 +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 generating curve c is an embedded arc that begins and ends at the boundary component of Īκ R corresponding to the axis l 0. Thus by the argument in the proof of Proposition 2.3 the surface must be a sphere. If 4H 2 + κ 0, the condition 4H 2 + κ cos 2 (θ) > 0 implies that sin(θ) does not vanish anywhere along c. In fact, sin(θ) / [ sin(θ 0 ), sin(θ 0 )] where θ 0 := arcsin( 1 + 4H 2 /κ ), and the function θ(s) is asymptotic to one of the stationary solutions θ 0 or θ 0 of equation (7). This explains the claim about the range of the normal angle θ, and again, reasoning as in the proof of Proposition 2.3, the surface must be homeomorphic to a disk. Moreover, if 4H 2 + κ < 0, the asymptotic normal angle θ 0 is different from 0. In other words, sin θ(s) is bounded away from 0, and thus the radius function r is a regular parameter for the generating curve c as well; the range of this parameter is the entire half axis I κ = [0, ). Finally, equation (7) implies that ( ) ( s θ(s) ± θ0 = κ 4H sin θ0 sin θ(s) ) (sin θ 0 + sin θ(s) ) Hence θ(s) converges exponentially to θ 0 or θ 0, respecticely. Thus the generating curve c has real asymptotes rather than merely some asymptotic slope and the generated surface is asymptotically conical as claimed. On the other hand, if 4H 2 + κ = 0, we find that θ 0 = 0, and θ(s) decays only 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 function r is still a regular parameter along c, and its range is still the entire half axis I κ = [0, ). Remark 2.6. W.-Y. Hsiang has already computed both, the area and the enclosed volume of the embedded cmc spheres S 2 H H2 R. In [20] Pedrosa has extended these computations to embedded cmc spheres S 2 H S2 R and has then used this information to determine candidates for the isoperimetric profiles of the product spaces M 2 κ R. Proposition 2.7. Let Σ 2 M 2 κ R be a connected, rotationally invariant cmc surface that intersects the axis l 0. Then the traceless part of the tensor field q introduced in (1) vanishes identically, and so does the quadratic differential Q of Σ 2. More precisely, (8) q = 2H h Σ κ dξ 2 Σ = ( 2H κ cos2 (θ) ) g where g denotes the induced Riemannian metric on the surface Σ 2. As pointed out in the introduction, it is this simple proposition that has been the key to finding the proper expression for the quadratic differential of cmc surfaces in the product spaces M 2 κ R.

10 10 UWE ABRESCH AND HAROLD ROSENBERG Proof. If H = 0, we consider some point c(s) on the generating curve where the surface intersects the axis l 0. At such a point sn κ (r(s)) = 0, and thus L = 0. It follows from (6) that cos θ(s) 0, and therefore Σ 2 must be one of the totally-geodesic leaves Mκ 2 {ξ 0 } described in Remark 2.2i. For these surfaces equation (8) holds, as both its sides evidently vanish identically. If H 0, we may simplify the expression for the second fundamental form h Σ from (3) using equations (4) and (7): ( H + κ h Σ = 4H ) cos2 (θ) 0 (9) 0 H κ 4H cos2 (θ) Combining this formula with the expression for dξ 2 Σ from (3), we again arrive at equation (8). Remark 2.8. The rotationally invariant cmc spheres S 2 H M 2 κ R of Pedrosa are not convex if 0 < 4H 2 < κ. In fact, the principal curvatures computed in (9) have opposite signs at all points on S 2 H M 2 κ R that are closer to the antipodal axis ˆl 0 than to l 0 itself SO(2) invariant cmc surfaces Σ 2 Mκ 2 R of catenoidal type. In this subsection we are going to describe another family of rotationally invariant cmc surfaces with vanishing quadratic differential Q. For this purpose we study the solutions of (4) with first integral L = 4H/κ. In case κ > 0, the surfaces corresponding to solutions with L = 4H/κ are precisely the surfaces that intersect the antipodal axis ˆl 0 rather than l 0. As explained in Remark 2.4, they are congruent to the spheres that correspond to the solutions with L = 0 and that have been studied by Pedrosa. In case κ < 0, however, the surfaces obtained from the solutions of the ODE system (4) with L = 4H/κ are obviously not congruent to the cmc spheres of Hsiang, anymore. Yet, it is conceivable that they still have Q 0. We shall establish this property in Proposition Using the expression for the first integral from (5), the condition L = 4H/κ reads 0 = κ cos(θ) sn κ (r) + 4H [1 κ sn κ ( 1 2 r)2] In particular, we find that the radius r is bounded away from 0, and hence we may rewrite this equation in a form similar to (6) : (6 cat) κ cos(θ) = 2H ct κ ( 1 2 r) In fact, there is an extremely close relationship between the equations (6) and (6 cat ). For instance 2, either one of them implies that ct κ (r) = 1 ( 2 ctκ ( 1 2 r) κ ct κ( 1 2 r) 1) = H cos 1 (θ) κ 4H cos(θ) 2 Another consequence of either one of equations (6) and (6cat) is that the Pfaffian ( 4H 2 + κ cos 2 (θ) ) dr + 4H sin(θ) dθ vanishes along the corresponding solutions of (4). This in effect explains how to recover (6) and (6 cat) and thus eventually the ODE system (4) from the differential equations (23) when doing the classification in Section 4.

11 A HOPF DIFFERENTIAL FOR CMC SURFACES IN S 2 R AND H 2 R 11 ξ 1 κ H=0.033 κ 0 3 r κ 1 κ H=0.275 κ H=0.121 κ Figure 3. The meridians of the surfaces C 2 H of catenoidal type. Inserting this expression into the third equation in (4), we obtain of course the same differential equation for θ as in the proof of Proposition 2.5 : s θ = 1 ( (7 4H 4H 2 + κ cos 2 (θ) ) cat ) Thus it should not be surprising that most arguments in the proof of Proposition 2.5 can be carried over to this situation. As explained above, we are only interested in studying the case κ < 0. Proposition 2.9. Let κ < 0, and let Σ 2 Mκ 2 R be a connected, SO(2) invariant cmc surface whose generating curve can be described as a solution of (4) with first integral L = 4H/κ. Then the surface is an embedded annulus CH 2 with two asymptotically conical ends. It is generated by rotating a strictly concave curve with asymptotic slopes dr dξ = tan(θ 0) where θ 0 := arccos(2h/ κ). Moreover, the range of θ is the interval ( θ 0, θ 0 ). The name CH 2 has been picked for this family of surfaces, since their shapes resemble the shape of a catenoid or rather the shapes of the analogous surfaces in hyperbolic 3 space. We shall refer to them as the rotationally invariant cmc surfaces of catenoidal type. When H approaches zero, they converge to the double covering of some leaf Mκ 2 {ξ 0 } < 0 with a singularity at the origin. This collapse is indicated in Figure 3; it is pretty similar to the collapse of the minimal catenoids described in [23, Section 6] or [19]. Proof. Since r I κ by construction and since κ < 0 by hypothesis, equation (6 cat ) implies that H cos(θ) > 0 and 4H 2 + κ cos 2 (θ) < 0. Hence the maximal existence interval for the solutions of the differential equation (7 cat ) is the entire real axis. The range of the function s θ(s) is as claimed, and because of the first two equations in (4) all the other assertions are straightforward consequences hereof. Proposition Let κ < 0, and let Σ 2 Mκ 2 R be a connected, SO(2) invariant cmc surface whose generating curve can be described as a solution of (4) with first integral L = 4H/κ. Then the traceless part of the tensor field q introduced in (1) vanishes identically, and so does the quadratic differential Q of Σ 2. More precisely, 2H h Σ κ dξ 2 Σ = ( 2H κ cos2 (θ) ) g

12 12 UWE ABRESCH AND HAROLD ROSENBERG 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 context as well. Moreover, the differential equation for θ obtained in (7 cat ) happens to coincide with (7), as has been observed beforehand. Thus the proof of Proposition 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, we want to compute the derivative of the quadratic differential Q defined in formula (2) in Section 1 and verify that Q indeed vanishes identically. The key ingredients for this computation are, firstly, a formula that expresses the operator in terms of covariant derivatives and, secondly, the Codazzi equations for surfaces Σ 2 in the product spaces M 2 κ R. These are somewhat more complicated than the Codazzi equations for surfaces in space forms. Yet, all this is still essentially standard material, and so we shall summarize the necessary details in Lemmas 3.1 and 3.4 as we go along Geometry and Complex Analysis on Riemann surfaces. In this subsection we recall the basic facts about the complex analytic structure on the surface (Σ 2, g). In particular, we explain how the operator that comes with this complex structure is related to covariant differentiation. Passing to a 2 fold covering if necessary, we may assume that Σ 2 is oriented. Thus there exists a unique almost complex structure J C (End T Σ 2 ) that is compatible with the Riemannian metric g and the given orientation. A celebrated theorem first stated by C.-F. Gauß [10] guarantees the existence of isothermal coordinates. In such a coordinate chart ψ : U Σ 2 C = R 2 the almost complex structure J is given as multiplication by i, and the metric g is of the form e 2λ g 0 for some C function λ: U R ; here we have used g 0 to denote the standard euclidean metric on R 2. The transition functions between such charts are clearly orientation-preserving conformal maps and are thus holomorphic. In other words, the existence of isothermal coordinates turns the surface (Σ 2, J) into a complex 1 dimensional manifold Σ. Its complex tangent bundle T Σ, its cotangent bundle K := T Σ, and all its 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 surface Σ 2 Mκ 2 R is a section in a subbundle of the bundle of complex-valued symmetric bilinear 2 forms on the real tangent bundle of Σ 2, a subbundle that can be canonically identified with K 2. As mentioned above, we want to express the operator in terms of the Levi-Civita connection associated to the Riemannian metric g. The basic link between the real and the complex picture is the fact that on Riemann surfaces J vanishes 3 identically. Lemma 3.1. Let η C (K m ), m Z, be a section in the m-th power of the canonical bundle on a Riemann surface (Σ 2, g, J). Then η can be 3 In higher dimensional case this fact does not hold anymore for an arbitrary complex manifold with a hermitian inner product on the tangent bundle. In fact, the condition J = 0 is one way of saying that (M n, g, J) is Kählerian.

13 A HOPF DIFFERENTIAL FOR CMC SURFACES IN S 2 R AND H 2 R 13 computed in terms of Riemannian covariant differentiation as follows: ( ) η (X) = 1 (10) 2( X η + i JX η ), X C (T Σ 2 ). The lemma is an immediate consequence of a couple of basic facts. Firstly, whenever a holomorphic vector bundle E over some complex manifold is equipped with a hermitian inner product g, there exists a unique metrical connection ˆ such that ( ) η (X) = 1 2( ˆ X η + i ˆ JX η ). This connection is known as the hermitian connection of (E, g). On the tangent bundle of a Kähler manifold the hermitian connection ˆ coincides with the Levi Civita connection (see [24]). The extension of this identity to arbitrary tensor powers is then done using the standard product rules for and ˆ. By definition the operator depends only on the almost complex structure J, i.e. on the conformal structure and the orientation, whereas depends also on the choice of the metric g representing the conformal class. Clearly, this dependence must cancel when taking the particular linear combination of covariant derivatives appearing on the r.h.s. of (10). This observation can be 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 replaced by euclidean differentiation d with respect to that particular isothermal chart. In such a coordinate system the metric is by definition conformal to the euclidean metric g 0, i.e. g = e 2λ g 0, and the Christoffel formula specializes to X Y d X Y = d X λ Y + d Y λ X g 0 (X, Y ) grad g0 λ Because of the product rules for and it is sufficient to handle the case m = 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 the complex cotangent bundle K = T Σ. Thus a straightforward computation based on the Christoffel formula shows that: 2 ( η ) (X; Y ) ( X η Y + i JX η Y ) = (d X η X η) Y + i (d JX η JX η) Y = η( X Y d X Y ) + i η( JX Y d JX Y ) = (d X λ + i d JX λ) η(y ) + d Y λ (η(x) + i η(jx) ) ( g 0 (X, Y ) + i g 0 (JX, Y ) ) η(grad g0 λ) = g 0 (X, grad g0 λ) η(y ) + g 0 (JX, grad g0 λ) η(jy ) g 0 (X, Y ) η(grad g0 λ) g 0 (JX, Y ) η(j grad g0 λ) In order to explain the third equality sign, we simply observe that the coefficient of the factor d Y λ vanishes, as η is of type (1, 0). Picking an orthonormal basis e 1, e 2 = Je 1 such that Y is a multiple of e 1, we find that g 0 (X, Y ) η(grad g0 λ) = g 0 (X, e 1 ) g 0 (e 1, grad g0 λ) η(y ) + g 0 (X, e 1 ) g 0 (e 2, grad g0 λ) η(jy ) g 0 (JX, Y ) η(j grad g0 λ) = g 0 (JX, e 1 ) g 0 (e 1, grad g0 λ) η(jy ) g 0 (JX, e 1 ) g 0 (e 2, grad g0 λ) η(y )

14 14 UWE ABRESCH AND HAROLD ROSENBERG These identities reveal that the last line in the preceding display does indeed vanish Surface theory in the product spaces Mκ 2 R. There are two aspects in regard to which the surface theory in the product spaces Mκ 2 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 are more involved. Except for some inevitable changes caused by these differences, the computation of the derivative Q of the quadratic differential introduced in formulas (1) and (2) follows the same basic pattern as the corresponding computation in the case of space forms. Clearly, Q(Y 1, Y 2 ) = H ( Y 1, AY 2 JY 1, A JY 2 ) ih ( JY 1, AY 2 + Y 1, A JY 2 ) (11) 1 2 κ (dξ(y 1 ) dξ(y 2 ) dξ(jy 1 ) dξ(jy 2 ) ) iκ (dξ(jy 1 ) dξ(y 2 ) + dξ(y 1 ) dξ(jy 2 ) ) 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 bundle K 2 K. This means in particular that Q(X; JY 1, Y 2 ) = Q(X; Y 1, JY 2 ) = i Q(X; Y 1, Y 2 ) Q(JX; Y 1, Y 2 ) = i Q(X; Y 1, Y 2 ) In other words, Q is invariant under some T 3 action, and we are going to arrange the terms in the subsequent computations accordingly. The differential dξ of the height function is clearly a parallel 1 form with respect to the Levi Civita connection D of the target manifold M 2 κ R. However, its restriction is not parallel w.r.t. the induced Levi Civita connection on the surface; one rather has Lemma 3.3. The covariant derivative of the restriction of dξ to any surface Σ 2 M 2 κ R can be expressed in terms of the gradient of the height function ξ, the unit normal field ν of the surface, and its Weingarten map A = D ν as follows 4 : X (dξ) Y = ν, grad ξ X, AY. For intuition, we think of the term dξ(ν) = ν, grad ξ as the sine of some angle θ. Proof. Extending Y to a tangential vector field, a straightforward computation shows that X (dξ) Y = d X ( dξ Y ) dξ ( X Y ) = D X (dξ) Y + dξ ( D X Y X Y ) = D X (dξ) Y + dξ(ν) ν, D X Y = D X (dξ) Y ν, grad ξ X, A Y 4 The symmetric endomorphism A = D ν is sometimes also called the shape operator, and the associated bilinear form is the second fundamental form h Σ.

15 A HOPF DIFFERENTIAL FOR CMC SURFACES IN S 2 R AND H 2 R 15 The lemma follows when taking into account that the vector field grad ξ is globally parallel, i.e. that D X (grad ξ) and D X (dξ) vanish. Differentiating formula (11) using Lemmas 3.1 and 3.3, we find that (12) Q(X; Y 1, Y 2 ) = H T 1 (X; Y 1, Y 2 ) + κ ν, grad ξ T 2 (X; Y 1, Y 2 ) where T 1 (X; Y 1, Y 2 ) := 1 2 [ Y 1, X A Y 2 JY 1, X A JY 2 + JY 1, JX A Y 2 + Y 1, JX A JY 2 ] 1 2 i [ JY 1, X A Y 2 + Y 1, X A JY 2 Y 1, JX A Y 2 + JY 1, JX A JY 2 ], T 2 (X; Y 1, Y 2 ) := 1 4 [ Y 1, AX dξ(y 2 ) + dξ(y 1 ) AX, Y 2 JY 1, AX dξ(jy 2 ) dξ(jy 1 ) AX, JY 2 + JY 1, A JX dξ(y 2 ) + dξ(jy 1 ) A JX, Y 2 + Y 1, A JX dξ(jy 2 ) + dξ(y 1 ) A JX, JY 2 ] 1 4 i [ JY 1, AX dξ(y 2 ) + dξ(jy 1 ) AX, Y 2 + Y 1, AX dξ(jy 2 ) + dξ(y 1 ) AX, JY 2 Y 1, A JX dξ(y 2 ) dξ(y 1 ) A JX, Y 2 + JY 1, A JX dξ(jy 2 ) + dξ(jy 1 ) A JX, JY 2 ] Clearly, T 1 is just four times the (2, 1) part of A. It is the same expression that appears when proving that the standard Hopf differential for cmc surfaces in space forms Mκ 3 is holomorphic. It will be evaluated in Lemma 3.5 using the Codazzi equations. The other term, however, is new. Since in dimension 2 the traceless part A 0 of the Weingarten map anticommutes with the almost complex structure J, it follows that the tensor T 2 depends only on the mean curvature H but not on the traceless part A 0 := A H id of the Weingarten map A. Thus (13) T 2 (X; Y 1, Y 2 ) = H T red 2 (X; Y 1, Y 2 ) where (14) T red 2 (X; Y 1, Y 2 ) = 1 2 [ Y 1, X dξ(y 2 ) + dξ(y 1 ) X, Y 2 JY 1, X dξ(jy 2 ) dξ(jy 1 ) X, JY 2 ] 1 2 i [ JY 1, X dξ(y 2 ) + dξ(jy 1 ) X, Y 2 + Y 1, X dξ(jy 2 ) + dξ(y 1 ) X, JY 2 ] The expression on the r.h.s. in this formula is actually the simplest tensor of type (2, 1) that can be constructed from.,. dξ and that is symmetric w.r.t. interchanging Y 1 and Y 2.

16 16 UWE ABRESCH AND HAROLD ROSENBERG Lemma 3.4. The Codazzi equations for surfaces Σ 2 M 2 κ R are: X A Y Y A X, Z = R D (X, Y )ν, Z = κ ν, grad ξ ( Y, Z dξ(x) X, Z dξ(y ) ) Here R D denotes the Riemannian curvature tensor of the product space M 2 κ 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 (M 2 κ R) is given by K(span{X, Y }) = X Y 2 R D (X, Y )Y, X. Proof. In principle the first half of the claimed formula is well-known. However, in order to make sure that the sign of the curvature term is correct, it is easiest to include the short computation. Working locally, we may assume that Σ 2 is embedded, and thus we may extend X, Y, and ν as smooth vector fields onto a neighborhood of the surface: X A Y Y A X, Z = X (A Y ) Y (AX) A [X, Y ], Z = D X (A Y ) D Y (AX) A [X, Y ], Z = D X (D Y ν) D Y (D X ν) D [X,Y ] ν, Z = R D (X, Y )ν, Z The second equality sign in the claimed formula is obtained upon computing the curvature tensor R D for the product manifolds M 2 κ R. Indeed, the bilinear form g dξ 2 represents the metric on the leaves M 2 κ {ξ 0 }; hence R D = κ (g dξ 2 ) (g dξ 2 ) = κ (g g 2 g dξ 2 ) Like in the case of space forms, the tensor g g vanishes on the relevant combinations of arguments, and therefore R D (X, Y )ν, Z = 2κ g dξ 2 (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 Σ 2 M 2 κ R the tensor field T 1 introduced in the context of formula (12) can be evaluated as follows: T 1 (X; Y 1, Y 2 ) = κ ν, grad ξ T red 2 (X; Y 1, Y 2 ) where T red 2 is the field introduced in (14). Proof. On cmc surfaces it is evident that the tensor w A is traceless and therefore anticommutes with the almost complex structure J for any vector w T Σ 2. We conclude that the real part of T 1 can be rewritten as R T 1 (X; Y 1, Y 2 ) = Y 1, X A Y 2 + JY 1, JX A Y 2, and that the sum Y 1, Y2 A X + JY 1, Y2 A JX vanishes for all vectors X, Y 1, and Y 2. Subtracting this sum from the expression for R T 1, we may use the Codazzi equations from the Lemma 3.4 in order to evaluate the

17 A HOPF DIFFERENTIAL FOR CMC SURFACES IN S 2 R AND H 2 R 17 differences Y 1, X A Y 2 Y2 A X and JY 1, JX A Y 2 Y2 A JX. Hence we find that R T 1 (X; Y 1, Y 2 ) = κ ν, grad ξ [ Y 1, Y 2 dξ(x) + JY 1, Y 2 dξ(jx) 2 X, Y 1 dξ(y 2 ) ] By construction, T 1 is a tensor field of type (2, 1). Firstly, this implies that T 1 (X; Y 1, Y 2 ) = T 1 (X; JY 1, JY 2 ), and thus R T 1 (X; Y 1, Y 2 ) = 1 [ 2 R T1 (X; Y 1, Y 2 ) R T 1 (X; JY 1, JY 2 ) ] = κ ν, grad ξ [ X, JY 1 dξ(jy 2 ) X, Y 1 dξ(y 2 ) ] Moreover, the field T 1 is clearly symmetrical w.r.t. the permutation of Y 1 and Y 2, and thus we may symmetrize the r.h.s. accordingly. We conclude that R T 1 = κ ν, grad ξ R T red 2, as claimed. Referring to the invariance properties of type (2, 1) tensors one more time, we can deduce that the imaginary parts are equal, too. Proof of Theorem 1. Combining equations (12) and (13), one obtains Q = H [ T 1 + κ ν, grad ξ T2 red ], 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 might appear at first: The tensors T 1, T 2, and T2 red are the (2, 1) parts of basic geometric objects like A or the tri-linear form.,. dξ. When applying the Codazzi equations to T 1, we evidently obtain a tensor of type (2, 1) that is a sum of terms of the form κ ν, grad ξ.,. dξ. The upshot of such structural considerations is that Q must be a multiple of κh ν, grad ξ T2 red with some factor that is a universal constant. There is also an independent argument that this constant factor must indeed be zero. One simply considers the rotationally invariant cmc spheres SH 2 M κ 2 R of Hsiang and Pedrosa and observes that their quadratic differential Q vanishes identically. Therefore Q 0, too. On the other hand, however, neither the function κh ν, grad ξ nor the field T2 red vanish on any set of positive measure in SH 2 M κ 2 R. 4. Cmc surfaces with vanishing quadratic differential Q Our goal is to classify the cmc surfaces Σ 2 M 2 κ R with vanishing quadratic differential Q, unless H and κ vanish simultaneously. By the very definition of the quadratic differential Q in equations (1) and (2) we shall in effect classify complete surfaces in M 2 κ R such that (15) 2H h Σ κ dξ dξ Σ = ( 2H κ dξ Σ 2) g At this point it gets very clear why we have to exclude the case that H and κ vanish simultaneously; the preceding condition holds trivially despite the fact that there is an ample supply of interesting minimal surfaces in euclidean 3 space. The remainder of the minimal surface case on the other hand is easy: if H = 0 and κ 0, it follows directly from equation (15) that dξ dξ Σ = 0, and thus dξ Σ vanishes identically. Therefore Σ 2 must be a totally-geodesic

18 18 UWE ABRESCH AND HAROLD ROSENBERG leaf Mκ 2 {ξ 0 }. These considerations tie in nicely with the fact that by general theory the height function ξ has to be harmonic, provided that Σ 2 is compact. From now on we shall concentrate on the case H 0. In this case equation (15) can be solved for the second fundamental form h Σ. Clearly, for any surface Σ 2 Mκ 2 R, the vector fields (grad ξ) tan := grad ξ ν, grad ξ ν and 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 and that, moreover, these level curves have constant curvature. The proper way to formalize the treatment of equation (15) is to prolong the system once and interpret (15) as a problem about integral surfaces of a suitable distribution E H in the unit tangent bundle of Mκ 2 R. This prolongation will be defined in 4.1. In 4.2 we reap the easy consequences that are implied by the still fairly large isometry groups of the product spaces Mκ 2 R in the presence of the rotationally invariant examples described in Section 2. The properties of the distribution E H will then be analyzed in full detail in 4.3 based on the simple observations described in the preceding paragraph. The argument culminates in the proof of Theorem 3 on page The Gauss section of cmc surfaces with Q 0. In order to handle equation (15) in the non-minimal case in a conceptually nice way, we find it better not to work with the immersion F : Σ 2 M 2 κ R itself, but to think of the unit normal field ν as the primary unknown. It will be easiest to consider the Gauss section ν simply as an immersion of Σ 2 into the total space N 5 := T 1 M 3 of the unit tangent bundle of the manifold M 3 := M 2 κ R. Clearly, F = π ν where π : N 5 = T 1 M 3 M 3 denotes the standard projection map. Thus the immersion F can be recovered, once its lift ν : Σ 2 N 5 is known. It is well-known that any affine connection on a manifold M can be understood as a vector bundle isomorphism T T M π (T M T M), which identifies the bi-tangent bundle T T M with a more familiar vector bundle over T M. In particular, the Levi Civita connection D on M 3 = M 2 κ R induces an injective bundle map (16) Φ D : T N 5 π (T M 3 T M 3 ) such that Φ D ( s ν(s)) = ( s (π ν(s), D s ν(s)) for any smooth curve s ν(s) N 5. The image of this map is the 5 dimensional subbundle given by Φ D ( Tv N 5) = { (w 1, w 2 ) T π(v) M 3 T π(v) M 3 v, w 2 = 0 } With these preparations it is now possible to translate the classification problem for cmc surfaces in M 3 := Mκ 2 R with mean curvature H 0 and vanishing quadratic differential Q into the classification problem for integral surfaces of some 2 dimensional distribution E H T N 5. Proposition 4.1. Let Σ 2 M 3 = Mκ 2 R be a cmc surface with H 0 and vanishing quadratic differential Q. Then its Gauss section ν : Σ 2 N 5 := T 1 M 3 is necessarily an integral surface of the 2 dimensional distribution E H T N 5 given by ( ) { Φ D (EH ) v = (w, Av w) w v }

19 A HOPF DIFFERENTIAL FOR CMC SURFACES IN S 2 R AND H 2 R 19 where A v w := κ 2H w, (grad ξ)tan (grad ξ) tan + [ H κ 4H ( 1 v, grad ξ 2 )] (w v, w v ) for all w T π(v) M 3 and where (grad ξ) tan := grad ξ v, grad ξ v. Conversely, for any number H 0 an integral surface of the distribution E H necessarily projects onto a surface Σ 2 M 2 κ 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 E H with H 0 are integrable. Proof. Each unit vector v clearly lies in the kernel of the corresponding symmetric endomorphism A v introduced in the proposition. Moreover, the restriction of A v to v is precisely the tensor that one obtains when solving equation (15) for the Weingarten map of the surface Σ 2 Mκ 2 R, provided that v is chosen to be the unit normal vector ν at the point under consideration. Thus the proposition immediately follows when reading this expression with the Weingarten equation A = D ν and with the definition of the isomorphism Φ D in mind Symmetries of the distributions E H. In many cases the integral surfaces of the distribution E H T N 5 can be determined quite easily using the invariance properties of E H, and the symmetry properties themselves can be established without much effort, too. In fact, there is an induced action G N 5 N 5 of the 4 dimensional Lie group G := Isom 0 (M 2 κ R) on the unit tangent bundle N 5 := T 1 (M 2 κ R). Since the vector field grad ξ is even invariant under all isometries of M 2 κ R that preserve the orientation of the second factor, it is clear that the function (17) Θ: N 5 [ 1 2 π, 1 2 π] v arcsin ( v, grad ξ ) is invariant under the action of G on N 5. Since the isotropy group G x at any point x Mκ 2 R acts on T x Mκ 2 R as the group of rotations preserving (grad ξ) x, it follows that the function Θ separates the G orbits in N 5. The fibers over 1 2 π and 1 2π correspond to 3 dimensional singular orbits with isotropy group isomorphic to SO(2), whereas all other fibers of Θ correspond to regular orbits; the principal isotropy group of the G action on N 5 is trivial. Lemma 4.2. For any H 0 the distribution E H T N 5 introduced in Proposition 4.1 is invariant under the action of G = Isom 0 (Mκ 2 R). Proof. We may think of v as a G invariant section of π T M 3 where π denotes the canonical projection N 5 = T 1 M 3 M 3 = Mκ 2 R. From this point of view, the fields (grad ξ) tan and A v are G invariant sections of π T M 3 and π End(T M 3 ), respectively. Proposition 4.3. Suppose that 4H 2 + κ > 0 and H 0. Then the distribution E H T N 5 introduced in Proposition 4.1 is integrable, and all its

20 20 UWE ABRESCH AND HAROLD ROSENBERG integral surfaces are congruent to Gauss sections of the embedded rotationally invariant cmc spheres S 2 H M 2 κ R. This proposition actually proves the part of the Theorem 3 about the classification in the case that 4H 2 + κ > 0. Proof. Let v 0 N 5 be arbitrary. The monotonicity properties established in the proof of Proposition 2.5i show that there exists some point c(s 0 ) on the generating curve of the sphere SH 2 M κ 2 R such that Θ(v 0 ) = θ(s 0 ) = Θ(ν(s 0 )). Thus there exists an isometry ψ G = Isom 0 (Mκ 2 R) that maps the point c(s 0 ) to the foot point π(v 0 ) and ν(s 0 ) to v 0 itself. By Proposition 2.7 the quadratic differential Q of the sphere SH 2 vanishes. Thus Proposition 4.1 implies that the Gauss sections of SH 2 and of all its isometric images are integral surfaces of the distribution E H, and the Gauss section of ψ(sh 2 ) is an integral surface through the given point v 0 N 5. Proposition 4.4. Suppose that 4H 2 +κ 0 and H 0. Then the distribution E H T N 5 introduced in Proposition 4.1 has complete non-compact integral surfaces through any point v 0 N 5 such that 4H 2 +κ cos 2 (Θ(v 0 )) 0. These integral surfaces are congruent to the Gauss sections of (i) one of the non-compact cousins DH 2 M κ 2 R of the rotationally invariant cmc spheres SH 2 M κ 2 R, or, (ii) one of the embedded, rotationally invariant cmc surfaces CH 2 M κ 2 R of catenoidal type. The first case occurs iff 4H 2 + κ cos 2 (Θ(v 0 )) > 0, whereas the second case occurs iff 4H 2 + κ cos 2 (Θ(v 0 )) < 0. Proof. The argument is essentially the same as in the proof of the preceding proposition. Note that by hypothesis 4H 2 + κ cos 2 (Θ(v 0 )) is non-zero. Looking at the ranges of the angle function s θ(s) that have been determined in Propositions 2.5ii and 2.9, we see that there exists a point c(s 0 ) on either the generating curve of DH 2 or the generating curve of C2 H such that Θ(v 0 ) = θ(s 0 ) = Θ(ν(s 0 )). Thus there exists an isometry ψ G = Isom 0 (Mκ 2 R) that maps the point c(s 0 ) to the foot point π(v 0 ) and ν(s 0 ) to v 0 itself. By Propositions 2.7 and 2.10 the quadratic differentials of both kinds of surfaces vanish identically, and so we have identified the integral surface through v 0 as the Gauss section of DH 2 or C2 H, respectively. The preceding proposition, however, does not finish the proof of Theorem 3. This is so despite the fact that we have identified the integral surfaces of the distribution E H through an open dense set of points v N 5. It remains to study the distribution E H on the hypersurface in N 5 given by the equation 4H 2 + κ cos 2 Θ = 0. As we shall see in the next subsection, the integral surfaces of E H that are contained in this hypersurface are not congruent to the Gauss section of any rotationally symmetrical cmc surface in M 2 κ R Integral surfaces of E H. In this subsection we shall determine all integral surfaces of the 2 dimensional distribution E H in the unit tangent bundles N 5 of the product spaces M 3 = M 2 κ R in a systematic way.

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