Free boundary CMC annuli in spherical and hyperbolic balls

Calc. Var. (2025) 64:37 https://doi.org/10.1007/s00526-024-02883-6 Calculus of Variations Free boundary CMC annuli in spherical and hyperbolic balls Alberto Cerezo1,2 ·Isabel Fernández2·Pablo Mira3 Received: 15 March 2024 / Accepted: 30 October 2024 © The Author(s) 2024 Abstract We construct, for any H∈R, infinitely many free boundary annuli in geodesic balls of S3 with constant mean curvature Hand a discrete, non-rotational, symmetry group. Some of these free boundary CMC annuli are actually embedded if H≥1/√3. We also construct embedded, non-rotational, free boundary CMC annuli in geodesic balls of H3, for all values H>1 of the mean curvature H. Mathematics Subject Classification 53A10 ·53C42 1 Introduction and statement of the results 1.1 Historical introduction For a long time, it was believed that the only closed, i.e., compact without boundary, surfaces with constant mean curvature (CMC) in Euclidean space R3were the round, totally umbilic spheres. Hopf [16] proved in 1951 the validity of this statement for the particular case of genus zero. However, the conjecture was unexpectedly solved in the negative by Wente [34] in 1986, by constructing CMC tori in R3with self-intersections and a discrete symmetry group. Subsequently, Abresch [1]andWalter[30,31] gave a more explicit construction, by prescribing that the CMC tori were foliated by planar curvature lines. A fundamental achievement of CMC theory springing from these works was the classification of all CMC tori in the space forms R3,S3and H3by Pinkall-Sterling [26](inR3), Hitchin [15] (for minimal tori in S3) and then by Bobenko [5] (for CMC tori in S3and H3), Communicated by Manuel del Pino. BPablo Mira [email protected] Alberto Cerezo [email protected] Isabel Fernández [email protected] 1Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain 2Departamento de Matemática Aplicada I, Instituto de Matemáticas IMUS, Universidad de Sevilla, Sevilla, Spain 3Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain 0123456789().: V,-vol 123 37 Page 2 of 44 A. Cerezo et al. using algebro-geometric methods from integrable systems. In these theorems, CMC tori were described in connection with finite type solutions of the sinh-Gordon equation. Roughly, for any natural N≥1, the space of type Nsinh-Gordon solutions is finite dimensional, all CMC tori are of finite type, and they can be detected within their associated finite dimensional space by explicit closing conditions. This classification did not detect the possible embeddedness of CMC tori in S3(in R3and H3there are no closed embedded CMC surfaces, by Alexandrov’s theorem). The fundamental achievement in this direction was obtained more recently by Brendle [6],whoprovedthatthe Clifford torus is the only embedded minimal torus in S3, thereby solving in the affirmative a long-standing conjecture by Lawson. The idea in [6] was then adapted by Andrews and Li [2] to show that any embedded CMC torus in S3is rotational; this solved a conjecture by Pinkall-Sterling. There is a natural boundary version of the classification problem for CMC tori discussed above: to classify all free boundary CMC annuli in geodesic balls of M3(ε) =R3,S3or H3. Here, we say that a compact CMC surface has free boundary in a geodesic ball B⊂M3(ε) if ⊂Band intersects ∂Borthogonally along ∂. This problem was already considered by Nitsche [25] in 1985. The solutions are critical points associated to a natural variational problem for the area funcional, see [25,28]. Nitsche proved in [25] that any free boundary CMC disk in B⊂R3is totally umbilic. Ros and Souam [28,29] extended this result to S3and H3.In[25], Nitsche claimed without proof that any free boundary CMC annulus in a ball B⊂R3should be rotational. This claim was proved incorrect by Wente in 1995 [35], through the construction of immersed free boundary CMC annuli with very large mean curvature in the unit ball of R3. However, two questions remained: the existence of non-rotational free boundary minimal annuli and of embedded free boundary CMC annuli in the unit ball. The first question was recently answered by Fernández, Hauswirth and Mira in [11], where they constructed immersed free boundary minimal annuli in the unit ball. For that, they used the Weierstrass representation, and a study of minimal surfaces in R3with spherical curvature lines; later on, Kapouleas-McGrath [17] presented an alternative, more analytical construction via doubling. The second question has also been recently answered by the authors in [8], by constructing embedded non-rotational free boundary CMC annuli in the unit ball of R3, through a study of an overdetermined system for the sinh-Gordon equation. This answered an open problem posed by Wente in 1995 [35]. These results still leave unsolved the critical catenoid conjecture, i.e., that any embedded free boundary minimal annulus in the unit ball of R3should be the critical catenoid. See e.g. [12,13,20]. This conjecture is conceived as the free boundary version of the aforementioned Lawson conjecture for minimal tori in S3. The problem of whether the only embedded free boundary minimal annulus in a geodesic ball of S3is a (spherical) critical catenoid has been studied in two interesting recent works by Lima-Menezes [22] and Medvedev [23]. In [22], the uniqueness of this critical catenoid is obtained among immersed annuli by imposing that its coordinate functions are first eigenfunctions of a suitable Steklov problem. In [23] it is shown that the Morse index of the critical catenoid is 4, that its spectral index is 1, and that any free boundary minimal annulus with spectral index 1 is a (spherical) critical catenoid. See e.g. [3,7,14,21,24,29,32] for other results on free boundary CMC surfaces in spherical or hyperbolic balls. 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 3 of 44 37 1.2 Main results Our aim in this paper is to show that there exist many non-rotational free boundary CMC annuli in geodesic balls of S3and H3, which in many cases are actually embedded. In particular, this shows that the class of minimal annuli in S3considered in [22,23] is nontrivial, and that the classification of all free boundary CMC annuli in geodesic balls of space forms is a very rich geometric problem, both in the immersed and embedded cases. Our main results are described in Theorems 1.1 and 1.2 below. They indicate that: (1) In S3there exist immersed, non-rotational free boundary H-annuli in geodesic balls B⊂S3,foranyH∈R. In particular, there exist free boundary minimal annuli in S3 with a finite symmetry group. This answers a problem in [23]. (2) For H≥1/√3, some of these free boundary CMC annuli in S3are embedded. (3) In H3, there exist embedded non-rotational free boundary H-annuli in geodesic balls B⊂H3,foranyH>1. (4) All these H-annuli come in 1-parameter families, and are foliated by spherical curvature lines. They can be seen as free boundary bifurcations from finite covers of adequate rotational examples (nodoids in H3, catenoids or nodoids in S3). We will also show that there exist embedded, non-rotational, capillary minimal annuli in geodesic balls of S3. We should note that the analytic results by Kilian and Smith in [18] prove that any free boundary CMC annulus in a geodesic ball of R3,S3or H3is associated to a finite type solution of the sinh-Gordon equation. The spherical curvature lines condition of our examples is very natural in this context, since they correspond to solutions of type N=2, see [26,27,33]. The study of CMC surfaces in R3with spherical curvature lines dates back to classical works by differential geometers of the 19th century, like Enneper, Dobriner or Darboux. See [30,31, 33] for more modern approaches, and [4,9] for studies on isothermic surfaces with spherical curvature lines. In the next theorems, we let H≥0andε∈{−1,1}so that H2+ε>0, and denote M3(1)=S3⊂R4and M3(−1)=H3⊂L4. Theorem 1.1 There exists an open interval J=J(H,ε) contained in (0,1)such that, for any irreducible q =m/n∈J∩Q, there exists a real analytic 1-parameter family Fq:= {Aq(η) :η∈[0, 0(q))}of compact annuli in M3(ε) with the following properties: (1) Each annulus Aq(η) has constant mean curvature H, and has free boundary in a geodesic ball B=B(q,η)⊂M3(ε) centered at e4=(0,0,0,1)∈M3(ε). (2) Aq(η) is symmetric with respect to the totally geodesic surface S:= M3(ε) ∩{x3=0}. (3) The closed geodesic S∩Aq(η) of Aq(η) has rotation index −minS. (4) Aq(0)is a finite m-cover of a compact embedded piece of a Delaunay surface in M3(ε). (5) If η>0,thenAq(η) is not rotational, and its symmetry group is generated by the symmetry with respect to S, and by the symmetries with respect to n >1equiangular totally geodesic surfaces of M3(ε) orthogonal to S. That is, the symmetry group of Aq(η) is prismatic of order 4n. (6) Each annulus Aq(η) is foliated by spherical curvature lines, so that both of its boundary components are elements of this foliation. Theorem 1.2 Assume in Theorem 1.1 that ε=−1or that ε=1and H ≥1/√3. Then, there exist elements in J∩Qof the form q =1/n, and for any such q, the free boundary H-annuli Aq(η) are embedded, for ηsufficiently small. When ε=−1, it actually holds 1/n∈Jfor any n ≥2. 123 37 Page 4 of 44 A. Cerezo et al. 1.3 Organization of the paper and sketch of the proof The strategy to prove Theorems 1.1 and 1.2 is inspired by our previous works [8,11]on free boundary CMC annuli in R3. We will start by constructing a family of immersions ψ(u,v):R2→M3(ε) with constant mean curvature H(with H>1ifε=−1), depending on three real parameters (a,b,c), obtained through special solutions of the sinh-Gordon equation. The main property of any such immersion ψis that, along each curve v→ ψ(u0,v), it intersects with a constant angle θ(u0)some totally umbilic surface Q(u0). Moreover, it also satisfies ψ(−u,v) =(ψ(u,v)),wheredenotes the symmetry with respect to the totally geodesic surface M3(ε) ∩{x3=0}. The proof of Theorems 1.1 and 1.2 is then based on proving that there exist parameter values (a,b,c)such that: (1) ψis periodic in the v-direction, and so ψcovers an annulus. (2) There exists u0>0 such that θ(u0)=π/2andQ(u0)is a sphere invariant by . (3) The annulus ψ([−u0,u0]×S1)⊂M3(ε) lies in the geodesic ball Bof M3(ε) bounded by Q(u0), and so, because of the properties above, it is a free boundary annulus in B. The idea of the proof will be to bifurcate from free boundary rotational examples (critical catenoids or nodoids) within the family associated to the (a,b,c)-parameters, to create some non-rotational examples, and to control their embeddedness. Let us remark that there appear however several sources of complication in the process when we consider S3or H3instead of R3as our ambient space, and their resolution requires new ideas. For instance, we cannot use the Weierstrass representation of minimal surfaces as in [11]. And, in contrast with the R3case in [8], in our spherical or hyperbolic setting we do not have an explicit expresion for the period map that controls the periodicity of the spherical curvature lines of our examples. Moreover, we cannot use CMC surfaces with planar curvature lines as a limit in order to control the centers of the spherical curvature lines, as we did in [8] for the Euclidean case, since such surfaces do not exist in S3or H3. We next explain the basic steps of the proof. In Sect.2we present some preliminaries on CMC surfaces in space forms foliated by spherical curvature lines, i.e. each curvature line of this foliation lies in some 2-dimensional totally umbilic surface of the ambient space. In Sect.3we recall our construction in [8] of some special solutions to an overdetermined problem for the sinh-Gordon equation. When this equation is viewed as the Gauss equation of a CMC surface in M3(ε) =S3or H3parametrized by curvature lines (with H>1in H3), these solutions yield complete CMC surfaces in M3(ε) foliated by spherical curvature lines. Along such curvature lines, the surface intersects the corresponding totally umbilic surface at a constant angle, by Joachimstal’s theorem. In this way, we end up with a family of conformal CMC immersions ψ(u,v):R2→M3(ε) paramerized by curvature lines, which depends on three parameters (a,b,c), and so that the curves v→ ψ(u,v)are spherical. In Sect.4we study the geometry of the immersions ψ(u,v). We prove that they are symmetric with respect to a horizontal totally geodesic surface S⊂M3(ε) and with respect to a number of vertical totally geodesic surfaces k⊂M3(ε) that, in adequate conditions, intersect along a vertical geodesic of M3(ε) that contains all the centers of the totally umbilic surfaces where the spherical curvature lines v→ ψ(u,v)of the immersion lie. We also define a real analytic period map (a,b,c)with the property that, when (a,b,c)=m/n∈Q, the spherical curvature lines v→ ψ(u,v) are periodic curves with rotation index m, while ngives the number of vertical symmetry surfaces k. In particular, when (a,b,c)=m/n, the restriction of ψto [−u0,u0]×Rcovers a CMC annulus 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 5 of 44 37 0=0(a,b,c,u0)in M3(ε) that intersects with a constant angle two (isometric) totally umbilic surfaces of M3(ε). Thus, our objetive will be to show that along some real analytic curves in the parameter space (a,b,c,u0), the resulting CMC annuli 0are free boundary in some geodesic ball B⊂M3(ε). That is, we will need to control simultaneously that both boundary curves of 0 lie in the same totally umbilic surface of M3(ε), that this surface bounds a compact geodesic ball B⊂M3(ε),that0is totally contained in B, and that the intersection angle along ∂0 with ∂Bis π/2. All of this while controlling the possible embeddedness of 0. To achieve this, the main idea will be to bifurcate from some rotational free boundary CMC surface, so that the spherical curvature lines condition is preserved. For this, we will need a quite detailed description, that will be carried out in Sect.5, on the existence of critical (spherical) catenoids in S3,andcritical nodoids in geodesic balls of S3and H3. Section 5is essentially independent from the rest of the paper, and the proofs there are postponed to an appendix. In Sect.6we will show that, when a=1, the immersion ψ(u,v)parametrizes a compact piece of a rotational CMC surface. More specifically, of either a nodoid (in H3), or a nodoid, a (spherical) catenoid or a flat torus in S3. The parameter cwill control the necksize of this example. The parameter bis needed to account for the fact that, in the rotational case, each curve v→ ψ(u,v)is a circle, and so there is a priori an infinite number of totally umbilic surfaces in M3(ε) that contain it. The parameter bdetermines a choice for such umbilic surfaces. In Sect.7we will give an explicit expression for the period map (1,b,c)in the case a=1 that allows for a good control on the periodicity of the parametrized curvature lines v→ ψ(u,v), which in this a=1 case are merely circles. In Sect.8we will restrict to a certain free boundary region of our parameter space, and control there, also for the case a=1, the situation in which 0(1,b,c,u0)covers a critical catenoid or nodoid in M3(ε). We will show that these rotational free boundary surfaces must appear along certain curves of the parameter domain in that a=1 case. In Sect.9we prove Theorems 1.1 and 1.2. For that we use our study of the rotational a=1 case in the previous sections in what regards the free boundary annulus structure of the examples, and induce it to the non-rotational a>1 case. The embeddedness is obtained for values of the period (a,b,c)of the form −1/n, with a>1 close to 1. Finally, in Sect. 10 we use some of the analysis of the previous sections to construct embedded capillary CMC surfaces in geodesic balls of S3, for all values of H. The authors are grateful to the referee of this paper, for valuable comments that helped improving the exposition of the results. 1.4 Open problems It is very natural to conjecture that the (spherical) critical catenoids are the only embedded free boundary minimal annuli in geodesic balls of S3;see[22,23]. Our results imply that the embeddedness assumption cannot be removed from this conjecture. More generally, aligned with the theorems in [8,11] for the Euclidean case, one can conjecture that any embedded free boundary CMC annulus in a geodesic ball of R3,S3or H3 should be foliated by spherical curvature lines. We note that the existence of an embedded free boundary minimal annulus with spherical curvature lines in a geodesic ball of S3is also open (in R3, the results in [11] show that these do not exist). 123 37 Page 6 of 44 A. Cerezo et al. In the immersed case, it would be interesting to construct a free boundary CMC annulus in a geodesic ball that is not foliated by spherical curvature lines, or to show that such an example cannot exist. We also do not know if there exist continuous deformations of free boundary CMC annuli in a fixed geodesic ball of R3,S3or H3, with a fixed mean curvature H. 2 Preliminaries Let M3(c0)denote the space form of constant curvature c0∈R. In the case c0= 0, we view M3(c0)in the usual way as a hyperquadric of R4 ε=(R4,,), ,=dx2 1+dx2 2+dx2 3+εdx2 4, where εis the sign of c0.Thatis,R4 εis either the Euclidean 4-space (if ε=1) or the Lorentz-Minkowski space L4(if ε=−1) and M3(c0)={x∈R4 ε:x,x=1/c0}, with x4>0ifε=−1. Let denote an immersed oriented surface in M3(c0)with constant mean curvature H∈R.LetNdenote the unit normal vector field of in M3(c0).Letζ:= u+ivdenote a conformal parameter for , so that its first fundamental form is I=e2ω(du2+dv2). Then, the Codazzi equation gives that the Hopf differential Q:= ψζζ,Nis holomorphic, and the Gauss equation for in the (u,v)-parameters is ω +(H2+c0)e2ω−4|Q|2e−2ω=0.(2.1) Assume that is simply connected and does not have umbilical points. Then, after a change of conformal parameter, we can assume that Qis constant. In that way, the Gauss equation (2.1)isoftheform ω +Ae2ω−Be−2ω=0,A,B∈R,B>0.(2.2) Conversely, let ω(u,v) :R2→Rsatisfy (2.2) with respect to constants A,B,andlet H,c0,Q∈Rbe constants, with Q= 0, so that A=H2+c0,B=4|Q|2(2.3) hold. Then, there exists an immersion ψ(u,v):R2→M3(c0) with constant mean curvature H, Hopf differential Q, and whose first and second fundamental forms are I=e2ω(du2+dv2), II =(He2ω+2Q)du2+(He2ω−2Q)dv2(2.4) The principal curvatures κ1>κ 2of ψare κ1=H+2|Q|e−2ω,κ 2=H−2|Q|e−2ω.(2.5) The smallest principal curvature κ2corresponds to the principal u-curves (resp. v-curves) if Q<0 (resp. Q>0). This surface is unique up to orientation preserving ambient isometries, or equivalently, up to prescribing the moving frame {ψ, ψu,ψ v,N}at some point (u0,v 0). Here, Nis the 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 7 of 44 37 unit normal of associated to the parametrization ψ. The Gauss-Weingarten formulas of ψ are ψuu =ωuψu−ωvψv+(He2ω+2Q)N−c0e2ωψ ψuv=ωvψu+ωuψv ψvv =−ωuψu+ωvψv+(He2ω−2Q)N−c0e2ωψ Nu=−(H+2Qe−2ω)ψu Nv=−(H−2Qe−2ω)ψv (2.6) We next analyze the property that a curvature line v→ ψ(u,v) is spherical, i.e. it lies in a totally umbilical surface of M3(c0). The totally umbilical surfaces of M3(c0)are given by the intersection of hyperplanes of R4 εwith M3(c0). They can be described by S[m,d]:={x∈M3(c0):x,m=d},(2.7) for some m∈R4 ε\{0}and d∈R(here mand dare determined up to a common multiplicative factor). The condition for S[m,d]being a (non-empty) surface is that m,m−c0d2>0 and that, if ε=−1andm,m≥0, dand the x4-coordinate of mhave opposite signs. When ε=−1(c0<0),wehavethatS[m,d]is a sphere (resp. horosphere, pseudosphere) in H3(c0)if m,mis negative (resp. zero, positive). If d=0, then S[m,d]is totally geodesic in M3(c0). Lemma 2.1 For each fixed u ∈R,thev-curvature line ψ(u,v)of ψis spherical if and only if there exist α(u), β(u)∈Rsuch that 2ωu=α(u)eω+β(u)e−ω.(2.8) In that situation, ψintersects S[m(u), d(u)]at a constant angle θ(u)along v→ ψ(u,v) and we have α=2| N|Hcos θ−c0d | N|sin θ,β=−4Qcos θ sin θ,(2.9) where | N|=m,m−c0d2.In particular, β=0if and only if cos θ=0, and α=β=0 if and only if cos θ=0and S[m,d]is totally geodesic in M3(c0). Proof If for a fixed u=u0the curve v→ ψ(u0,v)lies in S[m,d],thenm,ψ v=0. One sees from here and m,ψ=dthat  N:= m−c0dψ lies in TM3(c0)and is normal to S[m,d]along ψ(u0,v),since N,t=0foreveryt∈R4 ε orthogonal to both ψand m. Defining θby N, N=|  N|cos θ, one easily obtains that, changing θby −θif necessary, m=e−ω| N|sin θψ u+| N|cos θN+c0dψ. (2.10) Using this expression together with (2.6)andψvv,m=0, we obtain (2.8)forα= α(u0), β =β(u0)given by (2.9). 123 37 Page 8 of 44 A. Cerezo et al. Conversely, assume (2.8) holds along the line v→ (u0,v). Then, (2.8) together with (2.6), imply that the expression m=4Qe−ωψu−βN−(2Qα+Hβ)ψ (2.11) satisfies mv(u0,v) =0, i.e., mis constant along v→ (u0,v). Therefore, m,ψis also constant along v→ (u0,v), i.e., the curvature line ψ(u0,v)is spherical.  3 Special solutions of the sinh-Gordon equation In this section we recall the construction in our previous paper [8] of some special solutions of the elliptic sinh-Gordon equation, that will be used later on. We will make a more detailed discussion than in [8], in order to motivate their origin. We will also indicate some new additional properties of these solutions that will be important for our purposes here. 3.1 Wente’s overdetermined system We will seek solutions ρ(u,v):R2→Rto the overdetermined system ρ +coshρsinh ρ=0,(3.1) 2ρu=α(u)eρ+ β(u)e−ρ.(3.2) for functions α,  β:R→R. Note that the system (3.1)–(3.2) is precisely the system (2.2)– (2.8) for the choices A=B=1/4. We will write (3.2) in the alternative form ρu=y(u)coshρ+z(u)sinh ρ, (3.3) where y(u), z(u)are real functions, so that α=y+zand  β=y−z. The next discussion is taken from Wente [33], pages 9-11 and 16-18. To start, assume that ρ(u,v)satisfies (3.1)and(3.3). Then, y(u), z(u)should be a solution to the differential system y =(a−1)y−2y(y2−z2), z =az −2z(y2−z2), (3.4) with respect to some constant a, see equation (3.6)in[33]. Moreover, if we denote Z(u,v):= eρ(u,v), then it follows by a computation from equation (2.20) in [33] (choosing A=B=1/4 and making the change α=y+z,β=y−zas explained above) that 4Z2 v=p(u,Z), (3.5) where p(u,x):= −(1+(y+z)2)x4−4(y+z)x3+6γx2+4(y−z)x−(1+(y−z)2). (3.6) Here, we are denoting y=y(u),z=z(u),and6γ=6(y2−z2)−4(a−1/2). Thus, for each fixed uvalue, p(u,x)is a polynomial of degree four. This process can be reversed under some additional conditions, as explained in [33,Thm. 2.3]. Specifically, assume that we prescribe the initial values (y(0), z(0), y(0), z(0),a,ρ(0,0)). (3.7) 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 9 of 44 37 so that p(0,eρ(0,0))>0, where p(u,x)is as in (3.6). Then, by [33, Thm. 2.3], there exists a solution ρ(u,v) to (3.1)–(3.3) whose associated functions y(u), z(u)solve (3.4)forthe given constant a, with the given initial conditions y(0), z(0), y(0), z(0). The system (3.4) has a Hamiltonian structure. The basic Hamiltonian constants of (3.4) are y2−z2−(a−1)y2+az2+(y2−z2)2=h∈R(3.8) and (zy−yz)2+z2+z2(y2−z2−a)=k∈R,(3.9) see (3.7)in[33]. One can then follow the classical procedure to solve the Hamilton-Jacobi equations by separation of variables in order to write system (3.4) into a more adequate form; see [33, pg. 17]. Specifically, first we apply the change of variables y2=−(1−s)(1−t), z2=−st.(3.10) Using (3.10), the system (3.4) transforms into the autonomous first order system s(λ)2=s(s−1)q(s), (s≥1), t(λ)2=t(t−1)q(t), (t≤0), (3.11) where q(x)is the third-degree polynomial q(x)=−x3+(a+1)x2+(h−a)x+k,(3.12) and the parameter λof (3.11) is linked to uby (see [33, pg. 18]) 2u(λ) =s(λ) −t(λ) > 1.(3.13) 3.2 Constructions of special solutions to the system We now explain our construction in [8]. To start, we will fix y(0)=z(0)=0 (3.14) by geometrical reasons. More specifically, we intend to use (3.1) as the Gauss equation for a CMC surface in some space M3(c0), see Sect.4below. These initial conditions will determine that intersects orthogonally a totally geodesic surface of M3(c0). See also Lemma 2.1. In particular will have a useful symmetry plane. We are interested in the possibility of obtaining embedded examples of free boundary CMC annuli in geodesic balls of M3(c0). After fixing the initial conditions (3.14), a lengthy, detailed inspection of all the cases that we will not reproduce here seems to indicate that this embeddedness is only possible when p(0,x)in (3.6) has two positive roots and two negative roots. So, we should prescribe the values of y(0), z(0)andain a way that this property holds for p(0,x)in (3.6). We will also be interested in the situation where the two positive roots of p(0,x)collapse into one, since this situation will detect Delaunay examples in M3(c0). Because of this, it seems convenient to seek initial conditions y(0), z(0),aso that p(0,x) in (3.6) can be written in the factor form below, which will force it to have the desired root structure: p(0,x)=−x−a cx−1 ac(x+bc)x+c b,(3.15) 123 37 Page 16 of 44 A. Cerezo et al. Let ϕbe the stereographic projection of M3(ε) from −e4.So,ifε=1, ϕmaps S3\{−e4} into R3, while if ε=−1, ϕmaps H3into the unit ball B3⊂R3.Sincelies in {x3=0}, the curve γ:= ϕ◦is then a planar curve in {z=0}⊂R3,where(x,y,z)denote the Euclidean coordinates of R3. Definition 4.8 Using the above notation, we define the period map as :O−→R, (a,b,c):= 1 πσ 0 κγ||γ||dv, (4.16) where κγand ||γ|| denote the Euclidean curvature and the length of γ, respectively. Note that π represents the variation of the (Euclidean) unit normal of γ(v) along the interval [0,σ]. Proposition 4.9 The map =(a,b,c)in (4.16) is real analytic in O−. Proof It is an immediate consequence of the real analyticity of σ=σ(a,b,c)(Proposition 3.2)andρ=ρ(u,v;a,b,c), together with the analytic dependence of the Gauss–Weingarten system (4.3) with respect to initial conditions.  We explain next the geometry of when its associated period is a rational number. We start by describing the geometry of the planar geodesic (v) =ψ(0,v). Proposition 4.10 Assume that (a,b,c)=m/n∈Q, with n ∈N\{0}and m/n irreducible. Then (v +2nσ) =(v). In particular (v) is a closed curve. Moreover, (v) has rotation index m ∈Zand, if a >1, a dihedral symmetry group Dn with n ≥2. Proof Let us see first that γ=ϕ◦satisfies γ(v+2nσ) =γ(v), where as before ϕdenotes the stereographic projection of M3(ε) from −e4. Consider the totally geodesic surfaces k=Pk∩M3(ε) (see Proposition 4.5). Then, ϕmaps kinto vertical planes kcontaining the z-axis. Let Lk:= k∩{z=0}.Then{Lk:k∈Z}is a family of equiangular lines in {z=0}≡R2passing through the origin. It follows from Proposition 4.5 and the fact that ϕis conformal that the angle between Lkand Lk+1is the angle betweeen γ(0)and γ(σ ). We denote this angle by ϑ.Notethatγ(kσ)is orthogonal to Lk. This implies that π =2πl+ϑ(4.17) for some l∈Z. In particular, nϑ∈πZ. By (4.13), we have γ(kσ−v) =Tk(γ (kσ+v)), (4.18) where Tkdenotes the symmetry of R2that fixes Lk.IfRdenotes the rotation around the origin of angle 2ϑ,thenwehaveby(4.18)thatγ(v +2σ) =R(γ (v)).Fromhereand nϑ∈πZwe obtain γ(v+2nσ) =γ(v). Also from (4.18) we obtain that f:= ||γ||κγsatisfies f(kσ−v) =f(kσ+v) for all k∈Z. Observe that this implies that, for all k∈Z, =1 π(k+1)σ kσ κγ||γ||dv. 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 17 of 44 37 From here, =m/nand the 2σn-periodicity of γ(v), it follows that the rotation index of γ(v)is equal to m. Finally, we determine the symmetry group of (v) when a>1. In that case we know that X(v) in (3.23) only has critical values at the points of the form kσ,k∈Z. Also, by (4.4)and since ψ(u,v)intersects S=M3(ε) ∩{x3=0}orthogonally along (v),wehave κ(v) =κ2(0,v)=H−μ X(v)2, where κis the geodesic curvature of in S. Since the stereographic projection ϕpreserves the critical points of the geodesic curvature of regular curves (because it preserves curves of constant curvature, and hence the contact order with these curves), we deduce that κγ(v) only has critical points at the values v=kσ,k∈Z. In particular, γ(v)has a (finite) dihedral symmetry group, as it is symmetric with respect to the reflections T1,...,Tn. So, its isometry group is Dnfor some n≥n,sincem/nis irreducible. If n>n, there would exist some additional symmetry line Lfor γ(v)different from all Lk.So,(v0−v) =((v0+v)) for some v0/∈{kσ:k∈Z},where is the symmetry with respect to L. Thus, κγwould have a critical point at v0,whatisa contradiction. Hence, the symmetry group of γ(v) is Dn, and generated by the reflections T1,...,Tn. Therefore, the symmetry group of (v) is isomoprhic to Dn, as claimed. Finally, we show that n≥2. Indeed, if n=1, then all the Lk’s agree, and this contradicts that (a,b,c)∈O−,since{ν0,ν 1}would be collinear.  As an immediate consequence of Proposition 4.10,wehave: Corollary 4.11 Let (a,b,c)∈O−so that (a,b,c)=m/n∈Q,wheren∈N\{0}, with m/n irreducible. Then, ψ(u,v+2nσ) =ψ(u,v). 4.5 Construction of CMC annuli Following the results in Sect. 4.4,givenu0>0, we can define 0=0(a,b,c,u0)as the restriction of ψ(u,v)to [−u0,u0]×R. By Corollary 4.11,if(a,b,c)=m/n∈Q, we can view 0as a compact H-annulus in M3(ε) under the identification (u,v+2nσ) ∼(u,v). With this, we have our main conclusion of this section: Theorem 4.12 Let (a,b,c)∈O−so that (a,b,c)=m/n∈Q,wheren ∈N\{0}, with m/n irreducible. Then, for any u0>0, the following properties hold for the annulus 0=0(a,b,c,u0): (1) 0is symmetric with respect to S=M3(ε) ∩{x3=0}, and with respect to n ≥2totally geodesic surfaces 1,..., nof M3(ε) that intersect equiangularly along a geodesic Lof M3(ε) orthogonal to S. (2) Along each boundary component ∂i 0,i =1,2,0intersects at a constant angle θa totally umbilic surface Qiof M3(ε). Specifically, the intersection angle θis the same at both components, and Q1=(Q2),whereis the symmetry of M3(ε) with respect to S. (3) Assume u0=τ,whereτ=τ(a,b,c)is defined in Proposition 3.4.Thenθ=π/2, i.e., ∂i 0intersects Qiorthogonally. (4) Assume that m3(u0)=0,wherem 3denotes the x3-coordinate of the center map m in (2.10). Then both boundary curves ∂1 0,∂2 0lie in the same totally umbilic 2-sphere Q1=Q2of M3(ε). 123 37 Page 18 of 44 A. Cerezo et al. (5) If a >1, the symmetry group of 0is isomorphic to Dn×Z2, and generated by the symmetries in item (1). In particular, 0is not rotational. Proof Item (1) is a direct consequence of Proposition 4.10. Item (2) follows from the symmetry of 0with respect to Sand the fact that the boundary curves of 0correspond to the spherical curvature lines ψ(±u0,v). Item (3) follows from the definition of τin Proposition 3.4, together with (4.5)and(2.9). Regarding item (4), we first note that S[m(u0), d(u0)]is one of Q1,Q2; here, we follow the notation of Lemma 2.1.Ifm3(u0)=0, the center of this Qilies in {x3=0}.Since Q1=(Q2)by item (2), we have Q1=Q2=S[m(u0), d(u0)]. We show next that S[m(u0), d(u0)]is an umbilic 2-sphere if ε=−1, a property equivalent to m(u0)being timelike by the discussion before Lemma 2.1.LetPdenote the plane where m(u)lies, as specified in Proposition 4.3.Since(a,b,c)∈O−,Pis timelike, and by Proposition 4.5 it contains e3.Sincem(u0), e3=0, then m(u0)is timelike, as desired. To prove item (5), let denote an isometry of M3(ε) that leaves 0invariant. Under this isometry, we must have () =, since the points of represent the middle points of the (intrinsic) geodesics j∩0of 0. Therefore, the restriction of to Sis a symmetry of , and hence a composition Tof the symmetries Tjwith respect to some j, by Proposition 4.10. If takes each boundary component of 0to itself, we deduce then that =T. Otherwise =◦T,whereis the symmetry with respect to S(note that interchanges the boundary components of 0). This proves item (5).  Theorem 4.12 motivates the next definition and consequence. Definition 4.13 For any (a,b,c)∈W,wedefineh:W→Ras the map h(a,b,c):= m3(τ(a,b,c)),wherem3denotes the x3-coordinate function of the center m(u). Corollary 4.14 Let (a,b,c)∈O−∩Wso that (a,b,c)=m/n∈Q,wheren∈N\{0}. Assume that h(a,b,c)=0. Then, choosing u0=τ, both boundary components of the annulus 0in Theorem 4.12 intersect orthogonally the same umbilic 2-sphere Qof M3(ε). 5 Critical catenoids and nodoids in space forms In this section we introduce the family of rotational CMC surfaces in M3(ε) =S3or H3 following do Carmo-Dajczer [10], and prove that each element within a subfamily of them has a compact piece that is an embedded free boundary rotational annulus in an adequate geodesic ball of M3(ε). This section can be treated independently of the rest of the paper; the proofs are postponed to an appendix. Up to an isometry, any rotational immersion in M3(1)=S3can be expressed as ψ(s,θ)=(x(s)cos θ,−x(s)sin θ,1−x(s)2sin(φ(s)), 1−x(s)2cos(φ(s)))).(5.1) for some functions x=x(s),φ=φ(s). In the case of M3(−1)=H3, we will focus on rotational surfaces of elliptic type, that is, surfaces that are invariant by a compact, continuous 1-parameter subgroup of isometries of H3. Up to an isometry, any rotational surface of this type can be expressed as ψ(s,θ)=(x(s)cos θ,−x(s)sin θ,x(s)2+1sinh(φ(s)), x(s)2+1cosh(φ(s)). (5.2) If the immersion has CMC H≥0 and is not totally umbilic, and we choose sas the arclength parameter of its profile curve, it can be shown that x(s)is an analytic function satisfying the 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 19 of 44 37 following differential equation: x2=h(x) x2:= x2−εx4−(Hx2−δ)2 x2,(5.3) for some constant δ= 0. In order for (5.3) to have solutions, it is necessary that the biquadratic polynomial h(x)be non-negative for some x∈R. We will treat the cases ε=1andε=−1 separately. 5.1 Rotational CMC surfaces in S3 In this case, the polynomial h(x)in (5.3) is non-negative for some x∈Rif and only if δ∈H−μ 2,H+μ 2(5.4) where, as in the previous section, μ:= √H2+1. If δ=H±μ 2then h(x)is non-positive, with double roots at the values x=± √|δ|/μ. In this case, the only solutions x(s)of (5.3) are the constant ones, x(s)≡± √|δ|/μ (up to an isometry, we can assume that x(s)≡√|δ|/μ). For δ∈(H−μ 2,H+μ 2),δ= 0, h(x) has four simple roots {−xM,−xm,xm,xM}, with 0 <xm<xM≤1, and h(x)≥0for all x∈[−xM,−xm]∪[xm,xM]. In this case, (5.3) has two types of analytic solutions: the constant ones, and the non-constant ones, which oscillate either on the interval [−xM,−xm] or on [xm,xM]. The only solutions that give rise to CMC immersions in S3are the nonconstant ones. Up to isometries, we can always assume that x(0)=xm. Proposition 5.1 ([10]) Let ε=1,H≥0and δ= 0such that (5.4) holds. If δ=H±μ 2,let x(s):= √|δ|/μ, constant. Otherwise, let x(s)be the unique non-constant solution of (5.3) with initial condition x(0)=xm.Wedefine φ(s):= s 0 δ−Hx2 x(1−x2)ds.(5.5) Then, denoting S1≡R/(2πZ), the immersion ψ:R×S1→S3given by (5.1) defines a rotational surface in S3with constant mean curvature H ≥0such that ψs,ψ s≡1.The s-curves and θ-curves are curvature lines, with respective associated principal curvatures κs=H+δ/x2,κ θ=H−δ/x2.(5.6) Conversely, any rotational CMC surface in S3must be an open piece of either one of these examples, or of a totally umbilical round sphere. Definition 5.2 (Spherical nodoids, unduloids and catenoids) Let ε=1. For any H≥0and δ= 0 such that (5.4) holds, let S=S(ε, H,δ) be the rotational CMC surface in S3of Proposition 5.1. •If H>0and0<δ<(H+μ)/2 we will say that Sis a spherical nodoid. •If H>0and(H−μ)/2<δ<0 we will say that Sis a spherical unduloid. •If H=0 we will say that Sis a spherical catenoid. Since the change δ→−δjust gives a reparameterization of S, we can assume in this case that δ>0. •If δ=(H±μ)/2, the surface covers a flat torus. For H=0, it is a Clifford torus. 123 37 Page 20 of 44 A. Cerezo et al. 5.2 Rotational CMC surfaces in H3of elliptic type We recall that, in order for (5.3) to have solutions, it is necessary that h(x)be non-negative for some x∈R. This will happen in any of the following situations: (1) H<1, (2) H=1andδ>−1 2, (3) H>1andδ≥μ−H 2,whereμ:= √H2−1. In the first two cases, h(x)only has two roots −xm<0<xm,andh(x)≥0forallx∈ (−∞,−xm]∪[xm,∞). In the third case, h(x)has four roots −xM≤−xm<0<xm≤xM, and xm=xMif and only if δ=μ−H 2. In this situation, h(x)≥0on[−xM,−xm]∪[xm,xM]. If H>1andδ=μ−H 2,thenxm=xM=H−μ 2and the only analytic solutions of (5.3) are the constants x=±xm. The resulting surfaces are flat hyperbolic cylinders in H3. In the rest of cases, (5.3) has two types of analytic solutions: the constants given by the roots of h(x), and non-constant solutions. The only ones that give rise to actual CMC immersions are those which are not constant. More specifically, if H>1, then x(s):R→Roscillates on either [−xM,−xm]or [xm,xM].IfH≤1, however, x(s):R→Ris unbounded, taking values on either (−∞,−xm]or [xm,∞). In all three cases, up to isometries in H3, we can always assume that x(0)=xm,where xmis the smallest positive root of h(x).Wehavethen: Proposition 5.3 ([10]) Let ε=−1,H≥0,δ= 0.IfH >1and δ=μ−H 2,letx(s)≡xm. Otherwise, let x(s)be the unique nonconstant solution of (5.3) with initial condition x(0)= xm.Wedefine φ(s):= s 0 δ−Hx2 x(x2+1)ds.(5.7) Under these conditions, the immersion ψ:R×S1→H3in (5.2) is a rotational surface in H3with constant mean curvature H ≥0and ψs,ψ s≡1, with principal curvatures given by (5.6). Conversely, any rotational CMC surface of elliptic type in H3must be an open piece of either one of these examples, or of a totally umbilical surface of H3. Definition 5.4 We will say that the immersion ψin Proposition 5.3 is a hyperbolic nodoid (resp. unduloid)ifH>1andδ>0 (resp. μ−H 2<δ<0), and denote it by S=S(ε, H,δ). 5.3 Existence of free boundary nodoids and catenoids Our goal in this section is to show that the nodoids and catenoids given in Propositions 5.1,5.3 for δ>0 (see also Definitions 5.2 and 5.4) are free boundary in a certain ball of M3(ε).The geodesic balls of M3(ε) will be described as B[m,d]:={x∈M3(ε) :x,m≥d}.(5.8) Here m∈M3(ε) is the center of the ball, while d∈Rsatisfies |d|<1whenε=1and d<−1whenε=−1. The fact that x(0)=xmand φ(0)=0 along with (5.3), (5.5), (5.7) imply that the function x(s)is symmetric, while φ(s)is antisymmetric. A geometric consequence of this is that the immersions ψ(s,θ)given in (5.1), (5.2) are symmetric with respect to the totally geodesic 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 21 of 44 37 surface S={x3=0}∩M3(ε). Moreover, the rotation axis of these examples is given by the geodesic L:= {x1=x2=0}∩M3(ε). We note that the balls B[e4,d]centered at the point e4∈M3(ε) are also symmetric with respect to Sand invariant under rotations with axis L. Given s0>0, we define S0as the compact annulus ψ([−s0,s0]×S1)⊂M3(ε),where we have identified the points (s,θ +2π) ∼(s,θ)in the obvious way. Consider the profile curve of S0, s→ ψ(s,0)=(x(s), 0,x3(s), x4(s)). (5.9) We are interested in studying whether S0is free boundary in a certain ball. By the symmetries of S0, it can be shown that its two boundary components are contained in the totally umbilical sphere S[e4,εx4(s0)]=∂B[e4,εx4(s0)]⊂M3(ε). So, if S0happened to be free boundary in some geodesic ball of M3(ε),itwouldnecessarily be in B[e4,εx4(s0)]. Proposition 5.5 Let ε∈{−1,1},H≥0,δ>0.Ifε=1, assume further that δ< H+μ 2.For any s, denote by sthegeodesicinM3(ε) with initial conditions ψ(s,0)and ψs(s,0). (1) There exists s= s(H,δ) > 0such that the compact annulus S0:= ψ([− s, s]×S1)is embedded and free boundary in the ball B := B[e4,εx4(  s)]⊂M3(ε), with x4(  s)>0. Moreover, the principal curvature associated to the profile curve is strictly decreasing on a certain interval [0, s+),>0. (2) The map (H,δ) → s(H,δ) is analytic. Moreover, if ε=1, sextends continuously to the boundary curve δ=H+μ 2by  sH,H+μ 2=π 2√2μ(H+μ) .(5.10) (3) On a neighbourhood I⊂Rof s= s(H,δ), the rotational axis Lof S0and the geodesic smeet at a unique point p(s)with p4(s)>0. Moreover, the function p:I→Lis analytic with p3(  s)=0and p3(  s)>0(here p3(s), p4(s)denote the third and fourth coordinates of p(s)respectively). Proof See Appendix A. Remark 5.6 The first item of the previous Proposition omits the limit case δ=H+μ 2,which corresponds to rotational flat tori. However, in this case the functions x(s),x3(s)can be computed explicitly: x(s)≡μ+H 2μ,x3(s)=μ−H 2μsin 2μ(μ +H)s.(5.11) With this, one can check that the embedded annulus ψ([− s, s]×S1),where  s=π 2√2μ(μ+H), is free boundary in B[e4,0]. 6 Delaunay surfaces via double roots We now consider the case a=1 in our discussion of Sect.4, i.e., we choose (1,b,c)∈O and let =(1,b,c)be the CMC surface in M3(ε) of Definition 4.1. This means that 123 37 Page 22 of 44 A. Cerezo et al. p(0,x)in (3.15) has a double root at x=1/c. As explained after (3.24), in this case we have eρ(0,v) =1/c. Therefore, ρv(0,v)≡0 and by uniqueness of the solution to the Cauchy problem for (3.1)wehavethatρ=ρ(u), i.e., ρv≡0. It follows then by (4.2) that the coefficients of I,II for only depend on u,andsois invariant under a 1-parameter group of ambient isometries of M3(ε). Moreover, each curve v→ ψ(u,v) is an orbit of such 1-parameter group. In our situation, (v) := ψ(0,v)is such an orbit, which actually lies in the totally geodesic surface S:= M3(ε) ∩{x3=0}of M3(ε), see Lemma 4.4.Sinceintersects Sorthogonally along (v), the geodesic curvature of (v) as a curve in Sis given by the principal curvature κ2(0,v), which by (4.4) is constant, and equal to H−μc2. Therefore, in the case ε=1, is a rotational CMC surface in S3.Theu-curves (resp. v-curves) of correspond to the s-curves (resp θ-curves) in the parametrization ψ(s,θ)of rotational surfaces of Sect.5.Inthisway,ψ(u,0)isaprofilecurveof. The principal curvature κ1(u)associated to this profile curve ψ(u,0)is always positive, by (4.4). Thus, if H= 0, is the universal cover of either a flat torus or a spherical nodoid of S3.IfH=0, covers a spherical catenoid or a Clifford torus. See Sect.5. Remark 6.1 If a=c=1, then y(0)=0by(3.20), and so y(u)≡0. This implies by (3.21) and (3.3)thatρ(u)≡0. Thus, a=c=1 corresponds to the case where covers a flat CMC torus in S3. In the case ε=−1, we have that is a generalized rotational surface in H3, i.e., it is invariant by either hyperbolic, elliptic or parabolic rotations in H3. We will be interested in the elliptic case, i.e. the case where the orbits of these rotations are (compact) circles. This happens if and only if the geodesic curvature of the orbits is greater than 1 in absolute value, that is, if and only if (H−μc2)2>1.(6.1) Thus, if (6.1) holds for our choice of (H,c),thenwill be a Delaunay surface in H3with constant mean curvature H>1. Again, since κ1is positive by (4.4) and describes the geodesic curvature of the profile curve of , we deduce that is a hyperbolic nodoid.See Sect.5. An equivalent form of (6.1)is H<μ 2c2+1 c2,(6.2) wherewehaveusedthatμ2=H2−1ifε=−1. This motivates the following definition. Definition 6.2 We let R⊂R3be the open subset of O∩{a=1}given by R:= {(1,b,c)∈O:(H−μc2)2+ε>0}. Note that Ris just O∩{a=1}if ε=1, and that the inequality defining Ris (6.1)if ε=−1. We thus have from the discussion above: Proposition 6.3 Assume that (1,b,c)∈R. Then, =(1,b,c)is the universal cover of the (spherical or hyperbolic) nodoid (H = 0) or catenoid (H =0)inM3(ε) with neck curvature given by κ=H−μc2. Remark 6.4 If (1,b,c)∈R∩O−, then the rotation axis of (1,b,c)is the geodesic L described in item (1) of Theorem 4.12. This follows from the fact that, in this case, (v) = 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 23 of 44 37 ψ(0,v)is a (compact) circle, and any of the symmetries in that item (1) must leave the center of (v) fixed, i.e. the center lies in the intersection of the symmetry planes k. Since both the rotation axis and kare orthogonal to S, we conclude from there that the rotation axis agrees with L=∩ k∈Zk. If follows from the above construction that the parameter cdetermines uniquely the immersion ψ(u,v) that defines (1,b,c). On the other hand, the role of the parameter bis to determine the initial values (3.7)ofsystem(3.4)via(3.19), (3.20). More conceptually, since ψ(u,v) is rotational, each curve v→ ψ(u,v) is a circle that can be seen as contained in infinitely many 2-dimensional totally umbilic surfaces S[m(u), d(u)]of M3(ε). The choice of bin (1,b,c)determines the values of m(u), d(u)in this description. We will next describe the behavior of the solutions to system (3.4) associated to our solution ρ(u,v) in our current case a=1. So, we consider the initial conditions (3.14), (3.19), (3.20) for system (3.4), that give together with (3.21) the solution ρ(u,v) to (3.1) starting from (1,b,c)∈O−constructed in Theorem 3.1. The Hamiltonian constants h,kin (3.8)and(3.9) can be expressed in terms of (b,c)using a=1, (3.19)and(3.20). This lets us write q(x)in (3.12) in terms of (b,c)as q(x)=−(x−r1)2(x−r3), (6.3) where r1=−(b−1)2 4b≤0(6.4) and r3=1 4c+1 c2 ≥1.(6.5) Let (s(λ), t(λ)) be the solution to (3.11) obtained after the change of coordinates (3.10) from our initial solution (y(u), z(u)) to (3.4). We list the following properties, that were obtained in [8], and that will be used later on: i) (s(λ), t(λ)) is defined for all λ∈R, and up to a translation in the λparameter it satisfies the initial conditions s(0)=1, t(0)=0. ii) s(λ) takesvaluesin[1,r3], while t(λ) takes values in [r1,0]. Moreover, s(λ) ≡1ifand only if r3=1andt(λ) ≡0 if and only if r1=0. iii) s(λ) is 2l-periodic, where l=r3 1 dx √x(x−1)q(x)<∞. In this way, s(2l)=s(0)=1ands(l)=r3. iv) If r1<0, then t(λ) is strictly decreasing, with t(λ) →r1as λ→∞. 7 The period map for nodoids and catenoids In this section we will assume, as in Sect.6,thata=1, and keep the same notations. Therefore, ρ=ρ(u),andis the rotational example of Proposition 6.3. In particular κ2(0,v)=H−μc2. As explained in Sect.6,ifε=1, (v) is a circle in S3∩{x3=0}. Also, if ε=−1, (v) is acurveinH3∩{x3=0}≡H2of constant curvature H−μc2. This curve will be a (compact) 123 37 Page 24 of 44 A. Cerezo et al. circle if and only if (6.1) holds, i.e., if and only if (1,b,c)∈R(see Definition 6.2). We also recall that the set O−and the period map were introduced in Definitions 4.6 and 4.8 respectively. We prove next: Theorem 7.1 Let (1,b,c)∈R∩O−. Then, (1,b,c)=−(H−c2μ)2+ε cμ1 c+bc1 c+c b.(7.1) Proof Consider the planar curve γ(v) =ϕ((v)) defined above (4.16). The metric on S\{−e4}can be written via the inverse stereographic map ϕ−1as !(dx2+dy2), ! := 4 (1+ε(x2+y2))2,(7.2) where (x,y)are Euclidean coordinates in R2.Fromhereand(4.2), we have ||γ|| = X 2μ√!,(7.3) where we have used our usual notation X(v) =eρ(0,v). On the other hand, by standard formulas of conformally related Riemannian metrics, the geodesic curvature κof (v) in S, i.e., the geodesic curvature of γ(v)with respect to the metric (7.2), is related to the Euclidean geodesic curvature κγof γby κγ=∇√!, n √!+√!κ,(7.4) where nis the unit normal of γin R2, and both ∇,,are Euclidean. Recall that κ= κ2(0,v)=H−μc2<0. Since (v) is a horizontal circle, its stereographic projection γ(v) parametrizes a circle of a certain radius r>0 in the plane (or in the unit disk of R2,if ε=−1), which is negatively oriented since κ<0. Along γ(v)we have √!=2 1+εr2,∇√!=−4ε (1+εr2)2γ(v), (7.5) and n=1 rγ(v). Then, it follows from (7.4), (7.5)andκγ=−1/rthat r=εH−c2μ±(H−c2μ)2+ε. Here, we should recall that (6.1) holds when ε=−1, since (1,b,c)∈R. Moreover, taking into account that r>0ifε=1andr∈(0,1)if ε=−1, we deduce that, actually, r=εH−c2μ+(H−c2μ)2+ε.(7.6) We now compute the value of =(1,b,c)using (4.16). First note that, by (7.3), (7.5) and X(v) =1/c,wehave ||γ|| = 1+εr2 4cμ. 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 25 of 44 37 Thus using that κγ=−1/rand (3.26), we have from (4.16) (1,b,c)=−(1+εr2) 2cμr1 c+bc1 c+c b. Using (7.6) in this expression, we obtain (7.1).  Remark 7.2 For any p0=(1,b,c)∈R∩O−,c>1, a straightforward computation from (7.1) shows that c(p0)= 0. By the implicit function theorem and the analyticity of (a,b,c)(Proposition 4.9), it follows that the level set (a,b,c)=0,where0:= (p0) can be locally expressed around p0as a graph c=c0(a,b),wherecis real analytic with respect to a,b. Similarly, if p0∈R∩O−,b>1, an analogous computation shows that b(p0)= 0, and so in this case we can express locally the level set (a,b,c)=(p0)as an analytic graph b=b0(a,c). We now consider (r1,r3)givenby(6.4), (6.5). The map (b,c)→ (r1,r3)is a homemorphism from {(b,c):b≥1,c≥1}onto {(r1,r3):r1≤0,r3≥1}, and so we can view (1,b,c)as a map (r1,r3).Using(6.4), (6.5)andμ2=H2+ε, the expression for =(r1,r3)in (7.1) simplifies to 2=μ(2r3−1)−H 2μ(r3−r1).(7.7) This can be rewritten alternatively as r3=2 2−1r1+H+μ 2μ(1−2)(7.8) For any fixed =0, this equation represents the line with slope 2 0/(2 0−1)that passes through the point p0:= H+μ 2μ,H+μ 2μ.(7.9) As an immediate consequence of (7.1)and(7.7), we have: Corollary 7.3 (1,b,c)∈(−1,0)for any (1,b,c)∈R∩O−. We next show that in Theorem 7.1 we can simply assume (1,b,c)∈R. Proposition 7.4 R⊂O−. Proof Let (1,b,c)∈R. Then, (1,b,c)is a rotational surface in M3(ε) whose rotation axis is orthogonal to e2. After a linear isometry of R4 εthat fixes e2,e3, we can assume that this rotation axis is M3(ε) ∩{x1=x2=0}.Wenowconsider,aswedidinSect.4.4,the stereographic projection ϕof M3(ε) from −e4to R3, and the planar curve γ(v):= ϕ((v)). Define next the number  θ:= 1 πσ 0 κγ||γ||dv, just as in (4.16). We use here the new notation  θ,sincein (4.16) was only defined on O−; nonetheless, the right hand side of (4.16) makes sense in our case. Moreover, the computations in Theorem 7.1 show that  θis also given by the right hand side of (7.1). 123 37 Page 32 of 44 A. Cerezo et al. Fig. 3 Level lines ϒof the period map on the parameter domain (r1,r3)for the cases ε=1 (left) and ε=−1 (right). If 8H2−ε>0, there are infinitely many level lines for which a segment of ϒsatisfies the hypotheses of Theorem 8.4 First, it is clear from (9.2)and0∈Jthat ϒ(0)=(0,r3(0)) ∈ W, i.e. the second condition of Theorem 8.4 holds. Also, ϒ(r)∈ Wfor r≥0 small enough; see Fig. 3. On the other hand, ϒ(r)/∈ Wfor rbig enough; indeed, consider the inequality for r3in (8.1). It is clear that r3(r)>−εH+μ 2μfor any r≥0, since ϒ(r)has negative slope. However, we can prove that r3(r)<G(r):= (r1(r)−1)2 1−2r1(r) for rlarge enough. This follows directly from 2 0<1/3(since0∈J) and the inequalities 0<r 3(r)=2 0 1−2 0 <1 2=lim r→∞G(r). As a consequence, there is a first valuer>0 such that ϒ(r)/∈ W, for which the equality in (8.4) is satisfied. Thus, the segment ϒ(r):[0,r]→R2is in the conditions of Theorem 8.4, after the linear change r→ r/r. Thus, by Theorem 8.4, there exists r∗>0 such that ϒ(r∗)∈ W, with h(ϒ(r∗)) =0and so that h(ϒ(r)) changes sign at r∗.Now,let(1,b∗,c∗)be the point related to (r1(r∗), r3(r∗)) by the change (6.4), (6.5). Consider the function g(a,b):= h(a,b,c0(a,b)), (9.3) defined on a neighborhood of (1,b∗)∈R2,wherec0(a,b)is the analytic map defined in Remark 7.2, which parametrizes the level set (a,b,c)=0in a neighbourhood of (1,b∗,c∗)∈R3. We know by our previous discussion that g(1,b∗)=0 and, moreover, that gchanges sign at b∗, meaning that g(1,b)<0 (resp. g(1,b)>0) for b∈(b∗−, b∗) (resp. b∈(b∗,b∗+)), for >0 small enough. Since g(a,b)is analytic, there exists a real analytic curve ζ(η) := (a(η), b(η)),η∈[0,δ)for δsmall, satisfying g(ζ(η)) ≡0and ζ(0)=(1,b∗). This is a consequence of the classical fact that if a non-constant real analytic function f:U⊂R2→Rhas a non-isolated zero at p0∈U, then the set {f=0}is, around p0, the union of a finite number of real analytic arcs meeting at p0; see e.g. [19,Thm. 5.2.3]. Moreover, as b→ g(1,b)changes sign, then a(η) > 1forallη>0. We define then the curve C(η) := (a(η), b(η), c0(ζ(η))), (9.4) 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 33 of 44 37 Fig. 4 The level line ϒand the line Lmeet at a point ϒ(r)∈ W which by definition is contained in the level set (a,b,c)=0, and satisfies h(C(μ)) ≡0. Observe that we obtain a different curve C(η) for every 0in the countable set J∩Q.This concludes the proof of the first step in the case 8H2−ε>0. First step, case 8H2−ε≤0. This implies in particular that ε=1, and so μ>H.Now, take any l>0 such that l<μ−H μ+H, and consider the line L(r):= (−r,lr +1). Also, for any 0∈−μ−H 2μ,−l 1+l(9.5) we consider the level line of the period map ϒ=ϒ(r;0)in (9.2). The conditions on 0 in (9.5) imply that ϒand Lmeet at a point L(r)=ϒ(r),r=r(0)>0; see Fig.4.The value L(r)depends analytically on 0, and for the limit case 0=− μ−H 2μit holds r=0, that is, Land ϒmeet at (r1,r3)=(0,1). Note also that, by Lemma 8.5, the inequality τ(L(r)) > u(lr +1)holds for all r>0 sufficiently close to zero. In particular, there exists a smaller interval J:= −μ−H 2μ,  such that, for all 0∈J: (1) The point L(r)belongs to  W,andsoτ(L(r)) is well defined. (2) The inequality τ(L(r)) > u(lr +1)is satisfied. We now fix some 0∈J∩Qand define τ(r):= τ(ϒ(r)),h(r):= h(ϒ(r)). We will prove that there exists a value r∗∈(0,r)where τ(r∗)=u(r3(r∗)), with r3(r)as in (9.2), and so h(r∗)=0. By hypothesis, we know that ϒ(r)=L(r)∈ W, and in fact τ(r)>u(r3(r)), by item (2) above. Since r3(r)is strictly increasing with r3(0)<1andr3(r)>1, we can define rb:= r−1 3(1)∈(0,r). Moreover, ϒ(rb)=(−rb,1)does not lie in  W, since the inequality for r3in (8.1) does not hold at that point. Consequently, there exists a certain interval (rc,r](rb,r]such that ϒ(r)∈ Wfor all r∈(rc,r],andϒ(rc)∈∂ W;seeFig.4. By a similar argument to the one in the proof of Theorem 8.4, it is possible to check that limr→r+ cτ(r)=0. On the other hand, u(r3(r)) is a 123 37 Page 34 of 44 A. Cerezo et al. positive function defined at r=rc,so lim r→r+ c (τ(r)−u(r3(r))) =−u(r3(rc)) < 0. In particular, there exists some r∗∈(rc,r)where τ(r∗)=u(r3(r∗)). In fact, sinceuand τare analytic and they do not coincide, we can take r∗so that the function f(r):= τ(r)−u(u3(r)) changes sign at r=r∗, being negative (resp. positive) for any r<r∗(resp. r>r∗)close enough to r∗. From the definition of r∗and Proposition 8.3 it follows that h(r∗)=0. In fact, since f(r) changes sign at r∗, arguing as in the last part of the proof of Theorem 8.4, we deduce that h(r) also changes sign. Now, let (1,b∗,c∗)∈O−be the point associated with (r1(r∗), r3(r∗)) by (6.4), (6.5). We deduce that the analytic function b→ g(1,b), with g(a,b)given by (9.3), changes sign at b=b∗. Consequently, as explained in the proof of the case 8H2−ε>0 above, it follows from the local description of the zero set of real analytic functions ([19, Theorem 5.2.3]) that there exists an analytic curve ζ(η) =(a(η), b(η)) such that g(ζ(η)) ≡0, ζ(0)=(1,b∗)and a(η) > 1forallη>0. We deduce that the curve C(η) defined as in (9.4), contained in the level set (a,b,c)=0, satisfies h(C(μ)) ≡0. We remark that we obtain (at least) one such curve for every 0in the countable set J∩Q. This completes the first step of the proof. To sum up: so far, we have proved that for any values H≥0, ε=±1 with H2+ε>0, there is a countable number of curves Cq:= Cq(η) :[0,δ(q)) →W∩O−, each of them contained in the level set of the Period map (a,b,c)=q∈J∩Q⊂(−1,0)∩Q, with the property that hvanishes identically along Cq(η). Let us now define Aq=Aq(η) as the compact annulus 0=0(a,b,c,τ(a,b,c)) of Theorem 4.12 associated to the point (a,b,c)=Cq(η). Our goal is to prove that Aq(η) satisfies each of the properties listed in Theorem 1.1. Items (2) and (5) of Theorem 1.1 are a direct consequence of Theorem 4.12 and the fact that a(η) > 1forallη>0. Similarly, item (3) follows from Proposition 4.10 and item (4) from Proposition 6.3 and Remark 8.2. Item (6) holds by construction, as any surface =(a,b,c) has constant mean curvature Hand is foliated by spherical curvature lines. Let us now prove item (1). Since hvanishes along the curve Cq(η), we deduce by Corollary 4.14 that the annuli Aq(η) meet orthogonally a certain totally umbilic 2-sphere Q=Q(q,η) of M3(ε) along their boundary. Let us denote by B(q,η)thegeodesicballofM3(ε) whose boundary is the sphere Q(q,η); in the case ε=1 there are two such balls, and we choose the one for which ψ(0,0)∈B(q,η).Forη=0, we know that τ(Cq(0)) =u(Cq(0)), so the compact nodoid or catenoid Aq(0)is one of the examples constructed in item 1 of Proposition 5.5. In particular, this rotational annulus is contained in B(q,0). By real analyticity, we conclude that the (non-rotational) annuli Aq(η),η∈(0,δ 0(q)), will also be contained in their respective balls B(q,η), at least for some δ0(q)>0 small enough. This completes the proof of Theorem 1.1. 9.2 Proof of Theorem 1.2 The key idea is to study the level sets of the form (a,b,c)=−1/n,wheren∈N,n≥2. Suppose that either ε=−1orH≥1 √3. Then, the inequality μ−H 2μ≤1 n2(9.6) 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 35 of 44 37 is satisfied for some n≥2. More specifically, if ε=1andH≥1 √3, then it holds for a finite set of natural numbers, while if ε=−1itistrueforalln. We will split our analysis into two cases, depending on whether the inequality (9.6) is strict or not. Suppose first that (9.6) is strict for some n≥2. This means that we are in the case 8H2−ε>0 detailed in the proof of Theorem 1.1,andthatqn:= −1/nlies in the open interval Jdefined in (9.1). By the proof of Theorem 1.1, we deduce that there is a curve Cqn(η) :[0,δ(n)) →O−, contained in the level set (a,b,c)=qn, such that the associated annuli Aqn(η) are free boundary in a geodesic ball B(qn,η)of M3(ε). It just remains to check the embeddedness of these examples, which will be studied later. Suppose now that (9.6) holds for the equality case. In that situation, we cannot use Theorem 1.1 directly. By the equality in (9.6), and using (9.2), the level curve (1,b,c)=qn:= −1/nis expressed in terms of (r1,r3)as ϒ(r):= (r1(r), r3(r)) =−r,r n2−1+1. This is the equation of a line L(r):= ϒ(r)that is in the conditions of Lemma 8.5,soforany r>0 small enough, we have ϒ(r)∈ Wand τ(ϒ(r)) > u(r3(r)). However, it is possible to check that ϒ(r)/∈ Wfor r>0 large enough, by a similar argument to the first step of the proof of Theorem 1.1. Also following the proof of that first step, we can deduce that there exists a point r∗where u(r3(r∗)) =τ(ϒ(r∗)),andsoh(ϒ(r∗)) =0. In fact, we can show that there is a real analytic curve Cn(η) such that hvanishes identically along Cn,and from this we obtain a 1-parameter family of free boundary annuli An(η) that satisfies the properties stated in Theorem 1.1. In conclusion, for any n≥2 such that (9.6) holds, there exists a 1-parameter family of immersed, free boundary annuli Aqn(η). It just remains to prove that the annuli Aqn(η), η∈[0,δ(n)), are embedded for some δ(n)>0 small enough. For every η∈[0,δ(n)), let us denote by ψη(u,v)our usual parametrization by curvature lines of the compact annulus Aqn(η). We know by Corollary 4.11 that ψη(u,v)=ψη(u,v+ 2nσ),whereσdepends analytically on η. Identifying (u,v) ∼(u,v+2nσ) as usual (see Sect.4.5), we can view ψηas a parametrization ψη:[−τ(η),τ(η)]×S1→Aqn(η) of Aqn(η). Observe that for η=0, the annulus Aqn(0)is an embedding, since it is a trivial covering of a critical catenoid or nodoid Nembedded in M3(ε); see Remark 8.2. Thus, ψ0 is injective. Now, by the real analyticity of the family of compact annuli Aqn(η), we deduce that for all ηsufficiently close to zero, the parametrizations ψηare also injective, and so the annuli Aqn(η) are embedded. This completes the proof. 10 Embedded capillary minimal and CMC annuli in S3 In Theorem 1.2, we constructed embedded examples of non-rotational, free boundary CMC annuli in geodesic balls of H3(for H>1) and S3(for H≥1/√3). In this section we will show that if we relax the free boundary condition to capillarity, then there exist embedded non-rotational capillary CMC annuli in S3for any H≥0. Recall that a compact surface in M3(ε) is called a capillary surface in a geodesic ball B⊂M3(ε) if ⊂Bintersects ∂Bat a constant angle along ∂. We prove next: 123 37 Page 36 of 44 A. Cerezo et al. Theorem 10.1 For any H ≥0and any n ≥2there exists a real analytic 2-parameter family of embedded capillary annuli An(a,η)with constant mean curvature H in a geodesic ball B=B(n,a,η)of S3, with a prismatic symmetry group of order 4n. The rough idea behind this result is as follows: consider the level set of the period (a,b,c)=−1/n, and suppose that for some (a,b,c)∈O−with a>1inthislevel set there exists a value u∗>0 such that m3(u∗)=0, that is, the third coordinate of the center function vanishes. According to items (2), (4) and (5) of Theorem 4.12, the compact annulus 0(a,b,c,u∗)intersects along ∂0at a constant angle a totally umbilic sphere Q of S3,and0has a prismatic symmetry group of order 4n. Consequently, it suffices to check that the annuli 0are embedded and contained in a geodesic ball Bof S3whose boundary is Q. We will make use of the following lemma: Lemma 10.2 Suppose that for some (a0,b0,c0)∈O−there exists u0>0such that m3(u0)=0. Then, there exists a neighbourhood Vof (a0,b0,c0)and an analytic function u∗=u∗(a,b,c):V∩O−→Rsuch that u∗(a0,b0,c0)=u0and m3(u∗(a,b,c)) ≡0. Proof The Lemma is a direct consequence of the implicit function theorem if we prove that m 3(u0)= 0. Assume by contradiction that m 3(u0)=0. Since m3(u0)=0, we deduce that the function m3(u)in (4.7) satisfies m3(u0)=m 3(u0)=0, due to (4.8). Moreover, m3satisfies the differential Eq. (4.9), and so we conclude that m3(u), and hence, m3(u), vanish identically. This is a contradiction, as (2.10) and the initial conditions (4.11)imply that m3(0)=1.  10.1 Proof of Theorem 10.1 Let n≥2andε=1. We will distinguish two cases, depending on whether or not (9.6) holds. Assume first that (9.6) holds, and consider the level set (a,b,c)=−1/n=: 0. Following the proof of Theorem 1.2 in Sect. 9.2, we know that there is a point (1,b∗,c∗)in that level set such that h(1,b∗,c∗)=m3(τ(1,b∗,c∗)) =0. By applying Lemma 10.2 with u0=τ(1,b∗,c∗), we deduce the existence of a function u∗defined on a neighbourhood Vof (1,b∗,c∗)such that m3(u∗(a,b,c)) vanishes identically. Now, consider the analytic function (a,b)→ u∗(a,b):= u∗(a,b,c0(a,b)), where c0(a,b)is the analytic map in Remark 7.2.LetAn(a,b)≡ψ([−u∗,u∗]×S1)be the compact annulus associated to the parameters (a,b,c0(a,b)), where we identify the points (u,v)∼(u,v+2nσ)as in Remark 4.11. By construction, any of these annuli meets a totally umbilic sphere Q(n,a,b)of S3with constant angle along its boundary ∂An(a,b),so in order to prove Theorem 10.1 we just need to check that the annuli An(a,b)are embedded and contained in a geodesic ball of S3bounded by their corresponding sphere Q. Notice that in Theorem 1.2 we already proved this for the annulus An(1,b∗), so by real analyticity, there is an open neighbourhood G⊂R2of (1,b∗)such that the annuli An(a,b)are also embedded and contained in a geodesic ball B(n,a,b)of S3bounded by Q(n,a,b)for all (a,b)∈G. We also recall that there are two such geodesic balls in S3; we make the same choice for it that we did when proving Theorem 1.2. Take next some n∈N,n≥2, such that (9.6) does not hold, and consider the level curve (1,b,c)=0:= −1/n.Inthe(r1,r3)-coordinates, this curve is given by ϒ(r)in (9.2), 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 37 of 44 37 and so it meets the horizontal line r3=1 at a certain point ϒ(rα)=(−rα,1),whererα>0. Let us see that (1,bα,cα)≡ϒ(rα)is in the conditions of Lemma 10.2. Let denote the rotational H-surface associated to (r1,r3)=ϒ(rα)≡(1,bα,cα); note that cα=1sincer3=1, see (6.5). Thus, by Remark 6.1,covers a flat CMC torus in S3.Since=−1/n, we can consider for any u0>0 the compact immersed H-annulus 0=0(1,bα,1,u0)as defined in Sect.4, after the identification (u,v) ∼(u,v+2nσ). Under this identification, the v-curves ψ(u0,v):S1→S3are injective parametrizations of circles. We find next an explicit parametrization for the profile curve ψ(u,0)of 0,usingthe expresion of the flat H-torus in S3given in Remark 5.6 in terms of the parameters (s,θ). To start, note that it holds eρ(u)≡1for0due to c=1, and so the reparametrizacion u=u(s)in (8.2)isjustu=2μs. In addition, the rotation axis of the flat torus in Remark 5.6 is S3∩{x1=x2=0}, which agrees with the geodesic of S3that contains the centers m(u) of ; see Remark 6.4. Thus, by (5.11), we see that the profile curve of is ψ(u,0)=μ+H 2μ,0,μ−H 2μsin μ+H 2μu,μ−H 2μcos μ+H 2μu. (10.1) Let u0∈(0,u),whereu:= 2μ μ+Hπ. From the expression of the profile curve ψ(u,0) and the previous discussion, it follows that the parametrization ψ:[−u0,u0]×S1→S3of 0is injective, where we identify (u,v)∼(u,v+2nσ)as usual. Moreover, 0is contained in the ball B[e4,x4(u0)]. We now claim that there exists some u0∈(0,u)such that m3(u0)=0. To prove this, note first that f(u):= m(u), ψu(u,0)never vanishes; indeed, by (2.10), it holds f(u)= eρ 2μ| N|sin θ. But now, we have | N|>0 by construction, and sin θcannot be zero by (2.9), as βis bounded since the functions (s,t)in (3.11) are; see also (6.3). Therefore, f(u)does not change sign. On the other hand, observe that, by (10.1), f(0)=m(0), ψu(0,0)=m3(0) 2μ, f(u)=m(u), ψu(u,0)=−m3(u) 2μ. Since the sign of f(u)is constant in (0,u), we deduce that there is some u0∈(0,u)for which m3(u0)=0. So, by Lemma 10.2, there exists a function u∗(a,b,c)defined on a neighbourhood of (1,bα,1), with u∗(1,bα,1)=u0and m3(u∗(a,b,c)) ≡0. We consider the analytic function u∗(a,c):= u∗(a,b0(a,c), c)and define for any (a,c) in a neighborhood of (1,1)the compact H-annuli An(a,c):= 0(a,b0(a,c), c,u∗(a,c)), where b0(a,c)is the analytic map in Remark 7.2. By construction, An(a,c)meets a totally umbilic sphere Q(n,a,c)with constant angle along ∂An(a,c), according to Theorem 4.12. The annulus An(1,1)is equal to 0(1,bα,1,u0), which by our previous discussion, is embedded and contained in a geodesic ball of S3bounded by Q(n,1,1). Consequently, there is a neighbourhood of (1,1)such that the annuli An(a,c)are embedded capillary CMC annuli in a geodesic ball of S3bounded by Q(n,a,c). This completes the proof of Theorem 10.1. 123 37 Page 38 of 44 A. Cerezo et al. A Appendix: Proof of Proposition 5.5 In this Appendix we will prove separately the three items of Proposition 5.5. A.1 Proof of item 1 of Proposition 5.5 By the symmetries of S0, if this annulus were free boundary, it should necessarily be in the ball B[e4,εx4(  s)]. In such case, S0and S[e4,εx4(  s)]would meet orthogonally along ∂S0. This happens if and only if the geodesic  spasses through the point e4, i.e., the center of the ball. As we will see in Appendix A.3, this property holds for the first positive root sof the function F(s):= x3(s)x(s)−x 3(s)x(s), (A.1) where x,x3are defined in (5.9). Consequently, our goal will be to show the existence of such  s= s(H,δ). We will deal with several cases depending on the values ε∈{−1,1},H≥0, δ>0, but the general strategy is to prove that F(0)<0 and that there is some s0>0 with F(s0)>0. We first consider the case where ε=1andδ< H+μ 2,orε=−1andH>1. Then, we showed in Sect.5that the function x(s)in (5.3) takes values on the interval [xm,xM], where xm<xMare the two positive roots of h(x); we recall here our assumption that x(0)=xm>0. In this situation, we define s2>0 as the first positive value for which x(s2)=xM. Consequently, x(s)will be strictly increasing for all s∈[0,s2]. We remark that if x≥0, then h(x)≥0 if and only if x∈[xm,xM]. Additionally, if ε=−1, H≤1, then x(s)is unbounded, taking values on [xm,∞),wherexmis the unique positive root of h(x). The function x(s)will be strictly increasing for all s≥0. In this case, if x≥0, then h(x)≥0 if and only if x≥xm. Let us now show that F(0)<0. By construction, x3(0)=0, x(0)=xm>0and x(0)=0, so it remains to prove that x 3(0)>0. By (5.1), (5.5)forε=1and(5.2), (5.7)for ε=−1, this is equivalent to the fact that δ−Hx2 m>0, that is, x0:= √δ/H>xm. Assume by contradiction that x0≤xm. This implies that h(x0)≤0, as h(x)≤0in[0,xm].Now, a direct computation shows that h(x0)=x2 0−εx4 0. This quantity is clearly positive when ε=−1 and also when ε=1, since x0≤xm<1. We reach a contradiction, what proves that F(0)<0. We are going to show next that there is some s0>0 such that F(s0)>0, and so the existence of sfollows. We split our analysis into five different scenarios: the first three concern the spherical case (ε=1), while the fourth and fifth cover the hyperbolic case (ε=−1). Case 1: ε=1, δ∈(0,H). In this case, x0=√δ/Hsatisfies xm<x0<xM, so the function s→ δ−Hx(s)2, and consequently φ(s), changes sign on the interval (0,s2).Lets1∈(0,s2)be the value for which φ(s1)=0, so that φ(s)is increasing on [0,s1].Ifφ(s1)>π/2, then we define s0<s1as the value for which φ(s0)=π/2. Otherwise, let s0=s1. In any of the cases, we see that x(s0), x(s0)>0, x 3(s0)<0andx3(s0)≥0, so necessarily F(s0)>0. We note that the function x4(s)defined in (5.9) is strictly decreasing and positive for all s∈[0,s0). Case 2: ε=1, δ=H. A direct computation using (5.3) shows that xM=1. This implies that x3(0)=x3(s2)= 0, and since x 3(0)>0, there exists a first s0∈(0,s2)such that x 3(s0)=0. x3(s)is increasing on [0,s0], so in particular x3(s0)>0, and hence F(s0)>0. Note also that the function φ(s), which must be increasing on [0,s0], satisfies φ(s0)<π 2: otherwise, there 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 39 of 44 37 Fig. 5 Integration path Cnfor f(z) would be an intermediate value ssuch that φ(s)=π 2,andsox 3(s)<0 at that point, reaching a contradiction. A consequence of this fact is that x4(s)is decreasing and positive for all s∈[0,s0). Case 3: ε=1, δ>H. In this case, δ−Hx2>H(1−x2)≥0, so φ(s)is increasing for all s∈R.Ifwemanage to prove that φ(s2)>π/2, then there exists a unique s0∈(0,s2)such that φ(s0)=π/2. In particular, x3(s0)>0, x 3(s0)<0, and so F(s0)>0. Additionally, we deduce that x4(s)is decreasing and positive for all s∈[0,s0). Consider the change of variables w(s)=x(s)2on the integral that describes φ(s2). Defining wm=w(0)=x2 m,wM=w(s2)=x2 M,wegetusing(5.3)that φ(s2)=wM wm δ−Hw 2√w(1−w)w−w2−(Hw−δ)2dw. We analyze this integral via residues. Note that z−z2−(Hz −δ)2=−(H2+1)(z−wm)(z−wM)(A.2) by definition of wm,wM, and consider from there the meromorphic function f(z):= iδ−Hz 2√H2+1(1−z)√z√z−wm√z−wM defined on C\((−∞,0]∪[wm,wM])and with a pole at z=1. We will integrate along a sequence of closed paths CnshowninFig.5. Each such path Cncan be divided into five pieces: a circle arc C(1) nof radius n,acurveC(2) nenclosing the segment [wm,wM], another curve C(3) naround the interval (−∞,0]with the same boundary points as C(1) n,andapair of segments S(1) n,S(2) nwhich connect C(2) n,C(3) n. The segments S(1) n,S(2) ncoincide but have opposite orientations. We take C(2) nand C(3) nso that they converge to the intervals [wm,wM] and (−∞,0]respectively as ngrows to infinity. By the residue theorem, and using (A.2), Cn f(z)dz =2πiRes(f,1)=π. 123 37 Page 40 of 44 A. Cerezo et al. A careful analysis of f(z)shows that lim n→∞C(1) n f(z)dz =0, lim n→∞C(2) n f(z)dz =2wM wm iδ−Hw 2i√H2+1(1−w)√w√w−wm√wM−wdw=2φ(s2), lim n→∞C(3) n f(z)dz =20 −∞ iδ−Hw 2i3√H2+1(1−w)√−w√wm−w√wM−wdw=−M, for some positive constant M>0. On the other hand, since the segments S(1) nand S(2) nhave opposite orientations, we deduce that S(1) n f(z)dz =−S(2) n f(z)dz, and so we end up with π=2φ(s2)−M<2φ(s2), as we wanted to prove. From this, the existence of sis immediate. It is also possible to deduce that x4(s)is decreasing and positive for all s∈[0,s0). Case 4: ε=−1, H>0. In this case, we define s0as the first value for which x(s0)=x0,wherex0=√δ/H.Let us prove that s0exists. First, if H>1, then x(s)oscillates between xmand xM, and the fact that h(x0)>0 implies that x0∈(xm,xM),andsos0exists indeed. In the case H≤1, it holds x0≥xm.Sincefors≥0 the function x(s)is increasing and satisfies lims→∞ x(s)=∞, we see again that s0exists. Now, notice that φ(s)is an increasing function on the interval [0,s0],andφ(s0)=0. In particular, φ(s0)>0. A direct computation using (5.2)shows that F(s0)=x(s0)sinh(φ(s0)) x(s0)2+1 >0, as we wanted to prove. It is also immediate to prove that εx4(s)=−x4(s)is decreasing for all s∈[0,s0]. Case 5: ε=−1, H=0. In this case, φ(s)is an increasing function satisfying lims→∞ φ(s)=: φM<∞(the integral in (5.7) is convergent). On the other hand, x(s)increasing and unbounded, and satisfies (5.3). We deduce that lim s→∞ F(s)=lim s→∞xsinh φ √x2+1−δcoshφ √x2+1=sinh(φM)>0, In particular, it follows that F(s)vanishes for some  s>0, as we wanted to prove. It also follows that −x4(s)is decreasing for all s∈[0, s]. In any of the considered cases, we deduce the existence of a first value  s>0 such that F(  s)=0. As commented before, this means that S0meets orthogonally the boundary sphere S[e4,εx4(  s)](see Appendix A.3). Moreover, we also find that x(s), x(s), x3(s)>0, F(s)<0foralls∈(0, s). In addition, x(  s)>0, which shows by (5.6) that the principal curvature κs(s)associated to the profile curve must be strictly decreasing on [0, s+) for some >0. Another consequence of these inequalities is that x 3(s)>0on(0, s). Since the function x3(s)is odd, we obtain that x3(s) is injective on [− s, s]. As a result, the annulus S0must be embedded. 123 Free boundary CMC annuli in spherical and hyperbolic balls Page 41 of 44 37 Finally, we will show that S0is free boundary in B. It suffices to check that S0is contained in that ball. This is a consequence of the already proven monotonicity of εx4(s),since e4,ψ(s,θ)=εx4(s)≥εx4(  s), for all s∈[0, s], which shows that S0⊂B[e4,εx4(  s)]. A.2 Proof of item 2 of Proposition 5.5 We will first prove the analiticity of  s. This fact is a consequence of the implicit function theorem applied on the function F=F(s;H,δ) defined in (A.1). We remark that this function not only depends analytically on s, but also on the parameters H,δwhich define the functions xand x3. If we differentiate Fwith respect to sat s= s, we obtain using F(  s)=0 that F=−xxx 3 x−x 3x x2=−xxx 3 x.(A.3) Since x(  s), x(  s)>0, we just need to check that x 3 x= 0at  sto deduce analyticity. If ε=1, then by (5.3), (5.5) and the fact that F(  s)=0, we deduce after a long but direct computation that x 3 xs= s=−cos(φ)√1−x2(δ +Hx2) x2x2<0,(A.4) where the final quantity is negative since δ>0, H≥0and0<φ(  s)<π/2 (see Appendix A.1). Similarly, if ε=−1, then by (5.3), (5.7), we obtain at s= sthat x 3 xs= s=−cosh(φ)√x2+1(δ +Hx2) x2x2<0.(A.5) In any of the cases, we deduce that s= s(H,δ)is analytic. We will now prove that the extension of s(H,δ)along the boundary curve δ=H+μ 2given by (5.10) is continuous. We consider the function F(s;H,δ) :→Rin (A.1)definedon the set := (s,H,δ) :s∈R,H≥0,0<δ≤H+μ 2. We know that Fis continuous on since x,x3and their derivatives with respect to s depend continuously on the parameters H,δ. We now claim that F(s;H,δ) < 0forall 0≤s< s,andthats→ F(s;H,δ) changes sign at s= s. This is true when δ< H+μ 2, according to the results in Appendix A.1 and the fact that F(  s)>0; see (A.3), (A.4). In the remaining case δ=H+μ 2, it is possible to compute explicitly the functions x(s),x3(s), which are given by (5.11). This allows us to prove the claims on Fin this situation, where we are definingsH,H+μ 2as in (5.10). We now fix some H0≥0, and denote μ0:= H2 0+1. We need to prove that there exists lim (H,δ)→H0,H0+μ0 2 s(H,δ)=π 2√2μ0(H0+μ0)=: s0.(A.6) 123