Simply-connected minimal surfaces with finite total curvature in $\H^2\times\R$
aa r X i v : . [ m a t h . DG ] O c t Simply-connected minimal surfaces with finite totalcurvature in H × R Juncheol Pyo ∗ and M. Magdalena Rodríguez † November 9, 2018
Abstract
Laurent Hauswirth and Harold Rosenberg developed in [4] the theory of minimal sur-faces with finite total curvature in H × R . They showed that the total curvature of onesuch a surface must be a non-negative integer multiple of − π . The first examples ap-pearing in this context are vertical geodesic planes and Scherk minimal graphs over idealpolygonal domains. Other non simply-connected examples have been constructed recentlyin [6, 11, 14].In the present paper, we show that the only complete minimal surfaces in H × R oftotal curvature − π are Scherk minimal graphs over ideal quadrilaterals. We also constructproperly embedded simply-connected minimal surfaces with total curvature − kπ , for anyinteger k ≥ , which are not Scherk minimal graphs over ideal polygonal domains. Mathematics Subject Classification:
Primary 53A10, Secondary 49Q05, 53C42
In the classical theory of minimal surfaces in R , the ones better known are those with finitetotal curvature. We recall that the total curvature of a surface M is defined as C ( M ) = R M K ,where K denotes the Gauss curvature of M . If a minimal surface M of R has finite total ∗ Research partially supported by the CEI BioTIC GENIL project (CEB09-0010) and the Basic Science Re-search Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education,Science and Technology (2012-0007728). † Research partially supported by the MEC-FEDER Grant no. MTM2011-22547 and the Regional J. An-dalucía Grant no. P09-FQM-5088. | C ( M ) | < + ∞ ) then either M is a plane or it must be C ( M ) = − πk , forsome integer k ≥ , and the equality only holds for M being the catenoid or Enneper’s surface(see [13, Theorems 9.2 and 9.4]).In the last decade, the geometry of minimal surfaces in H × R has been actively studied,and many examples have been constructed (see for instance [1, 3, 9, 10, 12, 15]). Hauswirthand Rosenberg started in [4] the study of complete minimal surfaces of finite total curvaturein H × R . The only known examples at that moment were the Scherk minimal graphs overideal polygonal domains with an even number of edges, with boundary values ±∞ disposedalternately. Morabito and the authors constructed in [11, 14] non simply-connected properlyembedded minimal surfaces with finite total curvature and genus zero. Quite recently, in ajoint work with Martín and Mazzeo, the second author [6] has constructed properly embeddedminimal surfaces with finite total curvature and positive genus.The classification of minimal surfaces of finite total curvature in H × R arises very naturally.The first result of classification appearing in this theory was that the only complete minimalsurfaces with vanishing total curvature are the vertical geodesic planes (see [5, Corollary 5]).Quite recently, Hauswirth, Nelli, Sa Earp and Toubiana have proved in [7] that a completeminimal surface in H × R with finite total curvature and two ends, each one asymptotic to avertical geodesic plane, must be one of the horizontal catenoids constructed in [11, 14]. In thispaper, we show that the Scherk minimal graphs over ideal quadrilaterals (i.e. ideal polygonaldomains bounded by four ideal geodesics) are the only complete minimal surfaces of totalcurvature − π .It was expected that each end of a minimal surface with finite curvature in H × R wereasymptotic to either a vertical geodesic plane or a Scherk graph over an ideal polygonal do-main. We construct new simply-connected examples, that we call twisted Scherk examples , thathighlight this is not the case. They all have total curvature an integer multiple of − π , so wecannot expect a classification result for Scherk graphs over ideal polygonal domains boundedby k + 2 edges as the only simply-connected complete minimal surfaces in H × R with totalcurvature − kπ . We consider the Poincaré disk model of H ; i.e. H = { z ∈ C | | z | < } , with the hyperbolicmetric g − = −| z | ) | dz | . We denote by ∂ ∞ H the infinite boundary of H (i.e. ∂ ∞ H = { z ∈ C | | z | = 1 } ) and by the origin of H . Also t will denote the coordinate in R .Let M be a complete orientable minimal surface immersed in H × R . We define the totalcurvature of M as C ( M ) = R M K , where K ≤ denotes the Gaussian curvature of M . We saythat M has finite total curvature when | C ( M ) | < + ∞ .2n this section we summarize the geometric properties of minimal surfaces in H × R withfinite total curvature given by Hauswirth and Rosenberg in [4].We call the height function of M the horizontal projection h : M → R , and we denote by F the vertical projection of M over H . It is well-known that h is a real harmonic function on M and that F is an harmonic map from M to H . Given a conformal parameter w on M , SaEarp and Toubiana [15] proved that ( h w ) = − Q , where Q is the Hopf differential associated to F . Then the zeroes of Q are of even order and, up to a sign (which corresponds to a reflectionsymmetry with respect to H × R ), h = ℜ (cid:18) − i Z p Q (cid:19) , (1)see equation (3) in [4].We fix a unit normal vector field N on M . We now state the main theorem in [4]. Theorem 1. [4] Let M be a complete, orientable, minimal surface immersed in H × R withfinite total curvature. Then:1. M is conformally a closed Riemann surface M punctured in a finite number of points p , · · · , p n , called ends of Σ .2. Q is holomorphic on M and extends meromorphically to its ends p i . If we parameterizeconformally a neighborhood of p i in M by Ω = C \ D , where D is the open unit diskin C centered at the origin, then Q ( z ) = z m i ( dz ) , for some integer m i ≥ − .3. N = h N, ∂ t i converges uniformly to zero on each end p i .4. The total curvature of M is given by Z M K = 2 π − g − n − n X i =1 m i ! . (2) Remark 2.
Suppose p i is an end of M for which m i = − . If we want to close periods inequation (1) , then we have to choose Q ( z ) = − z − ( dz ) , z ∈ Ω . Assertion 3.
In the second item of Theorem 1, m i cannot equal − . roof. Suppose M (in the setting of Theorem 1) has an end p for which m = − . We knowthat a neighborhood E of p can be conformally parameterized on Ω = { z ∈ C | | z | ≥ } ,where Q ( z ) = − z − dz (see Remark 2). From (1) we then get h ( z ) = 2 ℜ (cid:0)R M dzz (cid:1) = 2 ln | z | .Therefore, E is a vertical annulus whose intersection with each horizontal slice H × { t } , t ≥ ,is a compact curve.The boundary of E (which corresponds to {| z | = 1 } ) consists of a horizontal compactcurve Γ at height zero. Consider R > big enough so that the disc D ⊂ H of radius R centered at the origin contains Γ in its interior. And let C be the complete vertical rotationalcatenoid constructed by Nelli and Rosenberg in [12] whose neck is ∂D . Since E intersects eachhorizontal slice in a compact curve, we deduce using the Maximum Principle with verticallytranslated copies of C that E must be contained in D × R . But this is not possible: If wetranslate C vertically up a distance π , we reach a contradiction by applying the MaximumPrinciple with the family of shrunk catenoids going from C to the 2-sheeted covering of thepunctured slice ( H − { } ) × { π } .We finish this section by describing the asymptotic behavior of a complete, orientable,minimal surface immersed in H × R with finite total curvature. Lemma 4. [4] Let M be a minimal surface in the hypothesis of Theorem 1, and p i an endof M . If m i ≥ is the integer associated to p i , as defined in Theorem 1, then p i correspondsto m i + 1 geodesics γ , . . . , γ m i +1 ⊂ H × { + ∞} , m i + 1 geodesics Γ , . . . , Γ m i +1 ⊂ H × {−∞} ,and m i + 1) vertical straight lines (possibly some of them coincide) in ∂ ∞ H × R , each onejoining an endpoint of some γ j to an endpoint of some Γ j . Given any two points p, q ∈ H ∪ ∂ ∞ H , we will denote by pq the geodesic arc joining p, q .We consider an even number of different points p , · · · , p k ∈ ∂ ∞ H (cyclically ordered), with k ≥ , and we call A i = p i − p i , B i = p i p i +1 , for any ≤ i ≤ k , where we consider the cyclicnotation p k +1 = p . Let Ω be the ideal polygonal domain bounded by A , B , · · · , A k , B k . Wecall Scherk minimal graph over Ω to a minimal graph over Ω with boundary values + ∞ overthe A i edges and −∞ over the B i edges (in [1, 12] it is proved that it exists and it is unique upto a vertical translation). In [1, 4] it is proved that such a graph has total curvature π (1 − k ) .Scherk graphs over ideal polygonal domains, together with the vertical geodesic planes, wherethe first known examples of minimal surfaces with finite total curvature.In [11, 14] other non-simply-connected examples where presented, called minimal k -noids .We briefly explain their construction: Consider an even number of points p , · · · , p k (cyclicallyordered) such that p i − ∈ H and p i ∈ ∂ ∞ H . We call A i = p i − p i and B i = p i p i +1 .4onsider the minimal graph Σ over the polygonal domain bounded by A , B , · · · , A k , B k withboundary values + ∞ over the A i edges and −∞ over the B i edges (it exists and is unique upto a vertical translation, by [1, 9]), which has total curvature π (1 − k ) (see [1]). The conjugateminimal surface Σ ∗ of Σ is a minimal graph contained in H × { t ≥ } , whose boundary consistsof k geodesic curvature lines in H × { } . (The conjugation for minimal surfaces in H × R was introduced by Daniel [2] and by Hauswirth, Sa Earp and Toubiana [5].) If we reflect Σ ∗ with respect to H × { } , we get a properly embedded minimal surface of genus zero, k endsasymptotic to vertical geodesic planes and total curvature π (1 − k ) . For k = 2 , the obtainedexamples are usually called horizontal catenoids , and have been recently classified by Hauswirth,Nelli, Sa Earp and Toubiana as the only complete minimal surfaces in H × R with finite totalcurvature and two ends, each one asymptotic to a vertical geodesic plane.Using a gluing method, the second author has recently constructed in a joint work withMartín and Mazzeo a wide range of properly embedded minimal surfaces with finite totalcurvature and finite topology (with possibly positive genus).We wondered if Scherk minimal graphs were, together with the vertical geodesic planes, theonly complete, embedded, simply-connected examples of finite total curvature. In this sectionwe explain the simple construction of other different complete, embedded, simply-connectedexamples, that we will call twisted Scherk examples . Let us first construct an example with total curvature − π . Let p , p be two points in ∂ ∞ H .Up to an isometry of H , we can assume p = 1 and p = e iθ , for some fixed θ ∈ (0 , π/ (see Figure 1). We call A = p , B = p p and C = p . Let ∆ be the geodesic trianglebounded by A ∪ B ∪ C . By the triangle inequality at infinity (see [1, Lemma 3]), we get that ∆ satisfies the Jenkins-Serrin condition for the existence of a minimal graph u over ∆ withboundary values + ∞ on A , −∞ on B and on C (see [1, Theorem 3] and [9, Theorem 3.3]).Now let us see that the graph surface Σ( u ) of u has finite total curvature: For any positiveinteger n , we denote r = 1 − / ( n + 1) and p ,n = r , p ,n = re iθ . By Theorem 3 in [12], thereexists a minimal graph u r ( n ) over the geodesic triangle of vertices , p ,n , p ,n taking boundaryvalues + n on p ,n , − n on p ,n p ,n and on p ,n . By the Gauss-Bonnet formula, the graphsurface of u r ( n ) has total curvature π . Since u r ( n ) converges uniformly on compact sets of ∆ to u as n → ∞ , the total curvature of Σ( u ) is at most π , and then finite.By rotating Σ( u ) an angle π about the horizontal geodesic ray p contained in its boundary,we obtain a minimal graph whose boundary consists of the vertical geodesic { }× R . We extendsuch a graph by rotation of angle π about its boundary, and we get a properly embedded simply-connected minimal surface Σ .Since Σ consists of four copies of Σ( u ) , then it has finite total curvature. Then equation (2)5igure 1: Left: The minimal graph over the triangle region with these prescrived values is thefundamental piece of a twisted Scherk example Σ with total curvature − π . Right: Verticalprojection of Σ .applies. In our case, g = 0 , n = 1 and m = 2 ( m = 2 follows from the fact that the intersectionof M with a horizontal slice H × { t } , for t > large enough, consists of three divergent curves,see Figure 1). Thus R Σ K = − π .Now, let us consider k ≥ . Let Ω be a polygonal domain whose vertices are and k − different ideal points p , · · · , p k − ∈ ∂ ∞ H . Assume that Ω satisfies the Jenkins-Serrincondition of Theorem 3 in [1] or Theorem 3.3 in [9]. The example below proves that there existsuch domains. We call Σ the minimal graph over Ω with boundary values + ∞ on p and on p i p i +1 , for ≤ i ≤ k − ; and −∞ on p i − p i , for ≤ i ≤ k − , and zero on p k − . Byrotating Σ an angle π about the vertical geodesic line { } × R in its boundary, we obtain aproperly embedded simply-connected minimal surface Σ k . Arguing similarly as for Σ , we canprove that R Σ k K = − kπ . Then we have proved the following theorem. Theorem 5.
For any integer k ≥ , there exists a properly embedded simply-connected minimalsurface Σ k of finite total curvature − kπ which is not a minimal (vertical) graph. Now let us construct a polygonal domain Ω in the above setting. For any θ ∈ (0 , π k ) , let Ω θ be the polygonal domain with vertices , e p = 1 , and p n = e i ( n − θ , ≤ n ≤ k + 1 . We mark by + ∞ the edge , e p and those of the form p i p i +1 ; by −∞ the edges of the form p i − p i ; and by the edge p k +1 . It is clear that Ω θ does not satisfy the Jenkins-Serrin condition(see Theorem 3 in [1] or Theorem 3.3 in [9]), as we can consider the inscribed polygonal domain6igure 2: Left: The fundamental piece of a twisted Scherk example Σ with total curvature − π . Right: Vertical projection of Σ . P ⊂ Ω with vertices , e p , p , p and any choice of disjoints horocycles H , H , H at e p , p , p respectively, for which dist H ( , H ) + dist H ( H , H ) = dist H ( , H ) + dist H ( H , H ) .To solve this problem, we consider a small perturbation of e p : Let Ω θ,β be the polygonaldomain with vertices p = e − iβ , for β ∈ (0 , π − kθ ] small, and p n defined as above, for ≤ n ≤ k + 1 . This domain Ω θ,β satisfies the Jenkins-Serrin condition if we label by + ∞ the edge , p and those of the form p i p i +1 ; by −∞ the edges of the form p i − p i ; and by the edge p k +1 .Let R be the reflection with respect to the geodesic containing p k +1 . Then Ω = Ω θ,β ∪ R (Ω θ,β ) is in the desired conditions. See Figure 2. Theorem 6. If M is a complete minimal surface of total curvature − π in H × R , then M isthe Scherk minimal graph over an ideal quadrilateral.Proof. Since the total curvature of M is − π , we have by equation (2) in Theorem 1 that − π = 2 π − g − n − n X i =1 m i ! . We already know that m i ≥ , by Assertion 3. And n ≥ , since a complete minimal surface in H × R cannot be compact. So the only possibility is g = 0 , n = 1 (hence the complete minimalsurface M is simply-connected) and m = 1 . 7s m = 1 , we know by Lemma 4 that there are four points p , p , p , p ∈ ∂ ∞ H , with p i = p i +1 for any i , such that the end of M corresponds to ( p p × { + ∞} ) ∪ ( p p × {−∞} ) ∪ ( p p × { + ∞} ) ∪ ( p p × {−∞} ) , together with the complete vertical geodesics { p i } × R in the ideal cylinder ∂ ∞ H × R joiningtheir endpoints.Let us now prove that the four points p i are all different. By the maximum principleusing vertical geodesic planes, we know that at least three of them are different as M cannotbe a vertical plane. Suppose p = p (the case p = p follows similarly). Also using themaximum principle with vertical geodesic planes, we get that the vertical projection π ( M ) of M is contained in the ideal geodesic triangle of vertices p , p , p . Even more, π ( M ) iscontained in a domain T ⊂ H bounded by p p , p p and a strictly concave (with respect to T ) curve α . We observe that the points in M projecting onto α have horizontal normal vector.Suppose that the vertical projection of the limit normal vector of M (that we also call N )along p p × { + ∞} points to T . We observe that the horizontal curves in M with endpointin { p } × R arrive orthogonally to ∂ ∞ H × R . In particular, N is constant along the verticalasymptotic line { p } × R . On one hand that implies, looking at the behavior of N along theasymptotic boundary of M (corresponding to the end) that the vertical projection of N along p p × {−∞} also points to T , and its projection along p p × {±∞} goes out from T . Onthe other hand, if we follow the projection of N along α , we obtain that it points to T along p p × {±∞} , a contradiction.We now claim that p , p , p , p are cyclically ordered. We define the solid cylinder C r,T = { ( z, t ) : | z | ≤ r, | t | ≤ T } , for r < close to one and T large, and consider M r,T = M ∩ C r,T ,which is a compact minimal surface bounded by two horizontal compact curves contained in { t = T } close to p p × { T } and p p × { T } , two curves on { t = − T } close to p p × {− T } and p p × {− T } , and four curves on {| z | = r } close to vertical lines. By the flux formula withrespect to the Killing vector field ∂ t (see [8, Proposition 3]), we have Z ∂M r,T h ν, ∂ t i = 0 , (3)where ν is the outward-pointing unit conormal to M r,T along ∂M r,T . We get from (3), takinglimits as r → and T → + ∞ , that | p p | + | p p | = | p p | + | p p | , where | • | denotes (as in [1])the hyperbolic length of the curve • outside some disjoint horocycles at the ideal points p i ,identifying H with the corresponding horizontal slice. By the triangle inequality at infinity [1,Lemma 3] we get that p , p , p , p must be cyclically ordered.We call Ω the ideal quadrilateral with vertices p , p , p , p . By the maximum principleusing vertical geodesic planes, we get that π ( M ) ⊂ Ω . On the other hand, the geometry of8igure 3: Left: The nodal domains between M and Γ × R at a point with horizontal normalvector. Right: The intersection curves between M and Γ ε × R .the end of M says that a neighborhood of ∂ Ω is contained in π ( M ) . Since M is complete andsimply-connected, we conclude π ( M ) = Ω .Now let us show that the normal vector of M is never horizontal. Suppose there exists apoint P ∈ M such that N ( P ) = 0 . Let Γ × R be the vertical geodesic plane tangent to M at P .Since M and Γ × R have first contact order at P , their intersection consists of k curves meetingat equals angles at P , with k ≥ . Thus, there are at least four branches of M ∩ (Γ × R ) leaving P (see Figure 3, left). Since M is simply-connected, we deduce using the maximum principlewith vertical planes that there cannot exists a compact cycle in M ∩ (Γ × R ) . Hence Γ cannotintersect two edges of Ω , so it must have some p i as an endpoint. Denote by γ = γ ( t ) , t ∈ R ,the arc-length parameterized geodesic of H orthogonal to Γ such that γ (0) = π ( P ) ; and by Γ t the geodesic of H passing through γ ( t ) orthogonally (in particular, Γ = Γ ). For ε > small, Γ ε intersects two edges of Ω , say p p and p p , and the number of intersection curves betweenthe vertical plane Γ ε × R and M is at least two (see Figure 3, right). But only one branchof the intersection curves can arrive to p p × { + ∞} (resp. p p × {−∞} ), the other branchshould be a compact loop, a contradiction.We have prove then that, for any point q ∈ Ω , the intersection of { q } × R with M istransverse. So the number of intersection points does not depend on q . For q near an edge of Ω this number is one. We conclude that M is a graph over Ω .9 eferences [1] P. Collin and H. Rosenberg, Construction of harmonic diffeomorphisms and mini-mal graphs , Ann. of Math., 172 (2010), 1879-1906. DOI:10.4007/annals.2010.172.1879,arXiv:math/0701547.[2]
B. Daniel , Isometric immersions into S n × R and H n × R and applications to minimalsurfaces, Trans. Amer. Math. Soc., (2009), 6255–6282. MR2538594, Zbl pre05638191.[3] L. Hauswirth, Minimal surfaces of Riemann type in three-dimensional product manifolds ,Pacific J. Math. 224 (2006), 91-117. arXiv:math/0507187, MR2231653, Zbl 1108.49031.[4] L. Hauswirth and H. Rosenberg,
Minimal surfaces of finite total curvature in H × R , Mat.Contemp., 31 (2006), 65-80. MR2385437, Zbl 1144.53323.[5] L. Hauswirth, R. Sa Earp and E. Toubiana, Associate and conjugate minimal immersionsin M × R , Tohoku Math. J. (2) 60 (2008), 267-286.[6] F. Martín, Rafe Mazzeo and M.M. Rodríguez, Minimal surfaces with positive genus andfinite total curvature in H × R , arXiv:1208.5253.[7] L. Hauswirth, B. Nelli, R. Sa Earp and E. Toubiana, Minimal ends in H × R with finitetotal curvature and a Schoen type theorem, prepint, arXiv:1111.0851.[8] D. Hoffman„ J. Lira and H. Rosenberg, Constant Mean Curvature Surfaces in M × R ,Trans. Amer. Math. Soc., 358:2 (2006), 491-507.[9] L. Mazet, M.M. Rodríguez and H. Rosenberg, The Dirichlet problem for the minimalsurface equation with possible infinite boundary data over domains in a Riemanniansurface , Proc. London Math. Soc., 102:3 (2011), 985-1023. DOI:10.1112/plms/pdq032.arXiv:0806.0498.[10] F. Morabito,
A Costa-Hoffman-Meeks type surface in H × R , Trans. Amer. Math. Soc.,363:1 (2010), 1-36.[11] F. Morabito and M.M. Rodríguez, Saddle Towers and minimal k -noids in H × R , J. Inst.Math. Jussieu, 11(2): 333-349 (2012). DOI:10.1017/S1474748011000107. arXiv:0910.5676.[12] B. Nelli and H. Rosenberg, Minimal surfaces in H × R , Bull. Braz. Math. Soc., 33:2(2002), 263-292. 1013] R. Osserman, A survey of minimal surfaces , Dover Publications, New York (1986).MR0852409, Zbl 0209.52901.[14] J. Pyo,
New complete embedded minimal surfaces in H × R , Ann. Global Anal. Geom.40:2 (2011) 167-176. arXiv:math/0911.5577.[15] R. Sa Earp and E. Toubiana, Screw motion surfaces in H × R and S × R , Illinois J.Math., 49:4 (2005), 1323-1362.Juncheol PyoDepartment of MathematicsPusan National UniversityBusan 609-735, Koreae-mail: [email protected] M. Magdalena RodríguezDepartamento de Geometría y TopologíaUniversidad de GranadaFuentenueva, 18071, Granada, Spaine-mail: [email protected]@ugr.es