DOCTEUR DE L UNIVERSITÉ PARIS XII

Transcription

1 THÈSE présentée par Mohamed Boussaïri JLELI pour obtenir le titre de DOCTEUR DE L UNIVERSITÉ PARIS XII Spécialité: Mathématiques Titre: Hypersurfaces à courbure moyenne constante. Soutenue le 1 janvier 004 devant le jury composé de: M. Ahmad ELSOUFI Président du jury M. Frédéric HELEIN Rapporteur M. Robert KUSNER Rapporteur M. Frank PACARD Directeur de thèse M. Harold ROSENBERG Examinateur M. Etienne SANDIER Examinateur

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3 A mes parents, A mes frères et mes sœurs, A ma femme et taati,...

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5 Remerciements Mes premiers remerciements vont à mon directeur, monsieur Frank PACARD, qui m a fait confiance en me proposant ce sujet. Son enthousiasme, son énergie, sa constante disponibilité et surtout ses qualités humaines ont décuplé ma motivation. Je pense avoir appris énormément à son contact, j espère apprendre encore avec lui. Je remercie messieurs Frédéric HELEIN et Robert KUSNER pour le temps qu ils ont consacré à rapporter cette thèse. Je souhaite de plus leur témoigner toute ma gratitude et mon respect. Mes vifs remerciements vont à messieurs Ahmad ELSOUFI, Harold ROSENBERG et Etienne SANDIER qui me font un grand honneur en acceptant d être membre du jury. Je tiens à remercier les deux occupants du bureau P3-437: Hassen Aydi et Aurelia Fraysse qui ont supporté mes humeurs et ma dissipation. Mes remerciements s adressent aussi à tout les membres du centre de Mathématiques de Paris XII, tant les enseignantschercheurs que les thésards. Enfin et dans un registre plus personnel, je remercie mes amis plus particulièrement Abdellatif, Abou Ayyoub, Fethi, Sahbi, Sami, ma famille et surtout mes parents et ma femme qui loin des Mathématiques m ont toujours apporté le soutien nécessaire.

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7 Contents 1 Notation 15 Constant mean curvature hypersurfaces of revolution 17 1 First parameterization Embedded constant mean curvature hypersurfaces Immersed constant mean curvature hypersurfaces Second parameterization The n-unduloids The n-nodoids Asymptotic properties 1 Compactness results The period of σ τ Expansion of the functions σ τ and κ τ as τ tends to The physical period of a n-delaunay hypersurface The Jacobi operator 35 1 The Jacobi operator about a n-delaunay Geometric Jacobi fields Indicial roots associated to a n-delaunay The Jacobi operator about the sphere S n The Jacobi operator about the n-catenoid Maximum principle 43 1 Maximum principle for the hyperplane R n Maximum principle for the Jacobi operator about S n Maximum principle for the Jacobi operator about the n-catenoid Maximum principle for the n-delaunay hypersurface The indicial roots of a n-delaunay hypersurface 50 1 Positivity of the indicial roots The limit of the indicial roots as τ tends to Bifurcation results for n-nodoids 56 1 Preliminaries The spectrum of L τ when τ > The spectrum of L τ when τ < Equivariant eigenspaces The bifurcation result

8 5.1 Bifurcation from simple eigenvalue General bifurcation result Distinguishing the solutions by their symmetry group Moduli space theory 65 1 Preliminaries Weighted spaces A priori estimates Fredholm properties of the Jacobi operator Nondegenerate hypersurfaces From weighted Lebesgue spaces to weighted Hölder spaces Near a nondegenerate constant mean curvature hypersurface Σ Deformations of the ends of Σ Constant mean curvature hypersurface close to Σ(Ξ) Structure of the mean curvature operator Linear analysis of the n-delaunay Jacobi operator 89 1 The Jacobi operator about a half of n-delaunay hypersurface The Poisson operator associated to Constant mean curvature hypersurfaces close to a n-delaunay hypersurface The nonlinear elliptic equation Perturbation of a half n-delaunay hypersurface The Cauchy data mapping Perturbation of constant mean curvature hypersurfaces The mean curvature operator for vertical graphs Linear analysis Mapping properties of Λ A The Poisson operator associated to R n Nonlinear analysis Constant mean curvature hypersurfaces close to Σ Addition of ends with prescribed Delaunay parameters 11 1 Addition of Delaunay ends The nondegeneracy of the hypersurfaces constructed End-to-End connected sum 17 1 Regular ends of constant mean curvature hypersurfaces Connecting two constant mean curvature hypersurfaces together Mapping properties of the Jacobi operator about M m The perturbation argument An end-to-end construction for compact constant mean curvature hypersurfaces The balancing formula for constant mean curvature hypersurfaces Building blocks Type-1 hypersurfaces

9 . Type- hypersurfaces The construction The mean curvature of the approximate solution Extension of the elements of the deficiency spaces Compact constant mean curvature hypersurfaces with topology

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11 Introduction En 1841, C. Delaunay [3] a classifié toutes les surfaces de révolution de R 3 dont la courbure moyenne est constante égale à 1. Ces surfaces sont maintenant connues sous le nom de surfaces de Delaunay. Les génératrices de ces surfaces sont en fait des roulettes de coniques [4]. Une généralisation de cette classification pour des hypersurfaces de R n, n 3 qui ont certaines symétries a été élaborée en 1981 dans [8]. En dimension supérieure, il existe des hypersurfaces de révolution dans R n+1, pour n 3, dont la courbure moyenne est une constante (fixée égale à 1). Ces hypersurfaces forment une famille à un paramètre D τ pour τ (, 0) (0, τ ] Lorsque τ (0, τ ], l hypersurface D τ est plongée et est appelée n-onduloïde, alors que lorsque τ est négatif, l hypersurface D τ est simplement immergée et est appelée n-nodoïde. Dans les chapitres et 3, une analyse de la paramétrisation de ces hypersurfaces est faite. Une attention toute particulière est portée au comportement de ces hypersurfaces lorsque le paramètre τ tend vers une valeur limite (i.e. τ tend vers 0, ou τ ). L analyse du comportement des n-nodoïdes quand τ tend vers permet de montrer qu il existe, le long de la famille des n-nodoïds, une infinité de points de bifurcations. En particulier ceci assure l existence d une infinité de branches d hypersurfaces à courbure moyenne constante qui sont 1-périodiques, incluses dans un cylindre mais qui ne sont pas des hypersurfaces de révolution. Ce résultat généralise en toutes les dimensions le résultat de R. Mazzeo et F. Pacard [19] qui n était valable que pour les surfaces, n =. La démonstration de ce résultat est basée sur une étude précise du spectre de l opérateur de Jacobi (opérateur de courbure moyenne linéarisé) associé à un n-nodoide, lorsque le paramètre τ tend vers. On montre que, quand le paramètre τ tend vers, l indice de Morse des n-nodoïdes tends vers +. Une application du Théorème classique de M. Crandall et P. Rabinowitz [], [9], concernant les bifurcations à partir d une valeur propre simple nous permet alors de démontrer le Théorème 1: On considère G = O(n 1) O(1). Il existe l 0 0 et, pour tout l l 0, il existe τ l (, 0) et une branche régulière d hypersurfaces à courbure moyenne constante égale à 1 qui bifurque de la famille des n-nodoïdes en D τ en τ = τ l. Chaque élément de cette branche est simplement périodique et invariant par l action du groupe {±I} G mais il n est pas invariant par l action de {±I} O(n). Quand l tend vers +, on a une estimation du paramètre de bifurcation l(n + l) τ l := + O(1), n Un résultat semblable est valable si l on considère G = O(p) O(n p). En utilisant un résultat de bifurcation dû à J. Smoller et A. Wasserman [7], résultat basé sur l utilisation de l indice de Conley, le résultat précedent peut être généralisé 11

12 comme suit : Soit k N et G O(n), on note E k,g le sous-espace de E k (l espace propre du Laplacien sur S n 1 correspondant à la valeur propre λ k = k(n + k )) des fonctions invariantes sous l action des éléments de G. Théorème : Soit G O(n) tel que dim E k,g 0 pour une infinité de k N. Alors, il existe une suite (τ l ) l qui tend vers pour laquelle la famille des n-nodoïdes bifurquent. En particulier dans chaque voisinage de l hypersurface D τl, il existe une hypersurface à courbure moyenne constante égale à 1 qui est simplement périodique et invariante sous l action de {±I} G mais qui n est pas invariante sous l action de {±I} O(n). Dans le chapitre 8, nous étudions l ensemble des hypersurfaces complètes non compactes, à courbure moyenne constante, qui ont un nombre fini de bouts asymptotiques à des n-nodoïdes. Cette étude généralise en toutes les dimensions et pour des hypersurfaces dont les bouts peuvent être de type n-nodoïdes, l étude menée par R. Kusner, R. Mazzeo et D. Pollack [16]. Pour tout k N on note M k l espace des hypersurfaces de R n+1 dont la courbure moyenne est constante égale à 1 qui sont complètes non compactes et qui ont k bouts asymptotiques à des n-delaunays dont le paramètre τ (τ, 0) (0, τ ], où τ < 0 est choisi assez petit. On commence par étudier l opérateur de Jacobi L Σ associé à Σ M k, agissant sur des espaces à poids. Nous donnons dans le Chapitre 8 une démonstration détaillée de l analyse fonctionnelle des opérateurs de Jacobi définis sur de telles hypersurfaces car cette analyse n avait jamais été détaillée auparavent. Ensuite, une application (plus ou moins directe) du Théorème des fonctions implicites permet de démontrer le : Théorème 3 : On suppose que Σ est un élément non dégénéré de M k (i.e. si L Σ w = 0 et si w L (Σ) alors w = 0). Supposons de plus que les paramètres de Delaunay τ l des bouts de Σ appartiennent à (τ, 0) (0, τ ] où τ < 0 est choisi assez petit. Alors, il existe un ouvert U M k contenant Σ qui est une variété régulière de dimension k (n + 1). Insistons sur le fait que l on ne quotiente pas par le groupe des isométries de R n+1. De plus ce résultat n est valable que pour les hypersurfaces dont les bouts sont asymptotiques à des n-onduloïdes ou bien des n-nodoïdes dont le paramètre est proche de 0. Dans les années 80, une méthode de construction de surfaces à courbure moyenne constante à été mise au point par N. Kapouleas d une part pour construire des surfaces compactes [11], [1] d autre part pour construire de surfaces noncompactes complètes dans R 3 [10]. Cette méthode est basée sur une méthode analogue utilisée par R. Schoen pour construire des métriques à courbure scalaire constante sur la sphère privée d un nombre fini de points. Ces techniques ont ensuite été développées et systématisées [18], [0], [5]. En particulier, étant donnée Σ une surface à courbure moyenne constante nondégénérée, il est possible de construire une famille de surfaces à courbure moyenne constante, elles aussi nondégénérées, qui sont somme connexe de Σ et d un nombre fini de surfaces de Delaunay. Dans les chapitres 1 et 13, nous étendons ces résultats au cas des hypersurfaces et nous considérons aussi le cas où l hypersurface de départ Σ est dégénérée. Plus précisément, soit Σ une hypersurface à courbure moyenne constante (compacte ou bien élément de M k ). On note L Σ l opérateur de Jacobi associé à Σ et l on note 1

13 φ 1,..., φ N une base du noyau de L Σ dans L (Σ). Si p =: (p 1,..., p m ) sont m points de Σ, on note Φ(p) := (φ i (p j )) ij. Alors on a le : Théorème 4 : On suppose que les conditions suivantes sont vérifiées : et il existe a := (a 1,..., a m ) (R {0}) m tel que Rang (Φ(p)) = N, (1) Φ(p) t a = 0. () Alors pour tout τ > 0 assez petit, il existe une hypersurface à courbure moyenne constante à k + m bouts qui est obtenue en effectuant la somme connexe de m demi-hypersurfaces de type n-delaunay dont le paramètre de Delaunay est égal à τ i := a i a i (1 n)/n τ à Σ aux points p i Σ. Ces hypersurfaces sont (pour τ assez petit) nondégénérées. Dans le cas particulier où l hypersurface Σ est nondégénérée, il est toujours possible d ajouter un nombre fini de bouts de type n-delaunay et la position des points où le recollement a lieu est arbitraire. Un Corollaire immédiat de ce résultat est le : Théorème 5 : Pour tout k, l ensemble M k n est pas vide et contient des éléments nondégénérés. Une nouvelle technique de construction dite recollement de surfaces bout-à-bout, a été récemment introduite par J. Ratzkin [4]. Dans le Chapitre 14, nous donnons une généralisation de cette construction en dimension quelconque. Dans cette construction, on considère deux hypersurfaces Σ 1 M k1 et Σ M k dont on suppose qu elles ont des bouts E 1 Σ 1 et E Σ qui sont asymptotiques à la même n-delaunay de paramètre τ [τ, 0) (0, τ ). On peut alors aligner Σ 1 et Σ de telle sorte que les axes de E 1 et de E soient confondus (avec des directions opposées). On peut alors translater l une des surfaces le long de cet axe et construire un nombre dénombrable de familles d hypersurfaces dont la courbure moyenne est proche de 1. Ces hypersurfaces peuvent alors être perturbées en des hypersurfaces à courbure moyenne constante éléments de M k1 +k. Le principal avantage de cette construction (outre qu elle permet de construire de nouveau exemples) réside dans la simplicité de sa démonstration. Cette procédure de recollement permet de donner une méthode relativement élémentaire de construction d hypersurfaces compactes à courbure moyenne constante. Dans un premier temps, on utilise le résultat du chapitre 13 pour construire des hypersurfaces à courbure moyenne constante avec un nombre fini de bouts. Ensuite, on montre que la méthode de recollement bout-à-bout peut être mise en œuvre pour construire des hypersurfaces compacte dont la courbure moyenne est constante. L analyse nécessaire est essentiellement basée sur l analyse développée dans le chapitre 8 afin de démontrer le Théorème 4. En particulier nous obtenons le : Théorème 6 : Il existe un ensemble dénombrable d hypersurfaces compactes à courbure moyenne constante (géométriquement distinctes) qui ont la topologie de S n k H 1 (i.e. une sphère S n avec k anses attachées), k. En fait les hypersurfaces que nous construisons ont de nombreuses symétries et l on montre qu il existe un intervalle non vide I et une fonction f régulière définie sur I telle 13

14 que toute solution (x, m) I N de l équation f(x) + m x N, permet de construire une hypersurface compacte à courbure moyenne constante égale 1 dont la topologie est celle d une somme connexe de k copies de S n 1 S 1. 14

15 Chapter 1 Notation In this brief section, we record some notation which will be used in this work. To begin with, let us denote by θ e j (θ), for j N the eigenfunctions of the Laplace-Beltrami operator on S n 1 with corresponding eigenvalue λ j. That is S n 1 e j = λ j e j. We further assume that these eigenvalues are counted with multiplicity, namely λ 0 = 0, λ 1 =... = λ n = n 1, λ n+1 = n,... and λ j λ j+1, and that the eigenfunctions are normalized by for all j N. S n 1 e j(θ) dθ = 1, It will be convenient to define some orthogonal projections π I and π II acting on functions of L (S n 1 ) as follows : If the eigenfunction decomposition of the function φ L (S n 1 ) is given by φ = j N a j e j L (S n 1 ) (1.1) then, we set and We define π I (φ) := π II (φ) := n a j e j, j=0 j n+1 a j e j. This being understood, we define the linear operator ( ( ) ) 1 n δ j := λ j +. (1.) D θ : H 1 (S n 1 ) L (S n 1 ), by D θ (φ) := j N δ j a j e j, 15

16 if the eigendecomposition of the function φ is given by (1.1). In other words, we have ( ( ) ) 1 n D θ = S n 1 +. It is an easy exercise to check that D θ is well defined and bounded. We will agree that, given two functions f and g, the notation g = O C (f) means that, for all k N, there exists a constant c k > 0 such that k s f c k k s g on the domain where the functions f and g are defined. 16

17 Chapter Constant mean curvature hypersurfaces of revolution We give two different parameterizations of a one parameter family of hypersurfaces of revolution in R n+1, which are immersed or embedded and have constant mean curvature normalized to be equal to 1. These hypersurfaces, which were originally studied in [13], generalize the classical constant mean curvature surfaces in R 3 which were discovered by Delaunay in [3] in the middle of the 19-th century. 1 First parameterization We are looking for constant mean curvature hypersurfaces of revolution (say around the x n+1 axis). Such an hypersurface can be locally parameterized by X : (t 1, t ) S n 1 R n+1 (t, θ) (ρ(t) θ, t) where the function t ρ(t) is a smooth positive function which is defined over some interval (t 1, t ). The first fundamental form g of the hypersurface parameterized by X is given by g = (1 + ( t ρ) ) dt dt + ρ dθ i dθ j where dθ i dθ j denotes the first fundamental form of S n 1. Let assume that the orientation of this hypersurface is chosen so that the unit inward normal vector field is given by N := ( t ρ) ( θ, tρ). With this chosen orientation, the second fundamental form b of the hypersurface parameterized by X is given by b = ( t ρ) ( t ρ dt dt + ρ dθ i dθ j ). 17

18 It follows at once from the above expressions that the mean curvature H of the hypersurface parameterized by X (which is the average of the trace of the shape form) is given by H = n n ρ 1 + ( t ρ) 1 1 n (1 + ( t ρ) ) 3 t ρ. (.1) Hence, the condition that the mean curvature of the hypersurface parameterized by X is equal to some given function H, is given by the equation t ρ n 1 ( 1 + ( t ρ) ) + n H ( 1 + ( t ρ) ) ) 3 = 0. (.) ρ We introduce H(ρ, t ρ) := ρ n ( t ρ) H ρn. (.3) In the case where the function H is constant, it follows from a simple computation that H(ρ, t ρ) is constant along solutions of (.). This property will be extensively used to derive a priori estimates for solutions of (.). 1.1 Embedded constant mean curvature hypersurfaces Using the above computation, it is easy to show that there exists a one parameter family of embedded constant mean curvature hypersurfaces of revolution. Indeed, in the case where H = 1, there are two special solutions of (.) which can be immediately determined. The first is the constant solution ρ n 1 n S n 1 ( n 1 n which corresponds to the cylinder ) R R n+1. The other explicit solution is ρ 0 (t) := 1 t, which is defined for t ( 1, 1) and which corresponds to a sphere of radius 1 centered at the origin. These two solutions are the two end-points of a one parameter family of solutions of (.) which produce embedded constant mean curvature hypersurfaces. Let us now describe this family. It will be convenient to define τ := 1 n 1 (n 1) n. n Given τ (0, τ ], we define ε (0, n 1 n ] by the identity ε n 1 (1 ε) = τ n and we define the function ρ τ to be the unique solution of (.), with H = 1, such that ρ τ (0) = ε and t ρ τ (0) = 0. Using the fact that H(ρ τ, t ρ τ ) = τ n, is constant along solutions of (.), it is a simple exercise to show that the function ρ τ is periodic. We will denote by D τ the hypersurface of revolution which is parameterized by X τ (t, θ) = (ρ τ (t) θ, t). By construction, D τ is an embedded constant mean curvature hypersurface of revolution will be called an n-unduloid 1. 1 This has not to be confused with the terminology used in [7] where this terminology refers to surfaces with k ends 18

19 Figure.1: Graphs of ρ τ 1. Immersed constant mean curvature hypersurfaces This time, we construct a one parameter family of immersed constant mean curvature hypersurfaces of revolution. Given τ (, 0), we define ε (0, + ) by the identity ε n 1 (1 + ε) = ( τ) n and we define the function ρ τ to be the unique solution of (.), with H = 1 such that ρ τ (0) = ε and t ρ τ (0) = 0. Using the fact that H( ρ τ, t ρ τ ) = ( τ) n, is constant along solutions of (.), it is a simple exercise to show that, ρ τ is defined over some maximal interval ( T τ, T τ ) where T τ > 0. Furthermore, we have lim ρ τ = τ and lim t ρ τ = ± t ± T τ t ± T τ Now, for the same value of τ, we define ε (1, + ) by the identity ε n 1 (1 ε) = ( τ) n. This being done, we define the function ρ τ to be the unique solution of (.), with H = 1 such that ρ τ (0) = ε and t ρ τ (0) = 0. Using the fact that H( ρ τ, t ρ τ ) = ( τ) n, is constant along solutions of (.), it is a simple exercise to show that, ρ τ is defined in ( T τ, T τ ) where T τ > 0. Furthermore, we have lim ρ τ = τ and lim t ρ τ =. t ± T τ t ± T τ Finally, the graph of the function ρ τ and the graph of the function ρ τ (once translated along the x n+1 axis by T τ T τ ) can be glued together to produce a piece of constant mean curvature hypersurface of revolution. In turn, we can extend this piece of hypersurface by periodicity along the x n+1 axis to produce a complete immersed constant mean curvature surface. These surfaces will be referred to as n-nodoids. Figure.: Graphs of ρ τ and ρ τ Second parameterization The previous parameterization is probably interesting in order to investigate the geometric properties of the hypersurfaces. However, in our analysis, it will be more interesting to consider an isothermal type parameterization for which the geometric properties of the hypersurfaces are hidden but which will be more convenient for analytical purposes. 19

20 Hence, we are now looking for hypersurfaces of revolution which can be parameterized by X(s, θ) = (τ e σ(s) θ, κ(s)), for (s, θ) R S n 1. The constant τ being fixed, the functions σ and κ are determined by asking that the hypersurface parameterized by X has constant mean curvature equal to H and also by asking that the metric associated to the parameterization is conformal to the product metric on R S n 1, namely ( s κ) = τ e σ ( 1 ( s σ) ). (.4) We choose the orientation of the hypersurface parameterized by X so that, the unit normal vector field is given by ( N := ) sκ τe σ θ, sσ. This time, using (.4) the first fundamental form g of the hypersurface parameterized by X is given by g = τ e σ (ds ds + dθ i dθ j ), and its second fundamental form b is given by b = ( s κ s σ s κ ( s σ + ( s σ) ) ) ds ds + s κ dθ i dθ j. Therefore, the mean curvature H of the hypersurface parameterized by X is given by H = 1 nτ e σ ( (n 1) s κ s κ ( s σ + ( s σ) ) + s κ s σ ). This is a rather intricate second order ordinary differential equation in the functions σ and τ which has to be complimented by the equation (.4). In order to simplify our analysis, we use of (.4) to get rid of the factor τ e σ in the above equation. This yields ( s σ s κ = s κ 1 n + s σ + ( s σ) + n H s κ ( 1 ( s σ) ) ) 1. Now, we can differentiate (.4) with respect to s, and we obtain s κ s κ = τ e σ s σ ( 1 s σ ( s σ) ). The difference between the last equation, multiplied by s σ, and the former equation, multiplied by s κ, yields s σ + (1 n)(1 ( s σ) ) + n H s κ = 0. (.5) Hence, in order to find constant mean curvature hypersurfaces of revolution, we have to solve (.4) together with (.5). This is the content of the next two paragraphs..1 The n-unduloids Recall that we have defined τ := 1 n 1 (n 1) n. n 0

21 For all τ (0, τ ]. We define σ τ to be the unique smooth nonconstant solution of ( s σ) + τ ( e σ + e (1 n)σ) = 1, (.6) with initial condition s σ(0) = 0 and σ(0) < 0. Next we define the function κ τ to be the unique solution of s κ = τ ( e σ + e ( n)σ), with κ(0) = 0. (.7) It is easy to check that σ τ and κ τ solve (.4) and (.5), when H = 1. Moreover, the function κ τ is monotone increasing since s κ τ > 0. In particular, the hypersurface parameterized by X τ (s, θ) := (τe στ (s) θ, κ τ (s)), for (s, θ) R S n 1, is an embedded constant mean curvature hypersurface of revolution. This hypersurface coincides with the hypersurface defined in 1.1 and, as already mentioned, this hypersurface will be referred to as the n-unduloid of parameter τ. Observe that the extreme element in this family which corresponds to τ = τ, is the cylinder S n 1 ( n 1 n ) R. While, as τ tends to 0 the family of n-unduloids converges to a sequence of spheres arranged along the x n+1 axis.. The n-nodoids For all τ < 0, we define σ τ to be the unique smooth nonconstant solution of ( s σ) + τ ( e σ e (1 n)σ) = 1, (.8) with initial condition s σ(0) = 0 and σ(0) < 0. Next, we define the function κ τ to be the unique solution of s κ = τ ( e σ e ( n)σ), with κ(0) = 0. (.9) Again, it is easy to check that σ τ and κ τ solve (.4) and (.5), when H = 1. However, observe that this time the function κ τ is not monotone anymore since κ τ changes sign. Hence, the hypersurface parameterized by X τ (s, θ) := ( τe στ (s) θ, κ τ (s)) for (s, θ) R S n 1 is an immersed constant mean curvature hypersurface of revolution. This hypersurface coincides with the hypersurface defined in 1. and, will be referred to as the n-nodoid of parameter τ. 1

22 Chapter 3 Asymptotic properties In this chapter, we are concerned about the behavior of the hypersurfaces which have been defined in the previous chapter, as the parameter τ tends to the limit values, namely, as τ tends to 0 or to. 1 Compactness results We begin with the study of the behavior of σ τ as τ tends to 0. We define ϕ τ := τ e στ, and η τ := τ e (1 n)στ, Using (.4) and (.5), one can check that, according to the sign of τ, the functions ϕ τ and η τ are nonconstant solutions of and ( s η) = (n 1) η (1 (ϕ ± η) ), (3.1) ( s ϕ) = ϕ (1 (ϕ ± η) ), (3.) with a + when τ > 0 and a when τ < 0. In addition, we have ϕ n 1 τ η τ = τ n. (3.3) Our first Lemma states that the functions ϕ τ and η τ and their derivatives, are uniformly bounded with respect to τ, provided that τ remains bounded. Lemma 3.1. Assume that τ 0 < 0 is fixed. Then, for all k N, there exists a constant c k > 0 which only depends on τ 0 and k, such that for all τ [τ 0, 0) (0, τ ]. ϕ τ C k + η τ C k c k Proof: When τ > 0, observe that (3.) already implies that the functions ϕ τ and η τ are uniformly bounded by 1. When τ < 0 is bounded from below by τ 0, (3.) together with (3.3) imply that the functions ϕ τ and η τ are uniformly bounded by a constant only depending on τ 0. Now that we know that the functions ϕ τ and η τ are uniformly bounded. We use (3.1) and (3.) inductively to show that the same property is also true for the derivatives of the functions ϕ τ and η τ.

23 Assume that s l is a sequence of real numbers, and that τ l is a sequence which tends to 0. For all l N, we define ϕ l := ϕ τl ( s l ) and η l := η τl ( s l ). The previous result together with Ascoli s theorem allows one to extract from the sequence (ϕ l, η l ) l, a subsequence which converges, as l tends to, to (ϕ, η ) in C k topology on any compact sets. The following Lemma classifies the possible limits (ϕ, η ). Lemma 3.. Under the above hypothesis, the following holds : Either ϕ = η 0, or ϕ 0 and there exists s such that η = or η 0 and there exists s such that 1 cosh((n 1)( s )), ϕ = 1 cosh( s ). Proof: Passing the limit in (3.3) we get ϕ n 1 η 0. This implies that, at least one of the functions η and ϕ has to be identically equal to 0. It only remains to identify the possible nontrivial limits. If ϕ 0, we can pass to the limit in (3.1) and in the derivative of (3.1) with respect to s to get the equation satisfied by η Furthermore, we have s η = (n 1) η ( 1 η ). ( s η) = (n 1) η ( 1 η ). The nontrivial solutions of these equations are all of the form for some s 0 R. s 1 cosh((n 1)( s 0 )) Now, if η 0, we can pass to the limit in (3.) and in the derivative of (3.) to get equation satisfied by η s ϕ = ϕ ( 1 ϕ ). Furthermore, we have ( s ϕ) = ϕ (1 ϕ ). This time the only nontrivial solutions of this equation are of the form s 1 cosh( s 0 ), for some s 0 R. This completes the proof of the result. 3

24 The period of σ τ When τ τ, the function σ τ is periodic and not constant. Hence, given τ (, 0) (0, τ ), we can define s τ > 0 to be the least period of the function σ τ. When τ = τ, the function σ τ is constant and we extend the function τ s τ by continuity as τ tends to τ. The value of s τ can be computed explicitly. This is the content of the : Lemma 3.3. As τ tends to τ, we have lim s τ = τ τ π n 1. Proof: Assume that τ (0, τ ) and that σ τ is a solution of (.6). For the sake of simplicity in the notations, we drop the τ indices. We can write where we have defined s σ = 1 τ f(σ), (3.4) f(σ) = e σ + e (1 n)σ. (3.5) It is easy to see that f is convex and achieves its minimum at σ := 1 n log(n 1). Using Taylor s expansion of f about σ, we can rephrase (3.4) as ( s σ) = τ τ τ ( ) τ (n 1) f(σ) (σ σ ) τ where f is a smooth function which satisfies f(σ ) = 1. Now, we define the function ω by the identity τ σ = σ + τ ω, By construction, the function ω solves ( ) τ ( s ω) = 1 (n 1) ω ˆf(ω) (3.6) where the smooth function ˆf is defined by ( ˆf(ω) := f σ + τ τ τ τ In particular, ˆf 1 and this implies that ( ) τ (n 1) ω 1 (3.7) By definition, the period of σ (or ω) is given by s τ = ω+ (τ) ω (τ) τ 1 (n 1) τ dx ( τ τ ω ) ) x ˆf(x). 4

25 where ω (τ) < 0 < ω + (τ) are the roots of the equation ( ) τ (n 1) ω ˆf(ω) = 1 τ In view of (3.7), the function ω remains bounded as τ tends to τ. Hence, we obtain lim ω ± (τ) = ± 1. τ τ n 1 Finally, Lebesgue s dominated convergence theorem implies that lim s τ = τ τ 1 n 1 1 n 1 dx 1 (n 1)x The result then follows at once. We define the variable e (0, 1) by e := τ τ τ for all τ (0, τ ). It is easy to see that the solution σ τ depends smoothly on e at e = 0. Using the proof of Lemma 3.3, we show that, as τ tends to τ, the function σ τ can be expanded as e σ τ = σ cos( n 1 ) + O(e ). n 1 where This is the content of the : σ := 1 n log(n 1). (3.8) Lemma 3.4. The solution of (3.6) satisfies e ω e=0 = 1 n 1 cos( n 1 s). Proof: With our notations, we have σ = σ + e ω Hence, is remains to understand ω, the limit of ω as τ tends to τ. By (3.7), the function ω is bounded uniformly as τ tends to τ and, passing to the limit in (3.6), we conclude that ( s ω ) + (n 1) ω = 1. The result then follows at once from the fact that σ τ achieves its minimum at s = 0 and hence this is also the case for the function ω. In the next result, we study the asymptotic behavior of the function τ s τ, as the parameter τ tends to 0. 5

26 Proposition 3.1. As τ tends to 0, we have s τ = n n 1 log τ + O(1). (3.9) Proof: In the case where τ < 0, the function σ τ is the solution of (.8). We have the expression of the period s τ as s τ = σ where we have defined the function g by and σ < 0 < σ + are the solutions of σ + dσ 1 τ g(σ) g(σ) = e σ e (1 n)σ 1 τ g(σ ± ) = 1 τ. The function g being clearly a diffeomorphism from R into R, the expression of s τ can be written as 1 τ 1 dx s τ = g g 1 (x) 1 τ x We perform the change of variables y = τx, this yields s τ = τ g g 1 (y/τ) dy 1 y. We define t τ := τ +1 1 Expanding g at ±, we get Finally, the function ( ( ) ( )) dy 1 1 g g 1 (y/τ) = g 1 g 1 τ τ t τ = n n 1 log τ + O( τ n z z g g 1 (z) n 1 ). 1 converges to 1 as z tends to + and to n 1 as z tends to. This together with Lebesgue s dominated convergence theorem, implies that the difference s τ t τ satisfies lim (s τ t τ ) = τ 0 n n y dy 1 y y. In the case where τ < 0, the result follows from the estimate of t τ and the boundedness of the last limit. Let us now assume that τ > 0. This time, the function σ τ is the solution of (.6) and we can write σ+ dσ σ s τ = 1 τ f(σ) + dσ 1 τ f(σ) σ 6 σ

27 where f has been defined in (3.5), σ is defined in (3.8) and where σ < 0 < σ + are defined to be the solutions of f(σ ± ) = 1 τ, Using arguments which are similar to the one developed in the case where τ < 0, we obtain σ+ t + dσ τ := = f 1 (1/τ) + O(1) 1 τ f(σ) t τ := σ σ σ dσ = f 1 (1/τ) + O(1) 1 τ f(σ) where f+ 1 1 (resp. f ) is the inverse of the restriction of f to (σ, + ) (resp. to (, σ )). The proof of the result when τ > 0 follows from a similar argument as in the proof of the result when τ < 0, we leave the details to the reader. + 3 Expansion of the functions σ τ and κ τ as τ tends to 0 In the remaining of this section, we obtain some precise expansions of the functions σ τ and κ τ as the parameter τ tends to 0. It will be convenient to observe that the function satisfies the equation (n 1) στ ξ τ := e ( s ξ) = (n 1) ( ξ τ (1 ± ξ n n 1 ) ). (3.10) with a + when τ > 0 and a when τ < 0. Furthermore, this function achieves its minimum at s = 0, namely ξ(s) ξ(0) := ξ 0 for all s R, since s = 0 corresponds to the point where the function σ τ is minimal. Therefore, it is possible to define a function ω by the identity ξ(s) := ξ 0 cosh(ω(s)). This being understood, our first result gives an expansion of the function ξ τ over the interval [ s τ /, s τ /] as τ tends to 0. The functions σ τ and ξ τ being s τ periodic, this gives us an expansion of the function ξ τ in all R. Lemma 3.5. For all s [ sτ, sτ ] we have ξ τ τ cosh((n 1)s) = 1 + τ n n 1 OC (1) + τ n n 1 OC ((cosh s) ), uniformly in τ as τ tends to 0. Proof: Using (3.10), we see that the function ω satisfies ( )) ( s ω) = (n 1) 1 τ (ξ n 1 0 A(cosh ω) ± τ n n 1 ξ0 B(cosh ω) (3.11) where we have defined for all z > 1 A(z) := z n n 1 1 z 1 7

28 and B(z) := z n n 1 1 z 1. Now, using (.6) or (.8), together with the fact that s σ(0) = 0, we see that ) ξ 0 = τ (1 + O( τ n n 1 ), as τ tends to 0. Plugging this information into (3.11), and assuming that we obtain ω(s) (n 1) s 1 (3.1) s ω(s) = n 1 + τ n n 1 OC ((cosh s) n ) + τ n n 1 OC ((cosh s) ), in the range where (3.1) holds. Hence, as long as (3.1) is fulfilled, we can integrate the previous expansion and conclude that ω(s) = (n 1) s + τ n n 1 OC (1) + τ n n 1 OC ((cosh s) ). (3.13) Finally, remember that the result of Proposition 3.1 implies that s τ = n n 1 log τ + O(1). This expansion, together with (3.13) prove that (3.1) is fulfilled in some interval of the form [ sτ + s, sτ s], for some fixed s > 0, independent of τ (0, τ ]. Finally, we use (3.11) to extend this estimate to [ sτ, sτ ]. The result then follows from the expansions of ω and ξ 0. The previous expansion being understood, we make use of (.7) or (.9) to obtain the corresponding estimates for the function κ τ. For further use, it will be more convenient to use the variable r := τ e σ(s). instead of the variable s. In the next Lemma, we agree that the notation f = O C 0 (r l ) means that, for all k N, there exists a constant c k > 0 such that r k k r f c k r l on the domain where the function f is defined. This definition being understood, we can now state the : Lemma 3.6. As τ tends to zero the function κ admits this expansion κ τ (s) = c τ ± τ n n r n ( 1 + τ n O C 0 (r n ) + τ n O C 0 (r n ) ), (3.14) for all r ( τ n n 1, 1), with a + when τ > 0 and a when τ < 0. Here the constant c τ R satisfies c τ c τ n n 1, for some constant c > 0 which does not depend on τ. 8

29 Proof: We recall that s κ = ±τ e ( n)σ (1 ± e nσ ). with a + when τ > 0 and a when τ < 0. Then, using the result of the last Lemma we get ) s κ = ± τ n n n 1 (cosh(n 1)s) n 1 (1 + τ n n 1 OC ((cosh s) n )), and for s [0, sτ ] this equation becomes ( ) s κ = ± n n n 1 τ n 1 e ( n)s 1 + τ n n 1 OC (e ns ) + O C (e ( n)s ). We integrate this quantity over [0, s] to conclude that κ satisfy κ = c τ ± n n 1 τ n n 1 e( n)s n ) (1 + τ n n 1 OC (e ns ) + O C (e ( n)s ), (3.15) for some constant c τ R which is bounded by a constant, independent of τ times τ n n 1. Now, using the expansion of e σ which is given in the last Lemma together with the definition of r = τ e σ(s) we deduce that ( ) r = 1 n n 1 τ n 1 e s 1 + O C (e ( n)s ) + τ n n 1 OC (1) + τ n n 1 OC (e s ). In particular, this implies that c 0 τ n n 1 e s r C 0 τ n n 1 e s, for some positive constants c 0 < 1 < C 0 which do not depend on τ. Inserting this in (3.15) to obtain the result. The estimate of κ τ when s [ sτ, 0] can be established similarly. We end this chapter with the study of the behavior of s τ and σ τ when the parameter τ tends to. We assume that τ < 0 and we introduce an auxiliary function ( sτ ) γ τ (t) := τσ τ π t. With this normalization, the function γ τ is π periodic. We prove in the next result that γ τ converges to a multiple of the cosine function, as τ tends to. We also give the asymptotic of the period s τ as τ tends to. Lemma 3.7. The following expansions hold as τ tends to s τ = π ( 1 + O( τ 1 ) ) and τ s τ = π nτ nτ (1 + O( τ 1 )). (3.16) Furthermore, as τ, the function γ τ converges uniformly to the function cos t. More precisely γ τ (t) = 1 n cos t + O C ( τ 1 ) and τ γ τ = O C ( τ ). (3.17) 9

30 Proof: Step 1 We define The proof of the Lemma is decomposed into three steps. σ M := σ τ (s τ /), the maximal value of the function σ τ and s 0 (0, sτ ) denotes the value of s for which σ τ vanishes. The function σ τ being maximal when s = s τ /, we can define the function v by the identity σ τ (s) = σ M cos(τv(s)), for all s [s 0, sτ ] and, to ensure uniqueness of v, we require that v(s τ /) = 0 and s v(s τ /) > 0. The parameter τ being negative, the function σ τ satisfies (.8), hence the function v satisfies ( s v) = g(σ τ ) g(σ M ) στ σm (3.18) where we recall that the function g is defined by g(σ) := e σ e (1 n)σ. The function σ τ being maximal at s τ /, we have from (.8) e σ M e (1 n)σ M = 1 τ and this implies that σ M can be expanded as σ M = 1 nτ + O(τ ). Hence σ τ = O C ( τ 1 ). Using the first term of the Taylor expansion of the function g we conclude that g(σ τ ) g(σ M ) σ τ σ M = n + O( τ 1 ) and g(σ τ ) + g(σ M ) σ τ + σ M = n + O( τ 1 ). These expansions together with the integration of (3.18) yield ( v = n s s ) τ (1 + O( τ 1 ) ) (3.19) for s [s 0, sτ ]. Step Now, we define σ m = σ τ (0) and as above we write for s [0, s 0 ], with σ τ = σ m cos(τw(s)), τ w(0) = 0 and s w (0) > 0. Following the strategy developed in the first part of this proof, we show that and also that σ m = 1 nτ + O(τ ). w = n s ( 1 + O C ( τ 1 ) ). 30

31 for all s [0, s 0 ]. Step 3 We use the fact that σ M cos (τ v(s 0 )) = σ m cos (τ w(s 0 )) = 0. The expansions derived in the first two steps imply that ( τ n s O( τ 1 ) ) ( = τ n s 0 s ) τ (1 + O( τ 1 ) ) = π. Hence, we conclude that s τ = π ( 1 + O( τ 1 ) ). nτ Inserting this information into the expansions obtained in the first two steps, we obtain the expansion γ τ = 1 n cos t + O C ( τ 1 ) for all t [0, s τ /]. The result then follows from the fact that σ τ is even and periodic of period s τ. Finally, to obtain the estimate for the derivative of s τ, we remember that s τ = τ g g 1 (y/τ) dy 1 y. Then τ s τ = 1 τ s τ τ = 1 τ s τ τ ( 1 ) τ g g 1 (y/τ) dy 1 y g g 1 (y/τ) ydy (g g 1 (y/τ)) 3. 1 y Since, the function z g g 1 (z) (g g 1 (z)) 3 is bounded, we use the expansion of s τ established below to deduce that τ s τ = π nτ + O C ( τ 3 ). 4 The physical period of a n-delaunay hypersurface Recall that a n-delaunay hypersurface can be parameterized as X τ : (s, θ) (τ e σ(s) θ, κ(s)) where the function σ τ is s τ periodic. The physical period of this hypersurface is defined by T τ = κ τ (s τ ), 31

32 so that X τ (s + s τ, θ) = X τ (s, θ) + T τ (0, 1). We would like to study the behavior of T τ as τ varies. When τ τ, it follows from the analysis given in the previous Chapter that the function σ τ is strictly increasing for s (0, s τ /). Hence it can be used as a change of parameter and we can define It follows from (.6) and (.8) that K := κ τ σ 1 τ. σ K = τ e σ H(σ) 1 H (σ) where H(σ) := τ (e σ (1 n) + ι e σ) and ι is equal to + when τ > 0 and equal to when τ < 0. Then, we have the expression of T τ which is given by σ+ T τ = τ e σ H(σ) dσ σ 1 H (σ) (3.0) where σ < 0 < σ + are the roots of H(σ) = 1. Now, we study the asymptotic behavior of T τ as the parameter τ tends to 0. Proposition 3.. As τ tends to 0, we have T τ = + ι c n τ n n 1 + O(τ ), with c n := n x 1 n 1 1 x dx. Proof: We will distinguish two cases according to the sign of τ Case 1: Assume that τ < 0 and let us define the function h(y) : y y 1 n, which is increasing and can be used as a change of variable. Writing h(e σ ) = x the physical period will be write as τ in (3.0), with T τ := 1 1 G(x) := x G( x ) dx, (3.1) 1 x τ 1 h h 1 (x). 3

33 First, Lebesgue s dominated convergence Theorem implies that lim T τ =. τ 0 Then, there exists c > 0 such that for all x positive the function G satisfies the two estimates G(x) 1 c (1 + x) n, and x G (x) c (1 + x) n. Using these, satisfies and T 1 := 1 0 x G( x 1 x τ ) dx, T 1 c τ, τ T 1 c τ. Now, it is easy to verify that there exists a constant c > 0 such that for all x < 0, we have G(x) 1 n n n 1 x n 1 c (1 x) n 1, and x G (x) c (1 x) n n 1. Hence satisfies T + n 1 τ n n 1 T := x 1 x G( x τ ) dx, x 1 n 1 1 x dx c τ 1 τ 0 x 1 n 1 (1 + x) n n 1 Finally, we see that the last integral converges. Which finishes the proof in this case. Case : Assume that τ (0, τ ) and let us define the function Then, changing e σ by y, (3.0) becomes f(y) := y + y 1 n. dx. with T τ := τ y+ y f(y) 1 τ f (y) dy, f(y ) = f(y + ) = 1 τ. Now, it is clear that f 1 the restriction of f in (y, (n 1) 1 n ) is strictly decreasing and f the other restriction of f in ((n 1) 1 n, y + ) is strictly increasing. Hence, the physical period can be written as 33

34 where ( τ τ T τ = 1 As in the first case, we prove that x G 1( x 1 1 x τ ) dx + τ τ x 1 x G ( x τ ) dx ), G 1 (x) := f 1 f 1 1 (x) and G (x) := f f 1 (x). τ τ 1 x G 1( x 1 x τ ) dx = τ n n 1 n x 1 n 1 1 x dx + O(τ ), and 1 This completes the proof. τ τ x 1 x G ( x τ ) dx = 1 + O(τ ). Remark 3.1. In the case where τ < 0, T τ is monotone and τ T τ > 0. Indeed, thanks to (3.1) we have τ T τ = 1 1 x 1 x n(n 1) z n 1 dx ( ) (n 1) z n τ > 0 where z := h 1 ( x τ ). 34

35 Chapter 4 The Jacobi operator In this chapter, we define and study the Jacobi operator about a n-delaunay hypersurface, about the n-sphere and also about the n-catenoid, a minimal hypersurface of revolution which generalizes the -dimensional catenoid. Next, we give the expression of the geometric Jacobi fields. Finally, we define the indicial roots associated with the Jacobi operator about a n-delaunay hypersurface. 1 The Jacobi operator about a n-delaunay Recall that the n-delaunay hypersurface D τ can be parameterized as X τ = (ι τ e στ θ, κ τ ). (4.1) Assume that the orientation of D τ is chosen so that the unit normal vector field is given by ( N τ := ι ) sκ τ e θ, sσ τ. (4.) στ Any hypersurface, close enough to D τ, can be parameterized (at last locally) as a normal graph over D τ. Namely, by X ω = X τ + ω N τ, for some (small) smooth function ω. The hypersurface parameterized by X ω will be denoted by D τ (ω) and we define the mean curvature operator H(ω) to be the mean curvature of D τ (ω). It is well known [5] that the linearized mean curvature operator about D τ, which is usually referred to as the Jacobi operator, is given by L τ := τ + A τ where τ is the Laplace-Beltrami operator and A τ is the square of the norm of the shape operator A τ on D τ. Recall that we have defined in Chapter 3 the function ϕ τ := τ e στ and, in the above parameterization, the metric on D τ is given by g = ϕ τ (ds ds + dθ i dθ j ), 35

36 and the second fundamental form is given by b = ϕ τ ( (1 ± (1 n) τ n ϕ n τ ) ds ds + (1 ± τ n ϕ n ) τ ) dθ i dθ j, with a + when τ > 0 and a when τ < 0. Using this, we find the expression of the Jacobi operator in term of the function ϕ τ L τ := ϕ n τ s (ϕ n τ It will be convenient to define the conjugate operator s ) + ϕ τ S n 1 + n + n (n 1) τ n ϕ n τ. (4.3) L τ := ϕ n+ τ which is explicitly given in terms of the function ϕ τ by ( ) n L τ = s + S n 1 + L τ ϕ n τ, (4.4) n(n + ) 4 ϕ τ + n(3n ) 4 τ n ϕ n τ. (4.5) Since the operators L τ and L τ are conjugate, the mapping properties of one of them will easily translate for the other one. With slight abuse of terminology, we shall refer to any of them as the Jacobi operator about D τ. We finally define the operator ( ) n 0 := s + S n 1, (4.6) which appears in the expression of L τ. This is conjugate to the Jacobi operator about the hyperplane R n {0} R n+1 in polar coordinates. Indeed if r = e s and θ S n 1 then Geometric Jacobi fields 0 = e n+ R n e n. Some Jacobi field, i.e., solution of the homogeneous problem L τ ω = 0, can be explicitly computed when they correspond to a smooth one parameter family of constant mean curvature hypersurfaces C λ, for λ ( 1, 1) to which D τ belongs, for example C 0 = D τ. For λ small enough, the hypersurface C λ can be written (at least locally) as a normal graph over D τ, for some function w λ. Differentiation with respect to λ, at λ = 0, yields a Jacobi field. Multiplying the result by ϕ n τ, we obtain a solution of L τ ω = 0 which, with slight abuse of terminology, will be referred to as a Jacobi field. In the special case where the one parameter family of constant mean curvature hypersurfaces is given by the action of a one parameter family of rigid motions, the corresponding Jacobi field can be obtained by projecting onto the normal bundle of D τ, the Killing vector field associated to the one parameter family of rigid motions under 36

37 consideration. We now describe these Jacobi fields as well as another independent Jacobi field which arise by changing the Delaunay parameter τ. Translation along the axis To begin with, for τ (, 0) (0, τ ), we define Φ 0,+ τ to be the Jacobi field corresponding to the translation of D τ along the x n+1 axis. As explained above, this Jacobi field can be obtained by projecting onto the normal bundle, the constant Killing field Since we conclude that χ(x, x n+1 ) := (0, 1) R n R. χ N τ = s σ τ = ϕ 1 τ s ϕ τ Φ 0,+ τ = ϕ n 4 s ϕ (4.7) is a solution of L τ ω = 0. It is easy to check that Φτ 0,+ only depends on s and is periodic, in particular, this implies that this Jacobi field is bounded. Translations orthogonal to the axis Since we have n directions orthogonal to x n+1, there are n linearly independent Jacobi fields which are obtained by translating D τ in a direction orthogonal to its axis. Let χ(x, x n+1 ) := (a, 0) R n R denote the constant Killing field which correspond to the translations in the (a, 0) direction. Then, we have ( χ N τ = τ e σ ± e (1 n)σ) a θ Taking a to be any vector of the canonical basis of R n, we get for j = 1,..., n ( ) Φ j,+ τ = ϕ n ± τ n ϕ n e j (4.8) which are solution of L τ ω = 0. Again, we see that Φ j, τ all j = 1,..., n. is periodic (hence bounded) for Rotations of the axis Next, we compute the Jacobi fields which correspond to the rotation of D τ in a direction orthogonal to the axis. Given a R n, We define the Killing field χ(x, x n+1 ) := ( x n+1 a, a x) R n R and we obtain Φ j, τ ( s κ κ χ N τ = ϕ ) + sϕ a θ Taking a to be any vector of the canonical basis of R n, we obtain for j = 1,..., n ( ) (s, θ) = ϕ n 4 ϕ s ϕ + κ s κ e j (4.9) which are solutions of L τ = 0. Observe that Φ j,+ τ is not periodic (hence not bounded), but grows linearly. In particular, there a constant c > 0 (depending on τ) such that for all (s, θ) R S n 1. Φ j,+ τ c (1 + s ), 37

38 Change of Delaunay parameter Finally, The Jacobi field corresponding to a change of parameter τ (, 0) (0, τ ) will be denoted by Φτ 0,. It can be obtained by writing, for η small enough, the constant mean curvature hypersurface D τ+η as a normal graph over D τ for some function ω η and differentiating ω η with respect to η at η = 0. More precisely, for τ close to τ there exists a local diffeomorphism Φ τ such that In particular, we get differentiating ω τ X τ Φ τ = X τ + ω τ N τ. ω τ = (X τ Φ τ X τ ) N τ, with respect to τ at τ = τ we get τ ω τ τ =τ = τ X τ N τ + (DX τ ( τ Φ τ )) N τ. Since (DX τ ( τ Φ τ )) N τ 0, the corresponding Jacobi field takes the form Φ 0, τ = ϕ n 4 ( τ κ s ϕ τ ϕ s κ). (4.10) Observe that, in the first and in the last study, we have assumed that τ τ. Some changes are needed to handle the case where τ = τ. Indeed, D τ is the cylinder S n 1 ( n 1 n ) R and hence it is invariant under translation along its axis and the projection of the constant Killing vector field (0, 1) along the normal gives 0! This fact is a consequence of the parameterization of the set of Delaunays hypersurfaces we have chosen. Therefore, we can give an intrinsic definition of the Jacobi fields Φτ 0,± by considering a different parameterization of the set of Delaunay hypersurfaces in the neighborhood of the cylinder S n 1 ( n 1 n ) R. So far we have considered the hypersurfaces parameterized by ( ) X(s, θ) = τ e στ (s+t) θ, κ τ (s + t) where t R and τ (0, τ ]. Now, we define the parameters η := π τ t and e := τ κ(s τ ) τ so that (e, η) [0, 1) S 1 are in some sense polar coordinates in the space of parameters It follows from Lemma 3.4, that σ τ = σ Then, a simple exercise shows that e n 1 cos( n 1 s) + O C (e ). Φ 0,+ τ = cos( n 1 s) and Φ 0, τ = sin( n 1 s) and these two Jacobi fields are periodic (hence bounded) corresponding to differentiation with respect to e and t respectively. 38

39 3 Indicial roots associated to a n-delaunay Because of the rotational invariance of the operator L τ we can introduce the eigenfunction decomposition with respect to the cross-sectional Laplace-Beltrami operator S n 1. In this way we obtain the sequence of operators ( ) n L τ,j = s λ j + n(n + ) ϕ + 4 n(3n ) τ n ϕ n (4.11) 4 for j N. By definition, the indicial roots of the operator L τ,j characterize the rate of growth (or rate of decay) of the solutions of the homogeneous equation L τ,j ω = 0 at infinity (see [16]). To explain this, observe that the potential in the expression of L τ,j is given by δj n(n + ) + ϕ n(3n ) + τ n ϕ n 4 4 where δ j has been defined in (1.), and hence this potential is periodic of period s τ. We define a by matrix T j (τ) M (R) such that, for all ω solution of L τ,j ω = 0 in R, we have ( ) ( ) ω ω (s s ω τ ) = T j (τ) (0). s ω Let λ (τ, j) and λ + (τ, j) denote the roots of the characteristic polynomial of T j (τ). Assume that v 1 and v are the solutions of L τ,j v i = 0 with v 1 (0) = s v (0) = 1 and v (0) = s v 1 (0) = 0. We denote by W the Wronskian of v 1, v W := v 1 s v v s v 1 The Wronskian of v 1 and v being constant, we evaluate it at s = 0 and s = s τ, and we obtain det (T j (τ)) = λ (τ, j)λ + (τ, j) = 1. Observe that the matrix T j (τ) has real entries, hence This being understood, we can write Tr (T j (τ)) = λ (τ, j) + λ + (τ, j) R λ + (τ, j) = µ e iξ and λ (τ, j) = 1 µ e iξ (4.1) where ξ := ξ(τ, j) [0, π) and µ := µ(τ, j) 1 satisfy (µ 1) sin ξ = 0. Now, we distinguish two cases according to the value of ξ. Case 1 Assume that ξ (0, π) (π, π). Then λ + (τ) = e iξ and λ (τ) = e iξ are distinct. In particular, the matrix T j (τ) can be diagonalized there exists two independent solutions of L τ,j ω = 0, whose Cauchy data at t = 0 correspond, to the eigenvectors of T j (τ). These two solutions are bounded. 39