Differential Geometry - Dynamical Systems. Adina BALMUŞ. Biharmonic Maps and Submanifolds

Transcription

1 Differential Geometry - Dynamical Systems *** Monographs # 10 *** Adina BALMUŞ Biharmonic Maps and Submanifolds Geometry Balkan Press Bucharest, Romania

2 Biharmonic Maps and Submanifolds DGDS Monographs # 10 Differential Geometry - Dynamical Systems * ISSN x * Monographs # 10 Editor-in-Chief Prof.Dr. Constantin Udrişte Managing Editor Prof.Dr. Vladimir Balan University Politehnica of Bucharest Biharmonic Maps and Submanifolds, Adina BALMUŞ. Bucharest. Differential Geometry - Dynamical Systems * Monographs, 009 Includes bibliographical references. c Balkan Society of Geometers, Differential Geometry - Dynamical Systems * Monographs, 009 Neither the book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming or by any information storage and retrieval system, without the permission in writing of the publisher.

3 UNIVERSITÀ DEGLI STUDI DI CAGLIARI DIPARTIMENTO DI MATEMATICA E INFORMATICA DOTTORATO DI RICERCA IN MATEMATICA XX CICLO - ANNO 007 BIHARMONIC MAPS AND SUBMANIFOLDS Tutor Stefano MONTALDO Cezar ONICIUC Tesi di dottorato di Adina BALMU Settore scientico disciplinare: GEOMETRIA - MAT/03 OTTOBRE 007

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5 To my mother and brother, Elena and Sorin-Bogdan, in appreciation of their love and support

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7 Abstract Biharmonic maps generalize the notion of harmonic map and are dened as critical points of the bienergy functional. This work is devoted to the study of biharmonic maps and submanifolds. The main results are: - several new methods, inspired by the Baird-Kamissoko method, for constructing proper biharmonic maps starting with harmonic maps; - a geometric approach to the study of the proper biharmonic curves of the 3- dimensional unit Euclidean sphere S 3, and the study of biharmonic curves on 3-dimensional Berger spheres; - the study of the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres; - the complete classication of proper biharmonic hypersurfaces with at most two distinct principal curvatures in space forms; - the non-existence of proper biharmonic hypersurfaces in R 4 and H 4 ; - the classication of compact proper biharmonic hypersurfaces of S 4 ; - some rigidity properties for pseudo-umbilical biharmonic submanifolds of codimension and for biharmonic surfaces with parallel mean curvature vector eld in S n ; - the characterization and classication of proper biharmonic products of spheres in spheres; 5

8 - the characterization of submanifolds of the Euclidean space with biharmonic Gauss map; - the construction of examples of hypersurfaces of the Euclidean space with biharmonic Gauss map.

9 Abstract Le applicazioni biarmoniche generalizzano la nozione di applicazione armonica e sono denite come punti critici del funzionale bienergia. Questa tesi è dedicata allo studio delle applicazioni e delle sottovarietà biarmoniche. I risultati principali sono: - la determinazione, ispirandoci al metodo di Baird-Kamissoko, di nuovi metodi per la costruzione di applicazioni biarmoniche proprie a partire da applicazioni armoniche; - un'interpretazione geometrica delle curve biarmoniche nella sfera euclidea 3-dimensionale e la determinazione delle curve biarmoniche delle sfere 3-dimensionali di Berger; - lo studio del tipo, nel senso di B-Y. Chen, delle sottovarietà biarmoniche compatte con curvatura media costante; - la classicazione completa delle ipersuperci biarmoniche con al più due curvature principali distinte nelle forme spaziali; - la non-esistenza delle ipersuperci biarmoniche proprie in R 4 e H 4 ; - la classicazione delle ipersuperci biarmoniche proprie di S 4 ; - alcune proprietà di rigidità per le sottovarietà biarmoniche pseudo-ombelicali di codimensione e per le superci biarmoniche con curvatura media parallela in S n ; - la caratterizzazione e classicazione delle sottovarietà biarmoniche proprie di tipo prodotto di sfere in sfere; - la caratterizzazione dello sottovarietà dello spazio euclideo con applicazione di Gauss biarmonica; - la costruzione di esempi notevoli di ipersuperci con applicazione di Gauss biarmonica.

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11 Contents 1 Introduction 13 Preliminaries 5.1 Operators on vector bundles Harmonic maps The rst variation of the energy Examples of harmonic maps The second variation of the energy Harmonic morphisms Harmonic Riemannian submersions The biharmonic problem Physical problems formulated in terms of the biharmonic equation Biharmonic maps The rst variation formula of the bienergy Examples of biharmonic maps Examples of biharmonic submanifolds Obstructions to the existence of proper biharmonic maps and submanifolds New methods of construction for biharmonic maps Biharmonic maps and conformal changes Conformal changes of metric Isoparametric functions The Baird-Kamissoko method Conformal changes of metric on the codomain Biharmonic maps and warped product manifolds

12 10 Contents 4..1 Riemannian structure of warped products Ane functions The biharmonicity of the inclusion The case of product maps Axially symmetric biharmonic maps Axially symmetric biharmonic maps from R m \ {0} to R n \ {0} Axially symmetric biharmonic maps from R m \ {0} to S n \ {±p} Axially symmetric biharmonic maps from R m \ {0} to H n \ {0} Biharmonic curves in spheres Biharmonic curves on Riemannian manifolds Biharmonic curves on Euclidean spheres Biharmonic curves on S 3 via the stereographic projection Biharmonic curves on Berger spheres Berger spheres. Denition and Riemannian structure Biharmonic curves on S 3 ε Biharmonic submanifolds in space forms Generalities on some classes of submanifolds Pseudo-umbilical submanifolds Isoparametric hypersurfaces in spheres Hypersurfaces with at most two distinct principal curvatures Biharmonic submanifolds in space forms The main examples of biharmonic submanifolds in spheres The type of biharmonic submanifolds in spheres Biharmonic hypersurfaces in spheres Biharmonic hypersurfaces with at most two distinct principal curvatures Biharmonic hypersurfaces with at most three distinct principal curvatures Codimension biharmonic pseudo-umbilical submanifolds in spheres Rigidity results New examples of proper biharmonic submanifolds in spheres Biharmonic products of spheres Geometric properties of two classes of biharmonic submanifolds of S On the biharmonicity of the Gauss map The Grassmannians. Preliminaries

13 Contents Denition and dierential structure Tangent spaces to Grassmannians The compact Grassmannian as a symmetric space. Curvature The Gauss map associated to an m-dimensional submanifold in R m+n Denition. The pull-back bundle The harmonicity of the Gauss map The biharmonicity of the Gauss map The case of hypersurfaces Hypercones with biharmonic associated Gauss map Riemannian structure and second fundamental form of the cone generated by a submanifold in S m Hypercones with biharmonic Gauss map

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15 Chapter 1 Introduction The goal of this introduction is, rstly, to present some ideas that encouraged the study of the geometry of biharmonic maps and, secondly, to briey describe the new results gathered in the present thesis. Denote by C (M, N) the space of smooth maps ϕ : (M, g) (N, h) between two Riemannian manifolds. A map ϕ C (M, N) is called harmonic if it is a critical point of the energy functional E : C (M, N) R, E(ϕ) = 1 dϕ v g, and is characterized by the vanishing of the tension eld τ(ϕ) = trace dϕ. If we consider Imm(M, N) to be the space of Riemannian immersions in (N, h), then a Riemannian immersion ϕ : (M, ϕ h) (N, h) is called minimal if it is a critical point of the volume functional V : Imm(M, N) R, V (ϕ) = 1 v ϕ h, and the associated Euler-Lagrange equation is H = 0, where H is the mean curvature vector eld. Although they are conceptually dierent, there is a strong connection between these two variational problems: if ϕ : (M, g) (N, h) is a Riemannian immersion, then it is a critical point of the energy in C (M, N) if and only if it is a minimal immersion [53]. One can generalize harmonic maps and minimal immersions by considering the functionals obtained by integrating the squared norm of the tension eld or of the mean curvature vector eld, respectively. More precisely, biharmonic maps are the critical points of the bienergy functional E : C (M, N) R, E (ϕ) = 1 τ(ϕ) v g, 13 M M M

16 14 Chapter 1. Introduction and Willmore immersions are the critical points of the Willmore functional W : Imm(M (, N) R, W (ϕ) = H + K ) v ϕ h, M where K is the sectional curvature of (N, h) restricted to the image of M. Although the above variational problems produce natural generalizations of harmonic maps and minimal immersions, biharmonic Riemannian immersions do not recover Willmore immersions, not even when the ambient space is R n. In R 3 there exist no proper (non-minimal) biharmonic surfaces (see [48]), while all the spheres are Willmore surfaces. Also, the unique compact proper biharmonic surface of S 3 is the hypersphere S ( 1 ), while all the hyperspheres S (a), a (0, 1), are non-trivial Willmore surfaces. An example of proper biharmonic immersion which is not a Willmore immersion was provided by T. Sasahara in the class of Legendre immersions in S 5 (see [106]). As we shall see in Section 6., the biharmonic equation for Riemannian immersions in space forms is equivalent to the vanishing of both a tangential and a normal component, while the Willmore equation involves only a normal component. We also point out that the submanifolds with vanishing normal component of the bitension eld, which are called biminimal (see [79]), do not recover Willmore immersions. In his studies on nite type submanifolds (see [43]), B.Y. Chen dened biharmonic submanifolds M R n of the Euclidean space as those with harmonic mean curvature vector eld, that is H = 0, where is the rough Laplacian. By considering the denition of biharmonic maps for Riemannian immersions into the Euclidean space one recovers the notion of biharmonic submanifolds in the sense of B-Y. Chen. Thus biharmonic Riemannian immersions can also be thought of as a generalization of Chen's biharmonic submanifolds. All the results obtained by B-Y. Chen and his collaborators on proper biharmonic submanifolds in Euclidean spaces are non-existence results. Nevertheless, their techniques were adapted and led to classication results for proper biharmonic submanifolds in spheres. Biharmonic maps have been extensively studied in the last decade and there are two main research directions. On one hand, in dierential geometry, a special attention has been payed to the construction of examples and classication results (see, for example, [4, 7, 1, 14, 15, 9, 30, 3, 33, 46, 48, 49, 67, 13, 14]). On the other hand, from the analytic point of view, biharmonic maps are solutions of a fourth order strongly elliptic semilinear PDE and the study of their regularity is nowadays a well-developed eld (see, for example, [40, 74, 75, 110, 116, 117]). Complementary results can be found in [9, 10, 1, 35, 66, 89, 114]. We mention some other reasons that encourage the study of biharmonic maps. The theory of biharmonic maps is an old and rich subject, initially studied due to its implications in the theory of elasticity and uid mechanics. G.B. Airy and J.C. Maxwell

17 15 were the rst to study and express plane elastic problems in terms of the biharmonic equation (see [, 84]). Later on, the theory evolved with the study of polyharmonic functions developed by E. Almansi, T. Levi-Civita, M. Nicolaescu. Biharmonic and polyharmonic functions on Riemannian manifolds were studied by R. Caddeo and L. Vanheke [8, 34], L. Sario et all (see [105]) and others. The dierential geometric aspect of biharmonic submanifolds was also studied in the semi-riemannian case (see, for example, [46]). The variational problem associated by considering, for a xed map, the bienergy functional dened on the set of Riemannian metrics on the domain gave rise to the biharmonic stress-energy tensor (see [80]). This proved to be useful for obtaining new examples of proper biharmonic maps and for the study of submanifolds with certain geometric properties, like pseudo-umbilical and parallel submanifolds. There also exists an interesting connection between the biharmonic stress-energy tensor associated to the inclusion of a submanifold in the Euclidean space and the harmonic stress-energy tensor of the corresponding Gauss map. Proper biharmonic surfaces in the 3-dimensional sphere are also II-minimal surfaces, i.e. critical points of the volume functional associated to the second fundamental form (see [64]). It was proved that there exists no harmonic map from T to S (whatever the metrics chosen) in the homotopy class of Brower degree ±1 ([54]). The biharmonic maps are expected to exist where harmonic maps do not. The interest in the theory of biharmonic maps crossed the border of dierential geometry and analysis of PDE's. In computational geometry, more precisely in the eld of boundary based surface design, the biharmonic Bézier surfaces are studied (see [71, 87, 88]). The present thesis mainly deals with: the construction of new classes of biharmonic maps; the study and classication of proper biharmonic submanifolds in space forms; the study of submanifolds with proper biharmonic Gauss map in Euclidean spaces. The rst chapter is intended to establish the notations and to recall some basic results which will be used throughout the entire thesis. We focus on calculus on vector bundles and harmonic maps and we closely follow the monographs [8, 50, 115]. In the second chapter we introduce the notion of biharmonic map as it was suggested by J. Eells and J.H. Sampson. We recall the classical biharmonic problem and some of its applications in continuous media mechanics. We present fundamental results and examples of proper biharmonic maps and submanifolds (see [13]).

18 16 Chapter 1. Introduction The third chapter contains new methods for constructing proper biharmonic maps by using harmonic ones. We present the Baird-Kamissoko method (see [7]) and new results that are gathered in [11] and [14]. A natural way for constructing proper biharmonic maps is the following: given a harmonic map ϕ : (M, g) (N, h) one can conformally change the metric g, or h, in order to render ϕ proper biharmonic. This problem represents the object of the rst section in the present chapter. The behavior of the bitension eld, under the conformal change of the domain metric, was studied by P. Baird and D. Kamissoko in [7] and we present here the main results. A metric g, which renders the identity map 1 : (M, g) (M, g) biharmonic, was called a biharmonic metric with respect to g and it was proved that on Einstein manifolds the conformally equivalent biharmonic metrics are dened only by means of isoparametric functions. We study the biharmonicity of the identity after a conformal change of the codomain metric. The results are similar to those obtained in [7], but we underline the fact that, in general, the biharmonicity of 1 : (M, g) (M, e ρ g) is not equivalent to the biharmonicity of 1 : (M, e ρ g) (M, g) (see Remark 4.). The results gathered in the second section extend the idea of studying the eect of conformal changes of metric to the biharmonic equation by considering the situation of warped products. The rst problem consists in the study of the biharmonicity of the inclusion of a Riemannian manifold N into the warped product M f N and we obtain Corollary 4.9 ([14]). The inclusion map i x0 : N M f N is a proper biharmonic map if and only if x 0 is not a critical point for f, but is a critical point for grad f. With this setting we obtain new examples of proper biharmonic maps and recover some of the examples rst obtained in [9] and [3]. Ane functions play an important role in our study: Corollary 4.31 ([14]). Let (M, g) be a Riemannian manifold with a positive nontrivial ane function f and let (N, h) be an arbitrary Riemannian manifold. Then any inclusion i x : N M f N, x M, is a proper biharmonic map. We then consider the product of two harmonic maps φ = 1 M ψ : M N M N. By warping the metric on the domain or codomain we lose the harmonicity; nevertheless, under certain conditions on the warping function the product map remains biharmonic. In the case that the product map is the identity map 1 : M f N M N we shall call the warping function, which is a solution of the above problem, a biharmonic warping function. We characterize the biharmonicity: Theorem 4.37 ([14]). Let ψ : N N be a harmonic map and let f C (M) be a positive function. Then φ = 1 M ψ : M f N M N is a proper biharmonic

19 17 map if and only if f is a non-constant solution of trace g grad ln f + Ricci M (grad ln f) + n grad( grad ln f ) = 0. In the case that M is an Einstein manifold we show that the isoparametric functions provide examples of biharmonic warping ones: Proposition 4.44 ([14]). Let M be an Einstein space. If ρ C (M) is an isoparametric function, then it admits away from its critical points a local reparameterization f which is a biharmonic warping function. For the projection π : M f N M we obtain Proposition 4.47 ([14]). The projection π : M f N M is proper biharmonic if and only if f is a biharmonic warping function. We also give the complete classication of the biharmonic warping functions when M = R (see Example 4.46). Similar results are obtained when the codomain metric is warped and ψ is harmonic with constant energy density. In the last section we use the warped product setting to study axially symmetric biharmonic maps from R m \ {0} to an n-dimensional space form. We get a general characterization result Theorem 4.57 ([14]). Let ϕ : S m 1 S n 1 be an eigenmap of eigenvalue k 0. Then φ = ρ ϕ : R m \ {0} R f S n 1 is biharmonic if and only if ρ is a solution of F + m 1 F k ( f t t (ρ) + f(ρ)f (ρ) ) F = 0, where F = ρ + m 1 ρ k t t f(ρ)f (ρ). Then we discuss several examples and, when the target manifold is R n \ {0}, we give the complete classication of biharmonic axially symmetric maps. From the former classication we also deduce that the generalised Kelvin transformation φ : R m \ {0} R m \ {0}, φ(y) = y/ y l is a proper biharmonic map if and only if m = l +. The goal of the fourth chapter is to give a geometric approach for the study of the proper biharmonic curves of the 3-dimensional unit Euclidean sphere S 3, since it was proved in [30] that the proper biharmonic curves of S n, n 4, are essentially the ones of S 3 composed with the totally geodesic embedding of S 3 in S n. We also study the biharmonic curves on 3-dimensional spheres endowed with distorted metrics called Berger metrics. The new results of this chapter can be found in [1] and [19].

20 18 Chapter 1. Introduction The rst section contains general characterization results for biharmonic curves in Riemannian manifolds, which were obtained in [31] by expressing the biharmonic equation in terms of the Frenet frame associated to the curve. In the next section we present the classication result for proper biharmonic curves of S n, and by using the stereographic projection we obtain the image in R 3 of the biharmonic curves of S 3. The last section deals with proper biharmonic curves in Berger spheres. There are several ways to introduce the 3-dimensional Berger metrics; here we choose to dene them as distortions of the canonical metric on S 3 constructed by performing a biconformal change of metric with respect to the Hopf bration. The study of the behavior of biharmonic curves under these distortions comes as a natural question. In this study we use the same techniques used in [33], for the case of the Heisenberg space, which have been afterwards developed for the general case of biharmonic curves in SO()-isotropic 3-dimensional manifolds (see [31]). We obtain the explicit parametric equations of proper biharmonic curves in Berger spheres (see Theorem 5.19). If we restrict our interest only to the Legendre proper biharmonic curves we get Corollary 5.0. Let S 3 ε be a Berger sphere, ε (0, 1). Then the parametric equations of all proper biharmonic Legendre curves, parametrized by arc length, of S 3 ε are γ(s) = α α 1 + α exp( iα 1 s)e 1 + α α 1 + α exp(iα 1 s)e 3, where e 1, e 3 are unit orthogonal vectors of R 4 with e 3 orthogonal to Je 1, α 1, = 3 ε ± (1 ε )( ε ), and J is the canonical complex structure on R 4. All the known results on proper biharmonic submanifolds in non-positively curved spaces are non-existence results. For example, it was proved (see [45]) that a spherical submanifold M S n 1 cannot be biharmonic in R n. Also there exist no proper biharmonic curves, submanifolds of nite type, submanifolds of constant mean curvature, pseudo-umbilical submanifolds of dimension m 4, hypersurfaces with at most two distinct principal curvatures, conformally at hypersurfaces in the Euclidean space or proper biharmonic hypersurfaces in R 3 and R 4 ([48, 49, 63]). Similar results were obtained (see [30]) for curves, submanifolds of constant mean curvature, pseudo-umbilical submanifolds of dimension m 4 in the hyperbolic space or surfaces in H 3. The non-existence theorems for the case of non-positive sectional curvature codomains, as well as the Generalized Chen Conjecture. Biharmonic submanifolds of a non-positive sectional curvature manifold are minimal,

21 encouraged the study of proper biharmonic submanifolds in spheres and other curved spaces [9, 30, 55, 67, 86, 98]. The attempt to classify the biharmonic submanifolds in space forms was initiated in [48] and [9]. The rst achievement towards the classication problem is represented by the complete classication of proper biharmonic submanifolds of the 3-dimensional unit Euclidean sphere, obtained in [9]. Then, inspired by the 3-dimensional case, two methods for constructing proper biharmonic submanifolds in S n were given in [30]. Although important results and examples were obtained, the classication of proper biharmonic submanifolds in spheres is still an open problem. The fth chapter is fully devoted to the classication of proper biharmonic submanifolds in space forms and the new results presented here are contained in [15], [16], [17]. In the preliminary section we include denitions and results on some special classes of submanifolds in space forms. The second section contains fundamental characterization results on proper biharmonic submanifolds of space forms and, in particular, of the Euclidean sphere. We present the biharmonic equation for this particular case, non-existence results for nonpositively curved space forms and the main examples of biharmonic submanifolds in spheres. The section also contains some basic information on nite type Euclidean submanifolds. A partial classication result for biharmonic submanifolds in spheres was given in [94]. It was proved that proper biharmonic submanifolds M of constant mean curvature H in S n satisfy H (0, 1], with H = 1 if and only if M is minimal in S n 1 ( 1 ). Taking this further, by using this result, we study the type of compact proper biharmonic submanifolds of constant mean curvature in S n and prove that, depending on the value of the mean curvature, they are of 1-type or of -type as submanifolds of R n+1, Theorem 6.5 ([15]). Let M m be a compact constant mean curvature, H = k, submanifold in S n. Then M is proper biharmonic if and only if either (i) H = 1 and M is a 1-type submanifold of R n+1 with eigenvalue λ = m, or (ii) H = k (0, 1) and M is a mass-symmetric -type submanifold of R n+1 with the eigenvalues λ 1, = m(1 ± k). The third section is devoted to the study of proper biharmonic hypersurfaces with at most two distinct principal curvatures in space forms. We prove Theorem 6.7 ([15]). Let M be a hypersurface with at most two distinct principal curvatures in E m+1 (c). If M is proper biharmonic in E m+1 (c), then it has constant 19

22 0 Chapter 1. Introduction mean curvature. We obtain the full classication of such hypersurfaces in spheres: Theorem 6.30 ([15]). Let M m be a proper biharmonic hypersurface with at most two distinct principal curvatures in S m+1. Then M is an open part of S m ( 1 ) or of S m 1 ( 1 ) S m ( 1 ), m 1 + m = m, m 1 m. A similar classication is obtained for conformally at biharmonic hypersurfaces in spheres. In contrast, for the hyperbolic space H m+1 we prove a non-existence result. The problem of the biharmonic hypersurfaces with at most 3 distinct principal curvatures is also investigated. A non-existence result is obtained for hypersurfaces with constant mean curvature and exactly three distinct principal curvatures, Theorem 6.35 ([16]). There exist no compact proper biharmonic hypersurfaces of constant mean curvature and three distinct curvatures in the unit Euclidean sphere. and the full classication of compact hypersurfaces in S 4 is given, Theorem 6.38 ([16]). The only proper biharmonic compact hypersurfaces of S 4 are the hypersphere S 3 ( 1 ) and the torus S 1 ( 1 ) S ( 1 ). We also recover the non-existence result for proper biharmonic hypersurfaces in R 4 and prove that it also holds for hypersurfaces of H 4. In the fourth section we prove that the pseudo-umbilical biharmonic submanifolds in spheres have constant mean curvature and we give an estimate for their scalar curvature. Then we classify the proper biharmonic pseudo-umbilical submanifolds of codimension, Theorem 6.4 ([15]). Let M m be a pseudo-umbilical submanifold of S m+, m 4. Then M is proper biharmonic if and only if it is minimal in S m+1 ( 1 ). We also investigate surfaces with parallel mean curvature vector eld in S n, Theorem 6.45 ([15]). Let M be a proper biharmonic surface with parallel mean curvature vector eld in S n. Then M is minimal in S n 1 ( 1 ). The last section contains new examples of proper biharmonic submanifolds in spheres. We analyze biharmonic products of spheres, proving Theorem 6.47 ([17]). Consider T = S n 1 (a 1 ) S n (a )... S nr (a r ) S m+r 1 r r R m+r, where m = n k and = 1. Then k=1 k=1 a k (i) T is minimal in S m+r 1 if and only if for all k = 1,..., r. a k n k = 1 m, (ii) T is proper biharmonic in S m+r 1 if and only if there exists p = 1,..., r 1 such

23 1 that and a 1 n 1 =... = a p n p = a p+1 n p+1 =... = a r n r = 1 (n n p ) 1 m 1 (n p n r ) 1 m. Moreover, we study the geometric properties of two classes of biharmonic submanifolds of S 5. Ever since the characterization result obtained by E. Ruh and J. Vilms in [101] and the remarkable link with constant mean curvature hypersurfaces the study of submanifolds with associated harmonic Gauss map in Euclidean spaces has been a classical problem in harmonic maps theory. The sixth chapter proposes a generalization of this problem: the characterization of submanifolds with proper biharmonic Gauss map in Euclidean spaces. The new results of this chapter are gathered in [18]. In the rst section we recall the denition and the dierential structure of the Grassmannian (see [6]). We use the identication of the Grassmannian as a symmetric space in order to compute its curvature (see [99]). We then recall the denition of the Gauss map associated to a submanifold in an Euclidean space and compute its dierential and its tension eld (see [50, 51, 101]). In the third section we characterize the biharmonicity of the Gauss map associated to a submanifold in an Euclidean space, Theorem 7.3 ([18]). The Gauss map associated to an m-dimensional orientable submanifold M of R m+n is proper biharmonic if and only if H 0 and X H m A H (X) H + trace B( A ( ) H (X) A ) XH( ), trace R (, X) H trace( R )(, X)H = 0, for all X C(T M), where A denotes the Weingarten operator and H the mean curvature vector eld of M in R m+n. The biharmonic equation for the Gauss map simplies for the case of hypersurfaces, Theorem 7.4 ([18]). The Gauss map of a nowhere zero mean curvature hypersurface M m in R m+1 is proper biharmonic if and only if grad f 0 and grad f + A (grad f) A grad f = 0, where denotes the rough Laplacian on C(T M) and f and A denote the mean curvature function and, respectively, the shape operator of M in R m+1. The last section is devoted to the examples of hypercones with biharmonic Gauss map in the Euclidean space. We study the hypercones generated by constant mean curvature hypersurfaces in spheres and, in particular, by the hyperspheres,

24 Chapter 1. Introduction Theorem 7.7 ([18]). Let M be an orientable constant non-zero mean curvature hypersurface of S m+1. The Gauss map associated to the hypercone (0, ) t M is proper biharmonic if and only if m > and A = 3(m ), where A is the shape operator of M in S m+1. and Proposition 7.11 ([18]). Consider the hypercone (0, ) t S m (a) in R m+, a (0, 1). Its associated Gauss map is proper biharmonic if and only if m > and a = m 4m 6. Finally, we obtain a non-existence result for cones with biharmonic Gauss map in R 3, Theorem 7.8 ([18]). There exist no cones with proper biharmonic Gauss map in R 3.

25 3 Acknowledgements First and foremost, I would like to thank my supervisors Dr. Stefano Montaldo and Dr. Cezar Oniciuc for their invaluable guidance, useful ideas and constant support throughout this work. I thank Professor John C. Wood for carefully reviewing my thesis on a very short notice and for giving me insightful comments. Warm thanks go to Professors Renzo Caddeo and Vasile Oproiu for their encouragement and continuous interest in my research. I am grateful to the Istituto Nazionale di Alta Matematica "Francesco Severi" for the nancial support and to the Department of Mathematics, University of Cagliari, for hospitality. I would like to thank all my colleagues and friends for their support and for their innite patience.

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27 Chapter Preliminaries This chapter is intended to establish the notations and to recall some basic results which will be used throughout the entire thesis. We focus on calculus on vector bundles and harmonic maps and we closely follow the monographs [8, 50, 115]. The fundamental notions of dierentiable and Riemannian geometry, as dierentiable manifold, vector bundle, linear connection, Riemannian manifold, Riemannian immersion and submersion, etc, are considered to be known (see, for example, [36, 73, 99, 10]). Throughout this work we shall place ourselves in the C category, i.e. all manifolds, connections, metrics, maps are assumed to be C, unless otherwise stated. By (M m, g) we shall denote an m-dimensional connected manifold, endowed with a Riemannian metric g and by v g we shall denote its volume form. We shall use the Einstein summation convention for repeated indices..1 Operators on vector bundles Let ξ : V M be a smooth vector bundle of nite rank over M. We shall denote by C(ξ) or C(V ) the vector space of smooth sections of V, i.e. maps σ : M V satisfying ξ σ = 1 M, the identity map of M. Remark.1. (i) The real vector space C(V ) is innite-dimensional. (ii) If T M is the tangent bundle of M, then the space of smooth sections C(T M) is the Lie algebra of vector elds on M, where the Lie bracket is dened by [X, Y ]f = XY (f) Y X(f), for all X, Y C(T M) and f C (M), where, as usual, we denote by C (M) the space of smooth functions on M. 5

28 6 Chapter. Preliminaries If ξ : V M and η : W N are two vector bundles and ϕ : M N is a smooth map, we shall denote by (i) V the dual bundle of V, (ii) V W the direct (Whitney) sum of V and W, (iii) V W the tensor product of V and W, (iv) k V the tensor power of V, (v) k V the exterior power of V, k V the symmetric power of V, (vi) ϕ 1 W the pull-back bundle of W over M induced by ϕ, the bundle whose ber over p M is W ϕ(p), the bre of W over ϕ(p). We have the natural commutative diagram ϕ 1 W W Denition.. M ϕ N η (i) A Riemannian metric on a vector bundle V is a section a C( V ) which induces on each ber a positive denite inner product. We shall use the notation a(σ, ρ) = σ, ρ, for σ, ρ C(V ). (ii) A linear connection on a vector bundle ξ : V M is a bilinear map on spaces of sections, such that, for all f C (M), : C(T M) C(V ) C(V ), (X, σ) X σ, fx σ = f X σ, X (fσ) = (Xf)σ + f X σ. Denition.3. A Riemannian structure on a vector bundle V is a pair (, a), where a is a Riemannian metric, is a connection, and a = 0, i.e. ( a)(x, σ, ρ) = ( X a)(σ, ρ) = X σ, ρ X σ, ρ σ, X ρ = 0, for all X C(T M), σ, ρ C(V ).

29 .1. Operators on vector bundles 7 Denition.4. The curvature operator of a connection on a vector bundle ξ : V M is the map R : C(T M) C(V ) C(V ), dened by R(X, Y )σ = X Y σ Y X σ [X,Y ] σ. Given two metrics a, b and two connection V, W on V and W, respectively, we can dene the induced metrics and connections in the following cases: (i) For V, the metric is dened by α, β V = α, β V, where the musical isomorphisms between the bres V p and Vp dene : V p Vp by σ (ρ) = σ, ρ, σ, ρ V p are used. Indeed, we and : V p V p by = 1. The connection on V is dened by and its curvature operator is ( Xθ)(σ) = X(θ(σ)) θ( X σ), (R (X, Y )θ)(σ) = θ(r(x, Y )σ), for all X, Y C(T M), θ C(V ), σ C(V ). (ii) The metric on V W is dened by σ λ, ρ µ = σ, ρ + λ, µ, and the connection by X (σ λ) = V Xσ W X λ, for σ, ρ C(V ) and λ, µ C(W ). The curvature operator is in this case given by R(X, Y )(σ λ) = R V (X, Y )σ R W (X, Y )λ. (iii) For V W, the metric has the expression σ λ, ρ µ = σ, ρ λ, µ, the connection is given by X (σ λ) = ( V Xσ) λ + σ ( W X λ),

30 8 Chapter. Preliminaries and its curvature operator is R(X, Y )(σ λ) = (R V (X, Y )σ) λ + σ R W (X, Y )λ. The above product and connection induce products and connections on k V, k V and k V. (iv) For a given map ϕ : M N and a vector bundle η : W N with metric b, we can identify σ, ρ (ϕ 1 W ) p with σ, ρ W ϕ(p) and dene σ, ρ ϕ 1 W = σ, ρ W. Moreover, it can be proved that there exists a unique linear connection on ϕ 1 W, called the pull-back connection, such that for all p M, X T p M and λ C(W ), ϕ 1 W X (ϕ λ) = W dϕ p (X) λ, where dϕ p : T p M T ϕ(p) N is the dierential of ϕ, and ϕ λ = λ ϕ C(ϕ 1 W ). The curvature operator of the induced connection has the expression R p (X, Y )ρ = R W ϕ(p) (dϕ p(x), dϕ p (Y ))ρ(p), where ρ C(ϕ 1 W ). Note that if ( V, a) and ( W, b) are Riemannian structures on V and W, respectively, then the induced metrics and connections on the vector bundles under consideration give Riemannian structures. Example.5 (The tangent bundle T M). If g is a Riemannian metric on T M, the fundamental theorem of Riemannian geometry asserts that there exists a unique connection on T M, the Levi-Civita connection, satisfying g = 0 and T = 0, where, in general, the torsion T of a connection on T M is dened by T (X, Y ) = X Y Y X [X, Y ], X, Y C(T M). We recall that the Levi-Civita connection is characterized by the Koszul formula g( X Y, Z) = Xg(Y, Z) + Y g(z, X) Zg(X, Y ) g(x, [Y, Z]) + g(y, [Z, X]) + g(z, [X, Y ]). for all X, Y, Z C(T M). Throughout this thesis we shall always endow the tangent bundle of a Riemannian manifold (M, g) with the Levi-Civita connection.

31 .1. Operators on vector bundles 9 Note that, in general, for a given vector bundle, the torsion of a connection could not be dened and a metric does not determine an unique Riemannian structure. The sectional curvature of a -plane P of T p M is dened by Riem p (P ) = R(Y, X)X, Y p, where {X, Y } is an arbitrary orthonormal basis of P. The Ricci tensor at p is the map dened by Ricci p : T p M T p M R, Ricci p (X, Y ) = trace(z R(Z, X)Y ), and the Ricci operator has the expression Ricci p : T p M T p M, Ricci p (X), Y = Ricci p (X, Y ). For a smooth map ϕ : M N we shall consider on ϕ 1 T N the pull-back connection ϕ induced by the Levi-Civita connection on N. The following formula holds, ϕ X dϕ(y ) ϕ Y dϕ(x) = dϕ([x, Y ]), X, Y C(T M), (.1) where dϕ(x) C(ϕ 1 T N) is dened by dϕ(x)(p) = dϕ p X p, p M. From (.1) we deduce the symmetry of the section dϕ C( T M ϕ 1 T N) dened by dϕ(x, Y ) = ϕ X dϕ(y ) dϕ( M X Y ), (.) which is called the second fundamental form of ϕ. A map ϕ : (M, g) (N, h) with vanishing second fundamental form is said to be totally geodesic. Denote by A k (ξ) = C( k T M V ) the space of smooth k-forms on M which take values in a vector bundle ξ : V M. Clearly, A 0 (ξ) = C(V ). Denition.6. The exterior dierential operator d : A k (ξ) A k+1 (ξ) relative to the connection V is given by dσ(x 1,..., X k+1 ) = k+1 ( 1) i+1 V X i (σ(x 1,..., X i,..., X k+1 )) i=1 + i<j ( 1) i+j σ([x i, X j ], X 1,..., X i,..., X j,..., X k+1 ), where X i C(T M), i = 1,..., k + 1, and the symbols covered by are omitted. If the tangent bundle T M is equipped with any torsion-free connection, in particular with the Levi-Civita connection, then d can also be dened as the antisymmetrization of on C( k T M V ), k+1 (dσ)(x 1,..., X k+1 ) = ( 1) i+1 ( Xi σ)(x 1,..., X i,..., X k+1 ), i=1

32 30 Chapter. Preliminaries where ( X ρ)(x 1,..., X k ) = V X(ρ(X 1,..., X k )) k ρ(x 1,..., M X X i,..., X k ). Denition.7. The codierential operator relative to the Riemannian structures on V and T M, is dened by i=1 d : A k (ξ) A k 1 (ξ), (d ρ)(x 1,..., X k 1 ) = trace( ρ)(, X 1,..., X k 1 ) m = g ij ( ei ρ)(e j, X 1,..., X k 1 ), i,j=1 ρ A k (ξ), where {e i } m i=1 is a basis of T pm, X 1,..., X k 1 T p M and (g ij ) is the inverse of the matrix (g(e i, e j )). In particular, if ρ A 1 (ξ), then d ρ = trace ρ. Proposition.8. The codierential operator is the adjoint of d via the formula M dσ, ρ v g = where σ A k 1 (ξ) and ρ A k (ξ) are of compact support. The result used for proving Proposition.8 is the following M σ, d ρ v g, (.3) Proposition.9 (Divergence theorem). Let Ω be a compact domain with smooth boundary of a Riemannian manifold (M, g). Let ρ be a 1-form dened on a neighborhood of Ω. Then d ρ v g = ρ(ν)v i g, Ω Ω where ν denotes the outward pointing unit normal of Ω in M and i : Ω M the canonical inclusion. Denition.10. The Laplacian on V -valued dierential forms is dened by = dd + d d : A k (V ) A k (V ). Proposition.11 ([50]). The operator is strongly elliptic, selfadjoint and positive. Remark.1. Let f C (M). Then f = d df = trace df. Notice that this sign convention is dierent from the one usually encountered in classical papers on harmonic functions.

33 .. Harmonic maps 31 Denition.13. A k-form σ A k (V ) is called harmonic if σ = 0. Remark.14. By using (.3) it follows that, when M is compact, σ = 0 if and only if dσ = 0 and d σ = 0.. Harmonic maps Let (M, g) and (N, h) be two smooth, connected Riemannian manifolds of dimension m and n, respectively. The dierential dϕ of a smooth map ϕ : (M, g) (N, h) can be viewed as a section of the bundle T M ϕ 1 T N. By endowing T M ϕ 1 T N with the Hilbert-Schmidt norm, we have Denition.15. dϕ = trace g ϕ h. (i) The energy density of ϕ is the smooth function e(ϕ) : M [0, ), given by e(ϕ) = 1 dϕ. (ii) For a compact domain Ω M, the energy of ϕ over Ω is dened by E Ω (ϕ) = e(ϕ) v g. (.4) Note that E Ω (ϕ) 0, and E Ω (ϕ) = 0 if and only if ϕ is constant on Ω. If ϕ : (M, g) (N, h) is a smooth map, by a smooth variation of ϕ we mean a smooth map Φ : I M N, I = ( ε, ε) R, satisfying Φ(0, p) = ϕ(p), p M. We denote a smooth variation of ϕ by {ϕ t } t I, where ϕ t (p) = Φ(t, p), (t, p) I M, and say that the variation is supported in Ω M if ϕ t = ϕ on M \ Ω, for all t I, where Ω denotes the interior of Ω. Denition.16. A smooth map ϕ : (M, g) (N, h) is said to be harmonic if it is a critical point of the energy functional E Ω : C (M, N) R, for any compact domain Ω M, i.e. for any Ω M, compact, and for any smooth variation {ϕ t } t I of ϕ supported in Ω, we have D V E Ω (ϕ) = d E Ω (ϕ t ) = 0. dt t=0 Here V ϕ 1 T N denotes the variation vector eld of {ϕ t } t I, V (p) = d ϕ t (p), p M. dt t=0 Ω

34 3 Chapter. Preliminaries Remark.17. By construction, every variation supported in a compact subset Ω M has an associated variation vector eld supported in Ω. Conversely, given a section V C(ϕ 1 T N) supported in Ω there exists a variation (not necessarily unique) supported in Ω with variation vector eld V. A method to construct such a variation is provided by the exponential map on N. Since ϕ(ω) is compact there exists ε R such that exp ϕ(p) tv (p) is dened for all t ( ε, ε) and p Ω. This allows us to dene the smooth map Φ : I M N, I = ( ε, ε), which is the required variation. Φ(t, p) = exp ϕ(p) tv (p),..1 The rst variation of the energy Proposition.18 (First variation of the energy, [53]). Let ϕ : (M, g) (N, h) be a smooth map and let {ϕ t } t I be a smooth variation of ϕ supported in Ω. Then d E Ω (ϕ t ) = τ(ϕ), V v g, (.5) dt t=0 where is the tension eld of ϕ. Ω τ(ϕ) = d dϕ = trace dϕ C(ϕ 1 T N) (.6) Theorem.19 ([53]). A map ϕ : (M, g) (N, h) is harmonic if and only if Remark.0. τ(ϕ) = 0. (.7) (i) Equation (.7) is the Euler-Lagrange equation associated to the energy functional. (ii) Equation (.7) is called the tension eld equation or the harmonic equation and, in local coordinates, yields a semilinear elliptic system of PDE's which are quadratic in the rst derivatives. Indeed, if we consider (U, x i ) and (W, y α ) two local coordinate charts at p on M and in ϕ(p) on N, respectively, and denote by M Γ k ij and N Γ α βδ the Christoel symbols of the Levi-Civita connection of M and N, respectively, then ( τ(ϕ) = g ij ϕ α x i x j M Γ k ϕ α ij x k + N Γ α βδ ( = ϕ α + g ij N Γ α ϕ β ϕ δ βδ x i x j ϕ β ϕ δ ) x i x j y α ϕ ) ϕ. (.8) yα

35 .. Harmonic maps 33 Proposition.1. The second fundamental form and the tension eld of the composition of two maps ϕ : M N and ψ : N P are given by.. Examples of harmonic maps d(ψ ϕ) = dψ( dϕ) + dψ(dϕ, dϕ) (.9) τ(ψ ϕ) = dψ(τ(ϕ)) + trace dψ(dϕ, dϕ). (.10) Example. (Constant maps). Any constant map ϕ : (M, g) (N, h) is harmonic. In fact it is an absolute minimum for the energy functional. Example.3 (The identity map). The second fundamental form of the identity map 1 M : (M, g) (M, g) vanishes identically, thus 1 M is a harmonic map. Example.4 (Harmonic maps to Euclidean spaces). Let N = R n be the Euclidean space endowed with the canonical metric and let ϕ : M R n be a smooth map, ϕ(p) = (ϕ 1 (p),..., ϕ n (p)). Since N Γ α βδ = 0, the map ϕ is harmonic if and only if M ϕ α = 0, α = 1,..., n, i.e. the ϕ α are harmonic functions in the classical sense. Example.5 (Geodesics). Let M = (a, b) be an open interval of R. Then a map γ : (a, b) N is a curve on N and τ(γ) = 0 if and only if d γ α dt + N Γ α dγ β dγ δ βδ dt dt = 0. Thus, γ is harmonic if and only if it is a geodesic. Example.6 (Harmonic maps to spheres). (i) Let ϕ : M S n be a smooth map and i : S n R n+1 the canonical inclusion. Denote by φ : M R n+1 the composition φ = i ϕ. Then ϕ is harmonic if and only if φ = dφ φ. (ii) The restriction of a homogeneous harmonic polynomial function of degree k, F : R m+1 R, to the sphere S m is an eigenfunction of the Laplacian on the sphere with eigenvalue λ k = k(m + k 1). The function f = F S m : S m R is called spherical harmonic of order k and it is known that the spherical harmonics give all the eigenfunctions of the Laplacian. (iii) A smooth map ϕ : M S n is called an eigenmap if the components of φ : M ϕ S n i R n+1 are all eigenfunctions of the Laplacian on M with the same eigenvalue. It was proved (see, for example, [8]) that a smooth map ϕ : M S n is an eigenmap if and only if it is harmonic with constant energy density. Moreover, a map between spheres ϕ : S m S n is an eigenmap if and only if the components of φ : S m R n+1 are all spherical harmonics of the same order.

36 34 Chapter. Preliminaries Example.7 (Minimal immersions). Let ϕ : (M, g) (N, h) be a Riemannian immersion, i.e. ϕ h = g. By identifying X C(T M) with dϕ(x) C(ϕ 1 T N), we nd that the second fundamental form (.) can be written as dϕ(x, Y ) = N XY M X Y, which coincides with the second fundamental form of ϕ used in the theory of Riemannian immersions. Thus, from (.6), τ(ϕ) = trace dϕ = mh, where H is the mean curvature vector eld of ϕ, and ϕ is harmonic if and only if it is minimal...3 The second variation of the energy Let (M, g) and (N, h) be two Riemannian manifolds. Suppose, for simplicity, that M is compact and let ϕ : M N be a harmonic map. We consider a smooth two parameter variation {ϕ s,t } s,t I, i.e. a smooth map Φ given by Φ : I I M N, Φ(s, t, p) = ϕ s,t (p), where Φ(0, 0, p) = ϕ 0,0 (p) = ϕ(p), for all p M. The variation vector elds corresponding to this variation, V and W, are given by V (p) = d ϕ s,0 (p) = dφ ds (0,0,p) ( / s) T ϕ(p) N, s=0 and W (p) = d ϕ 0,t (p) = dφ dt (0,0,p) ( / t) T ϕ(p) N. t=0 By a procedure similar to that in Remark.17, given two sections V, W C(ϕ 1 T N), one can determine a smooth two parameter variation {ϕ s,t } s,t I of variation vector elds V and W. The Hessian of the energy E in its critical point ϕ is dened by H(E) ϕ (V, W ) = E(ϕ s,t ). s t (s,t)=(0,0) Proposition.8. The Hessian of a harmonic map ϕ has the expression H(E) ϕ (V, W ) = = = M M M ( ϕ V, ϕ W + trace R N (dϕ, V )dϕ, W )v g ϕ V + trace R N (dϕ, V )dϕ, W v g J ϕ (V ), W v g,

37 .. Harmonic maps 35 where ϕ denotes the rough Laplacian acting on C(ϕ 1 T N), dened by m ϕ = trace( ϕ ) = { ϕ E i ϕ E i ϕ Ei E i }, (.11) i=1 with respect to a local orthonormal frame eld {E i } m i=1 on M. Remark.9. (i) The Hessian H(E) ϕ : C(ϕ 1 T N) C(ϕ 1 T N) R is a symmetric bilinear form, thus completely determined by {H(E) ϕ (V, V ) : V C(ϕ 1 T N)}. (ii) The operator J ϕ is a linear elliptic self-adjoint dierential operator. Moreover, since ϕ is positive the Hilbert space of sections L (ϕ 1 T N) = { } V C(ϕ 1 T N) : (V, V ) = V v g < M splits as the orthogonal sum of the eigenspaces of J ϕ, each being nite dimensional, and the spectrum of J ϕ consists of a discrete sequence of real numbers λ 1 < λ <... < λ i <.... In analogy with the Jacobi operator for vector elds along geodesics, J ϕ is called the Jacobi operator associated to ϕ and the elements of its kernel are called Jacobi elds along ϕ. Denition.30. Let ϕ : M N be a harmonic map. The index of ϕ is the dimension of the largest subspace of C(ϕ 1 T N) on which H(E) ϕ is negative denite. The nullity of ϕ is the dimension of the kernel of J ϕ. Denition.31. A harmonic map ϕ : M N is said to be stable if index(ϕ) = 0 and otherwise is said to be unstable. Proposition.3 ([53]). If (N, h) has non-positive sectional curvature, then any harmonic map ϕ : (M, g) (N, h) is stable. Example.33 (Geodesics). If γ : S 1 N is a geodesic, then J γ (V ) = γ γ V + R N ( γ, V ) γ, for all V C(γ 1 T N), thus J γ (V ) = 0 if and only if V is a Jacobi vector eld along γ in the classical sense.

38 36 Chapter. Preliminaries Example.34 (The identity map). Let 1 M : M M be the identity map of M. Then the Jacobi operator, J 1 M : C(T M) C(T M), assumes the following form J 1 M (X) = X Ricci(X) = trace X Ricci(X). We underline the fact that not even the identity map is always stable as a harmonic map. For example, it was proved in [108] that when M = S m, m 3, the identity map is unstable with index(1 S m) = m Harmonic morphisms Denition.35. Let ϕ : (M, g) (N, h) be a smooth map. Then ϕ is said to be a harmonic morphism if, for every harmonic function f dened on an open subset V N with ϕ 1 (V ) non-empty, f ϕ is harmonic on ϕ 1 (V ). Denition.36. A map ϕ : (M, g) (N, h) is said to be horizontally weakly conformal if for any p M with dϕ p 0, the restriction of dϕ p to the orthogonal complement of ker dϕ p is conformal and surjective. We shall use the following notations: ker dϕ p = Tp V M (the vertical space) and (ker dϕ p ) = Tp H M (the horizontal space). If ϕ is horizontally weakly conformal, then, away from critical points, there exists a function λ : M (0, ) such that h ϕ(p) (dϕ p (X), dϕ p (Y )) = λ (p)g p (X, Y ) for all X, Y T H p M. Setting λ(p) = 0 at critical points, we continuously extend λ to the whole M. The extended function is called the dilation function of the horizontally weakly conformal map ϕ. Since λ = n e(ϕ), the square dilation, λ : M R, is smooth even at critical points. Denition.37. A horizontally weakly conformal map ϕ : (M, g) (N, h) is said to be horizontally homothetic if the gradient of its dilation is vertical at regular points. The fundamental result in the theory of harmonic morphisms is the following characterization theorem proved by B. Fuglede and T. Ishihara. Theorem.38 ([59, 68]). Let ϕ : (M, g) (N, h) be a smooth map between two Riemannian manifolds. Then ϕ is a harmonic morphism if and only if ϕ is harmonic and horizontally weakly conformal. If ϕ is non-constant, then ϕ is a submersion on an open dense subset of M. Moreover, if at a point p M, rank dϕ p < n, then dϕ p = 0. For a detailed account on harmonic morphisms see [8].

39 .. Harmonic maps Harmonic Riemannian submersions As a consequence of Theorem.38, harmonic Riemannian submersions provide the simplest example of harmonic morphisms. In the following we shall present the basic results needed in order to characterize the harmonicity of a Riemannian submersion. Let ϕ : (M, g) (N, h) be a Riemannian submersion. Then T p M = T H p M T V p M, where p M, T V p M = ker dϕ p, and T H p M = (T V p M), i.e. the orthogonal complement of T V p M in T p M with respect to g. Lemma.39. Let ϕ : (M, g) (N, h) be a Riemannian submersion. We have (i) dϕ T H M T H M = 0, (ii) dϕ T V M T V M in M, (iii) dϕ T H M T V M = 0 if and only if the bers of ϕ are totally geodesic submanifolds = 0 if and only if the horizontal distribution is integrable. Corollary.40. If ϕ : (M, g) (N, h) is a Riemannian submersion, then the following statements are equivalent: (i) ϕ is a totally geodesic map, (ii) the bers of ϕ are totally geodesic submanifolds and the horizontal distribution is integrable. Let i p : ϕ 1 (ϕ(p)) M denote the inclusion of the ber over p of ϕ in M. We have 0 = τ(ϕ i p ) = dϕ(τ(i p )) + trace dϕ(di p, di p ) = dϕ(τ(i p )) + τ(ϕ), and, since τ(i p ) is horizontal, we have Theorem.41. A Riemannian submersion ϕ : (M, g) (N, h) has constant energy density e(ϕ) = n, and is harmonic if and only if its bers are minimal submanifolds. Corollary.4. If ϕ : (M, g) (N, h) is a Riemannian submersion, then the following statements are equivalent: (i) ϕ is a harmonic map, (ii) ϕ is a harmonic morphism, (iii) the bres of ϕ are minimal submanifolds.

40

41 Chapter 3 The biharmonic problem In this chapter we introduce the notion of biharmonic map as it was suggested by J. Eells and J.H. Sampson. Before doing this we shall recall the classical biharmonic problem and some of its applications in continuous media mechanics. 3.1 Physical problems formulated in terms of the biharmonic equation The classical biharmonic problem consists in nding a continuous function f : Ω R R, with continuous derivatives up to the fourth order, which is a solution of the homogeneous biharmonic equation and satises the boundary conditions ( f)(x) = 0, x Ω, f Ω = α and f n = β, i.e. f has prescribed values on the boundary and prescribed values of the outward normal derivatives. There are several independent mechanical phenomena modeled by the homogeneous or the inhomogeneous biharmonic equation. We present here three of them, following [58, 76, 85]. Example 3.1 (Thin plates). In the classical theory of thin plates the equilibrium for a thin plate can be characterized in terms of the biharmonic equation. Consider an elastic isotropic at plate of uniform thickness h. When it is bent the plate contains a neutral surface, that lies midway through the plate, on which there is 39

42 40 Chapter 3. The biharmonic problem no extension or compression. We consider a coordinate system with the origin on the neutral surface, the z-axis normal to the surface and the xy-plane as the plane of the undeformed plate and denote by ξ the vertical displacement of a point on the neutral surface, i.e. its z-coordinate. Then the total free energy of the deformed plate can be expressed as F (ξ) = D {( ξ) (1 σ) det(hess ξ) } dxdy, (3.1) where the constant D = Eh 3 /1(1 σ ) is called the exural rigidity of the plate and E and σ denote the Young's modulus and Poisson's ratio, respectively (see [76, p.13]). The equations of equilibrium can be derived from the minimum condition for the total free energy of the bent plate. By computing the rst variation of (3.1) and by considering P to be the load per unit area of the plate, we get to the inhomogeneous biharmonic equation D ξ = P, which, consequently, is the equation of equilibrium for a thin plate bent by external forces acting on it. Example 3. (The equilibrium in the plane deformation of a homogeneous isotropic elastic medium). The equations of equilibrium in the absence of external body forces, in the plane deformation of a homogeneous isotropic elastic medium reduce to j=1 σ ij x j = 0, i = 1, (3.) where {σ ij } i,j=1 denote the non-zero components of the stress tensor. The most general functions satisfying equations (3.) are of the form σ 11 = A x, σ 1 = A, x 1 x σ = A x, 1 where A : D R R From the compatibility condition satised by the non-zero components of the strain tensor, u 11 x + u x = u 1, 1 x 1 x u ij = 1 ( u i x j solution of + u j x i ), where u denotes the displacement vector, there follows that A is a i.e. it is a biharmonic function. A = 0,

43 3.1. Physical problems formulated in terms of the biharmonic equation 41 The function A is called the stress function or the Airy stress function. The reduction of plane elastic problems to the biharmonic equation is usually attributed to G.B. Airy [], but it was J.C. Maxwell (see [84]) the rst to present the equilibrium equation in the explicit form of the biharmonic equation. Example 3.3 (The -dimensional Stokes ow). The -dimensional Stokes ow can also be described in terms of the biharmonic equation. Consider a Newtonian incompressible uid and a set of axes moving with the local uid velocity u and another set xed in space. If the origin of the moving set of axes is at x = (x(t), y(t), z(t)) at time t, we can write its velocity as u(x, t) = dx dt. The steady ow of a Newtonian incompressible uid satises the Navier-Stokes equations div u = 0, (3.3) u, grad u = 1 ρ grad p ν u, where p denotes the pressure, ρ the density, ν the kinematic viscosity and no body forces are considered to be acting. The Stokes ow, sometimes referred to as creeping ow, describes the motion of an extremely viscous uid. In this case, the terms in (3.3) involving the squares of the velocities may be omitted compared with the viscous terms, and the uid motion may be approximated by using the Stokes equations div u = 0, (3.4) 0 = grad p µ u, where µ is the viscosity of the uid. If the ow in question is -dimensional with velocity eld u = ui + vj, the rst equation of (3.4) insures the existence of a function ψ satisfying u = ψ y and v = ψ x. Since ψ(x, y) = k, k R, provides a streamline for the motion, the function ψ is called the stream function. Taking the curl of the second equation in (3.4) and using div u = 0 we conclude that the stream function ψ satises the -dimensional biharmonic equation ψ = 0.

44 4 Chapter 3. The biharmonic problem 3. Biharmonic maps Let ϕ : (M, g) (N, h) be a smooth map between two Riemannian manifolds. As we have seen in (.6), the tension eld of ϕ is given by τ(ϕ) = trace dϕ. Denition 3.4. For a compact domain Ω M, the bienergy of ϕ over Ω is dened by E,Ω (ϕ) = 1 Ω τ(ϕ) v g. Denition 3.5. A smooth map ϕ is said to be biharmonic if it is a critical point of the bienergy functional E,Ω : C (M, N) R, for any compact domain Ω M The rst variation formula of the bienergy The rst variation formula and, thus, the Euler-Lagrange equation associated to the bienergy was obtained by R. Caddeo and V. Oproiu (unpublished) and by G.Y. Jiang in [69, 70]. We shall present a proof (see [97]) based on techniques introduced by J. Eells. Proposition 3.6 (First variation of the bienergy). Let ϕ : (M, g) (N, h) be a smooth map and let {ϕ t } t I, I = ( ε, ε), be a smooth variation of ϕ supported in Ω, where Ω M is compact. Then d E,Ω (ϕ t ) = dt t=0 Ω τ (ϕ), V v g, where V denotes the variation vector eld of {ϕ t } t I and is the bitension eld of ϕ. τ (ϕ) = ϕ τ(ϕ) trace R N (dϕ, τ(ϕ))dϕ (3.5) Proof. Let {ϕ t } t I, I = ( ε, ε), be a smooth variation of ϕ, i.e. Φ : I M N, satisfying { Φ(t, p) = ϕ t (p), (t, p) I M, Φ(0, p) = ϕ 0 (p) = ϕ(p), p M. a smooth map The variation vector eld V C(ϕ 1 T N) associated to the variation {ϕ t } t I is given by V (p) = d ϕ t (p) = dφ dt (0,p) ( / t) T ϕ(p) N, p M. t=0 We have d E,Ω (ϕ t ) = 1 { } dt t=0 Ω t τ(ϕ t), τ(ϕ t ) t=0 v g = Φ / t τ(ϕ t), τ(ϕ t ) v g. t=0 Ω

45 3.. Biharmonic maps 43 Let now {E i } m i=1 be a local orthonormal frame eld geodesic at p Ω, i.e. {E i} m i=1 is a local orthonormal frame eld with ( Ei E j ) p = 0, for all i, j = 1,..., m. With respect to {E i } m i=1 we have m Φ / t τ(ϕ t) = Φ / t dφ(e i, E i ) = = m i=1 m i=1 i=1 { } Φ / t [( Φ E i dφ)(e i )] For a given Z C(T M), since [ / t, Z] = 0, we get thus, at p, we obtain Φ / t τ(ϕ t) = = = = { } Φ / t Φ E i dφ(e i ) Φ / t dφ( E i E i ). Φ / t dφ(z) = Φ ZdΦ( / t) + dφ[ / t, Z] m i=1 = Φ ZdΦ( / t), { } Φ / t Φ E i dφ(e i ) Φ Ei E i dφ( / t) m Φ / t Φ E i dφ(e i ) i=1 m i=1 { Φ E i Φ / t dφ(e i) + Φ [ / t,e i ] dφ(e i) } +R Φ ( / t, E i )dφ(e i ) m { } Φ E i Φ E i dφ( / t) + R N (dφ( / t), dφ(e i ))dφ(e i ). i=1 Denote by ν the outward pointing unit normal of Ω in M and by i : Ω M the canonical inclusion. Now, by using the symmetries of the Riemann-Christoel tensor eld and the Divergence Theorem (see Proposition.9, Chapter ) for ρ = τ(ϕ), V and θ = τ(ϕ), V, d E,Ω (ϕ t ) = dt t=0 = = = trace V trace R N (dϕ, V )dϕ, τ(ϕ) v g Ω V, dτ(ϕ) v g ν V, τ(ϕ) v i g V, trace R N (dϕ, τ(ϕ))dϕ v g Ω Ω Ω V, trace τ(ϕ) v g + V, ν τ(ϕ) v i g Ω Ω V, trace R N (dϕ, τ(ϕ))dϕ v g Ω V, ϕ τ(ϕ) trace R N (dϕ, τ(ϕ))dϕ v g. Ω

46 44 Chapter 3. The biharmonic problem Theorem 3.7. A map ϕ : (M, g) (N, h) is biharmonic if and only if Remark 3.8. τ (ϕ) = 0. (3.6) (i) Equation (3.6) is called the bitension eld equation or the biharmonic equation and, in local coordinates, it becomes a 4-th order non-linear system of PDE's. If we consider (U, x i ) and (W, y α ) two local coordinate charts at p on M and in ϕ(p) on N, respectively, and denote by M Γ k ij and N Γ α βδ the Christoel symbols of the Levi-Civita connection of M and N, respectively, then τ (ϕ) = g ij( τ σ x i x j + τ α x j ϕ β +τ α ϕβ N Γ σ αβ x i x j ( τ σ M Γ k ij ϕ β N x j Γ σ αβ + τ α N x i x j Γ σ αβ + τ α ϕβ x i ϕ ρ x k + τ α ϕβ x k N Γ σ αβ N x j Γ ν αβ N Γ σ νρ ) τ ν ϕα ϕ β N x i x j Rβαν σ where τ(ϕ) = τ σ y ϕ has the local expression given by (.8). σ (ii) A map ϕ is biharmonic if and only if τ(ϕ) ker J ϕ, where J ϕ : C(ϕ 1 T N) C(ϕ 1 T N), J ϕ (V ) = ϕ V + trace R N (dϕ, V )dϕ, is, formally, the Jacobi operator corresponding to ϕ. (iii) Since J ϕ is linear, any harmonic map is biharmonic. ) y σ ϕ, (iv) Any harmonic map is an absolute minimum for the bienergy functional. Denition 3.9. A map between two Riemannian manifolds is said to be proper biharmonic if it is a non-harmonic biharmonic map. In the following we shall give some examples of biharmonic maps and submanifolds. 3.. Examples of biharmonic maps Example 3.10 (Biharmonic maps to Euclidean spaces). Consider a smooth map ϕ : (M, g) R n, ϕ(p) = (ϕ 1 (p),..., ϕ n (p)). In this situation the tension eld reduces to τ(ϕ) = ϕ = ( ϕ 1,..., ϕ n ) and the bienergy has the expression E,Ω (ϕ) = 1 Ω ϕ v g.

47 3.. Biharmonic maps 45 The biharmonic equation is given by τ (ϕ) = ϕ = ( ϕ 1,..., ϕ n ) = 0, thus a biharmonic map to R n is equivalent to an n-tuple of biharmonic functions on M. We now list some examples. (i) The simplest examples of biharmonic maps are provided by the polynomial maps of degree and 3 between Euclidean spaces. (ii) The Almansi Property (see [3]) provides a method for constructing proper biharmonic maps by using harmonic ones. The Almansi property states that if f : R m R is harmonic, then the product function r f : R m R is proper biharmonic, i.e. f = 0 = (r f) = 0. Here r : R m R denotes the distance function from the origin dened by r(x 1,..., x m ) = (x 1 ) (x m ). The map ϕ : R 4 R 3, ϕ(z, w) = ( z + w ) ( z w, zw ), is an example of proper biharmonic polynomial map of degree 4. It derives from applying the Almansi property to the Hopf map Ψ : R 4 R 3, Ψ(z, w) = ( z w, zw ). (iii) The inversion in the unit sphere ϕ : R m \ {0} R m \ {0}, ϕ(x) = x/ x, is proper biharmonic if and only if m = 4. The conclusion follows from a series of straightforward computations, but this example was obtained as a special case in the study of the behavior of the bitension eld under conformal changes of the domain metric for the identity map (see [7] and Chapter 4). Example 3.11 (Biharmonic maps from geodesics). A simple way of constructing examples of biharmonic maps to an arbitrary Riemannian manifold (N, h) is provided by using the geodesics. Consider γ : I N to be a geodesic and let t : J I, t = t(s), I, J R, be a change of parameter. It is not dicult to verify that γ = γ t : J N is a proper biharmonic map if and only if d 4 t/ds 4 = 0 and d t/ds 0.

48 46 Chapter 3. The biharmonic problem Example 3.1 (A family of biharmonic non-isometric curves on the Enneper surface). Any straight line passing through the origin in (R, dx + dy ) transforms into a nonisometric biharmonic curve after applying the conformal change of metric ( R, g = (r + 1) (dx + dy ) ), where r = x + y is the distance function from the origin (see [79]). The metric g is the metric, in local isothermal coordinates, of the Enneper minimal surface. Figure 3.1 is a plot of the Enneper surface in polar coordinates, so that radial curves on the picture are biharmonic. Figure 3.1: The radial curves of the Enneper surface through the origin are nonisometric proper biharmonic curves. Example 3.13 (Biharmonic projections from the tangent bundle). Two examples of proper biharmonic Riemannian submersions were constructed in [94] by using special "Sasaki type" metrics on the tangent bundle of a Riemannian manifold (M, g). Denote by V (T M) the vertical distribution on T M dened by V v (T M) = ker dπ v, v T M, where π : T M M is the canonical projection. We consider a nonlinear connection on T M dened by the distribution H(T M) on T M, complementary to V (T M), i.e. H v (T M) V v (T M) = T v (T M), v T M. For any induced local chart (π 1 (U), x i, y j ) on T M we have a local adapted frame in H(T M) dened by the local vector elds δ δx i = x i N j i (x, y), i = 1,..., m, xj where the local functions N j i (x, y) are the connection coecients of the nonlinear connection dened by H(T M). If T M is endowed with a Riemannian metric dened by S(X V, Y V ) = S(X H, Y H ) = g(x, Y ), S(X V, Y H ) = 0, then the canonical projection π : T M M is a Riemannian submersion and its biharmonicity can be expressed in the terms of the connection coecients. In the case the horizontal distribution is determined by the Levi-Civita connection on the base manifold, i.e. N j i = Γj ik yk, the metric S is the usual Sasaki metric and the projection π : T M M is harmonic and thus biharmonic.

49 3.. Biharmonic maps 47 Other non-trivial examples were generated: (i) Recall that two torsion-free ane connections and on a manifold M are said to be projectively equivalent if there exists a 1-form α on M such that X Y = X Y + α(y )X + α(x)y, X, Y C(T M). In local coordinates this condition is expressed as Γ j ik = Γ j ik + δj i α k + δ j k α i and we also say that is a projective change of connection or that the identity map 1 : (M, ) (M, ) is projective. Moreover, two torsion-free connections are projectively equivalent if and only they have the same pregeodesics (see [91, p.36]). If ξ is a unit Killing vector eld on (M, g) and the horizontal distribution is determined by the projective change of the Levi-Civita connection on (M, g) induced by ξ, i.e. N j i = Γj iky k = (Γ j ik + δj i ξ k + δ j k ξ i)y k, then the projection map π : (T M, S) (M, g) is proper biharmonic. (ii) A conformal change of metric on a Riemannian manifold g = e ρ g induces a change of the Levi-Civita connection X Y = X Y +X(ρ)Y +Y (ρ)x g(x, Y ) grad ρ, for all X, Y C(T M) (see [5]). In local coordinates Γ j ik = Γj ik + δj i α k + δ j k α i g ik α j, where α k = ρ/ x k. If ρ C (M) in a non-constant ane function and the horizontal distribution is determined by the conformal change of metric g = e ρ g, i.e. N j i = Γ j ik yk = (Γ j ik +δj i α k + δ j k α i g ik α j )y k, then the projection map π : (T M, S) (M, g) is proper biharmonic. Example 3.14 (Biharmonic maps to spheres). By using the expression for the curvature tensor eld for the unit Euclidean sphere, we conclude that ϕ : (M, g) S n is biharmonic if and only if ϕ τ(ϕ) + e(ϕ)τ(ϕ) trace dϕ, τ(ϕ) dϕ = 0. We now describe a method for constructing biharmonic maps to spheres from harmonic maps of constant energy density (see [81]). Let M be a compact manifold, let ψ : M S n 1 ( 1 ) be a non-constant map and denote by i : S n 1 ( 1 ) S n the canonical inclusion. The composition map ϕ = i ψ : M S n is proper biharmonic if and only if ψ is harmonic and e(ψ) is constant. We underline that the converse holds even when M is not compact. The previous construction can be applied to a series of remarkable examples of harmonic maps of constant energy density: (i) the generalized Veronese map ψ : S m ( (m + 1)/m) S m+p ( 1 ), where p = (m+1)(m ) ; (ii) the generalized Cliord torus ψ : S l ( 1 ) Sl ( 1 ) Sl+1 ( 1 ); (iii) the Hopf map ψ : S 3 ( ) S ( 1 ), ψ(z, w) = 1 (zw, z w ).

50 48 Chapter 3. The biharmonic problem 3..3 Examples of biharmonic submanifolds Example 3.15 (Biharmonic submanifolds in Euclidean spaces). In [44], B-Y. Chen classied the Riemannian immersions ϕ : M R n which satisfy the condition that their mean curvature vector eld in R n is an eigenvector of the Laplacian, i.e. H = λh, for some λ R. In particular, when λ = 0, that is when ϕ has harmonic mean curvature, from the formula of Beltrami, ϕ = mh, the condition H = 0 is equivalent to ϕ = 0. For this reason, the Riemannian submanifolds with harmonic mean curvature vector eld in R n were called biharmonic submanifolds by B-Y. Chen. As we have already seen, the tension eld of a Riemannian immersion has the expression τ(ϕ) = mh, (3.7) and, replacing (3.7) in (3.5), we get that the bitension eld of a Riemannian immersion in R n is τ (ϕ) = m H. We thus conclude that a Riemannian immersion ϕ : M R n is biharmonic if and only if it is biharmonic in the sense of B-Y. Chen. Example 3.16 (Biharmonic curves, [31]). Consider γ : I (N, h) to be a curve parametrized by arc length from an open interval I R to a Riemannian manifold. In this case, as we have seen, by denoting with T = γ, we get that τ(γ) = T T and the biharmonic equation (3.6) becomes 3 T T R(T, T T )T = 0. (3.8) A curve γ : I R (N n, h), parametrized by arc length, is called a Frenet curve of osculating order r, 1 r m, if there exist orthonormal vector elds {F i } r i=1 along γ such that F 1 = dγ( / s), γ / s F 1 = k 1 F, γ / s F i = k i 1 F i 1 + k i F i+1, i =,..., r 1, γ / s F r = k r 1 F r 1,

51 3.. Biharmonic maps 49 where the functions {k 1, k,..., k r 1 } are positive functions on I, called the curvatures of γ. A characterization of proper biharmonic curves of osculating order r in terms of the Frenet frame {F i } r i=1 and of the curvatures {k i} r 1 i=1 was given in [31]. The curve γ : I (N n, h), n, of osculating order r, is proper biharmonic if and only if k 1 = constant 0 k 1 + k = R(F 1, F, F 1, F ) k = R(F 1, F, F 1, F 3 ) k k 3 = R(F 1, F, F 1, F 4 ) R(F 1, F, F 1, F i ) = 0 i = 5,..., r R(F 1, F, F 1, ζ) = 0, ζ (span{f i } r i=1 ) γ 1 T N. More details on this problem can be found in Chapter 5. (3.9) Example 3.17 (Biharmonic curves on a surface of revolution, [3]). (i) On a surface of revolution with nonconstant Gauss curvature, obtained by rotating around the z-axis the curve α(u) = (f(u), 0, g(u)), parametrized by arc length, the biharmonic curves are the parallels u = u 0 =constant, where f (u 0 ) + f(u 0 )f(u 0 ) = 0. For example, the torus of revolution with its standard parametrization by arc length, X(u, v) = ( (a + r cos(u/r)) cos v, (a + r cos(u/r)) sin v, r sin(u/r) ), a > r, has two proper biharmonic curves which are the parallels given by u 1 = r arccos a + a + 8r, u = πr r arccos a + a + 8r. 4r 4r The proper biharmonic curves on the torus are plotted in Figure 3. Figure 3.: The proper biharmonic curves on the torus. (ii) The right circular cylinder, obtained when f is constant, and the surface of revolution generated by f(u) = ±c u, g(u) = u 4u c 4u c 4u c (8u 8 log + 8u c ) + c 1, 4u

52 50 Chapter 3. The biharmonic problem where c and c 1 are positive constants, are the only surfaces of revolution with all parallels biharmonic. Such a surface is plotted in Figure 3.3. Figure 3.3: Surface of revolution with all parallels proper biharmonic. (iii) The surfaces of revolution with positive constant Gauss curvature, K = 1/a, can be parametrized by ( ) u X(u, v) = b cos(u/a) cos v, b cos(u/a) sin v, 1 b a sin (s/a) ds, b > 0, and the biharmonic curves which are not parallels are obtained for ( c ± b c sin(a + u(t) = a arcsin b v(t) = t 0 ( sin u(s) a b cos u(s) a + 0 c b cos u(s) a ) ds, a t) ) where A, c R and b c > 0. The biharmonic parallels are obtained for u = ±aπ/4. Notice that a biharmonic curve on a sphere of radius r is a circle of radius r/ (see Figure 3.4). Example 3.18 (Biharmonic curves of Cartan-Vranceanu spaces). Let m be a real parameter. Denote by N the whole R 3 if m 0, and by N = {(x, y, z) R 3 : x + y < 1 m } otherwise. Consider on N the following two-parameter family of Riemannian metrics ds l,m = dx + dy ( [1 + m(x + y )] + dz + l ) ydx xdy [1 + m(x + y, (3.10) )] where l, m R. The family of metrics (3.10) includes all 3-dimensional homogeneous metrics whose group of isometries has dimension 4 or 6, except for those of constant negative sectional curvature.