Paper
0
The energy of unit vector fields on the 3-sphere
Authors
A. Higuchi,B. S. Kay
C. M. Wood,Tao Li,Qin Jim Chen,Pantaleo Rwelamila,Fen-May Liou,Veena Goswami,Daniela PIETROGIACOMI,Simonetta TUTI,Cindy Christiansen,Jose Rodriguez-Mirasol,Jeronimo Lopez-Martinez,Antonio Roque,José Rodríguez Mirasol, José María Fernández Sevilla,Ranjit Kumar Upadhyay,Lucia Del Bianco,Maowen Li,Manuel García León,Marta Estrada,Iñaki Berregi,Cristina Buzea,Christophe Lemaire,Anna AMBROSINI,sandra vitolo,Giorgio Gribaudo,Angelo Bisazza,Ana Poveda,Randolph Hall,Irina Gureviciene,Felicita Scapini,Michael Kanost,A. Joseph Layon,Laurent Michaud,Susanne Katharina Sauer,Ian Hall,William Seymour Brooks,Harry Van den Akker,Richard Kovacs,Nicholas Cook,KUO-YEN WEI,Alessandro Pesci,Sylvester Gates,Veronique Bernard,Włodzimierz Mrówczyński,Mercedes Atienza,Mahadevan Ganesh,Piotr Krutki,Laurence Mather,Frédéric Dardel,Javier Miñano Sánchez,Sheila Stager,Jordi Llorens,valery kalinchuk,Antonella Poggi,Pierre Muret,Hideto Kaba,Lotfi Belkhir,Isabel Liste,Simon Taylor,David Searls,Nebojsa Marinkovic,soledad garcía-tasende,Anjela Koblischka-Veneva,Agustin Sanchez,winston morgan,Theresa Coetzer,Niels H. Batjes,marian saling,Konstantin Smirnov,Robert Kolos,Stig S. Gezelius,William Staines,Leonor Martins Almeida,Chrissoleon Papadopoulos,N/A ,Ronit Avitsur,Chengji Zhou,Charles Myers,Emilio Molina Grima,Michael Lichten,Andreas Burkovski,NATALIA SEMENOVA,Илья Маркович Каганцов,Bin Ren,Gilberto Silva-Romo,Richard Roe,Jérôme Leveneur,Piet Mulders,Shu-Yen Wan,Gregory Bishop-Hurley,Luis Felipe Ribeiro Pinto,Shinya Oki,Paul Bates,Santiago Hernández-Cassou,Gilles Foret,Pradip Kumar Tewari,Julia Hargreaves,James Chalmers,Björn Zackrisson,Anne-Catherine Rolland,David Towey,Riccardo Caruso,Yang Luo,barbara barbini,Venkataranganna Marikunte,Aaldrik Tiktak,Piotr Drożdżewski,Cinzia Volonté,Gerardo Burton,José Dorado,valery astakhov,Julian Ross,Gunver Krarup Pedersen,Vera Lima,Winghong Chan,Michael Ondrusek,Jane-Lise Samuel,Akmal Djamaan,José Custódio,Izumi Shibuya,Christian Hönemann,Andrew Lam,Jin-Guang Teng,Laxmi Goparaju,Mitchell Anscher,Benoit DEFFONTAINES,Firoz Ahmad,Kazuya Koumoto,XAVIER SALA-BLANCH,Oleksandr Tereshchenko,Andrea Leoncini,elvira Moya de Guerra,Iosif Bena,Donald Marion,Roberto Pizzala,Jesús Casal,Philippe Dumée,Magny Thomassen,Steven Huddart,Jorge Rosa,Demetrio Raffa,Atsushi Higuchi,Teresa Granda,PAOLA VITERBORI,Andres Saul,Masahiko Demura,Marcos Hikari Toyama,massimo costantini,Andreas Lang,Nicoletta Brunello,Philippe Pondaven,Larysa Paniwnyk,Phillip Chalk,Giuseppe Di Vita,Marie-Odile Jauberteau,Elisabet Sarri,Luisa Schenetti,Guenter Krause,Junji Seki,Paul Freemont,Levente Kapas,Gabriel Dimitriou,Marco De Amici,Brigitte Malgrange,CARLO DE MICHELI,Terje Einar Michaelsen,Harry Vrieling,Ronald Clarke,Vanessa Beijamini,Osamu honmou,Shaocheng Ji,Vladimir Pankratov,Roberto Andreatini,Axel Michaelowa,DIEGO RODOLFO COLOMBO,Francisca Rebolledo-Vicente,alain IOST,FEDERICA COMPOSTELLA,Gamal Hassan,Francesco Felici,Giovanni Franchin,Gyula VATAI,Jennelle Kyd,Rogerio Gama,Atsuko Gunji,Rosamund Bryar,Xavier Mathew,Francisco J. 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arXiv:math/0005293v1 [math.DG] 31 May 2000
THE ENERGY OF UNIT VECTOR FIELDS ON THE 3-SPHERE
by
A. Higuchi, B. S. Kay & C. M. Wood
Abstract. The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent
bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed,
and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour
of the 3-dimensional Hopf vector field is compared and contrasted with that of the closely-related Hopf map.
Finally, it is shown that the Hopf vector fields are the unique global minima of the energy functional restricted
to unit vector fields on the 3-sphere.
1. Introduction
A smooth unit vector field σ on a compact Riemannian manifold (M, g) with Euler char-
acteristic zero may be regarded as a smooth mapping of Riemannian manifolds σ: (M, g) →
(U M, h), where U M is the unit tangent bundle and h is the restriction of the Sasaki metric
on the tangent bundle T M . The energy of σ may be defined accordingly. Since metrics
h and g are horizontally isometric, and σ is a section, it suffices to consider the vertical
energy functional:
Ev (σ) =
∫
M
|dvσ|2 dx (1-1)
where dvσ is the vertical component of the differential dσ. (Here, ‘horizontal’ and ‘vertical’
refer to the complementary distributions on T M defined by the Levi-Civita connection).
One says that σ is a critical point of Ev, or a harmonic section of U M , if Ev is stationary
at σ with respect to variations through unit vector fields. The (non-linear) Euler-Lagrange
equations for this variational problem are [17]:
∇∗∇σ − |∇σ|2 σ = 0, (1-2)
where ∇∗∇ is the trace (or rough) Laplacian:
∇∗∇σ = − Tr ∇2σ.
Further, one says that a harmonic section σ is Ev-stable if the second variation of Ev
at σ with respect to unit vector fields is non-negative. The second variation of Ev in
this constrained sense may be regarded as a quadratic form Hv
σ (the vertical Hessian) on
the space Vσ of appropriate variation fields; since σ is allowed to vary only through unit
vector fields, Vσ is the space of smooth vector fields on M which are pointwise orthogonal
to σ. Associated to Hv
σ is the vertical Jacobi operator J v
σ :
Hv
σ (α, β) =
∫
〈J v
σ (α), β〉 dx, for all α, β ∈ Vσ . (1-3)
THE ENERGY OF UNIT VECTOR FIELDS ON THE 3-SPHERE
by
A. Higuchi, B. S. Kay & C. M. Wood
Abstract. The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent
bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed,
and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour
of the 3-dimensional Hopf vector field is compared and contrasted with that of the closely-related Hopf map.
Finally, it is shown that the Hopf vector fields are the unique global minima of the energy functional restricted
to unit vector fields on the 3-sphere.
1. Introduction
A smooth unit vector field σ on a compact Riemannian manifold (M, g) with Euler char-
acteristic zero may be regarded as a smooth mapping of Riemannian manifolds σ: (M, g) →
(U M, h), where U M is the unit tangent bundle and h is the restriction of the Sasaki metric
on the tangent bundle T M . The energy of σ may be defined accordingly. Since metrics
h and g are horizontally isometric, and σ is a section, it suffices to consider the vertical
energy functional:
Ev (σ) =
∫
M
|dvσ|2 dx (1-1)
where dvσ is the vertical component of the differential dσ. (Here, ‘horizontal’ and ‘vertical’
refer to the complementary distributions on T M defined by the Levi-Civita connection).
One says that σ is a critical point of Ev, or a harmonic section of U M , if Ev is stationary
at σ with respect to variations through unit vector fields. The (non-linear) Euler-Lagrange
equations for this variational problem are [17]:
∇∗∇σ − |∇σ|2 σ = 0, (1-2)
where ∇∗∇ is the trace (or rough) Laplacian:
∇∗∇σ = − Tr ∇2σ.
Further, one says that a harmonic section σ is Ev-stable if the second variation of Ev
at σ with respect to unit vector fields is non-negative. The second variation of Ev in
this constrained sense may be regarded as a quadratic form Hv
σ (the vertical Hessian) on
the space Vσ of appropriate variation fields; since σ is allowed to vary only through unit
vector fields, Vσ is the space of smooth vector fields on M which are pointwise orthogonal
to σ. Associated to Hv
σ is the vertical Jacobi operator J v
σ :
Hv
σ (α, β) =
∫
〈J v
σ (α), β〉 dx, for all α, β ∈ Vσ . (1-3)
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