The bienergy of unit vector fields

Markellos, Michael; Urakawa, Hajime

doi:10.1007/s10455-014-9432-2

Cited by 5 publications

(6 citation statements)

References 17 publications

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“…As a consequence, this yields (in combination with the Table 2 in [17]), Theorem 3.1. Let G be the Lie group SU (2) equipped with a left-invariant Riemannian metric <, >.…”

Section: Examples Of Biharmonic Unit Sections Which Are Not Harmonic ...supporting

confidence: 54%

“…i=1 is a local orthonormal frame field of (M, g). Following the terminology used in [17], we get Theorem 2.3. Let (M, g) be an m-dimensional Riemannian manifold.…”

Section: Now We Give the Following Definitionsmentioning

confidence: 99%

“…Recall that V is a biharmonic unit section if and only if d dt E v 2 (V t )| t=0 = 0 for all variations {V t = e u V t } t∈I within X 1 (M ). We adopt the terminology of [17] and [18]. We consider the function…”

Section: Conformal Change Of the Metric And The Tension Field Of A Vementioning

confidence: 99%

“…The critical point condition for the bienergy functional E 2 restricted to X 1 (M ) has been determined in [17]. More precicely, Theorem 2.2.…”

Section: Now We Give the Following Definitionsmentioning

confidence: 99%

“…biharmonic unit sections). In [17], the authors derived the Euler-Lagrange equation for biharmonic unit vector fields and completely determined the class of invariant biharmonic unit vector fields on three-dimensional unimodular Lie groups and the generalized Heisenberg groups. We should mention that the majority of these examples are on non-compact Riemannian manifolds.…”

Section: Introductionmentioning

confidence: 99%

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The biharmonic homotopy problem for unit vector fields on 2-tori

Loubeau

Markellos

2019

Annali di Matematica

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The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form different sets. Working with surfaces, we first obtain general characterizations of biharmonic unit vector fields and biharmonic unit sections under conformal change of the metric. In the case of a 2-dimensional torus, this leads to a proof that biharmonic unit sections are always harmonic and a general existence theorem, in each homotopy class, for biharmonic unit vector fields.

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“…As a consequence, this yields (in combination with the Table 2 in [17]), Theorem 3.1. Let G be the Lie group SU (2) equipped with a left-invariant Riemannian metric <, >.…”

Section: Examples Of Biharmonic Unit Sections Which Are Not Harmonic ...supporting

confidence: 54%

“…i=1 is a local orthonormal frame field of (M, g). Following the terminology used in [17], we get Theorem 2.3. Let (M, g) be an m-dimensional Riemannian manifold.…”

Section: Now We Give the Following Definitionsmentioning

confidence: 99%