A Variational Problem Pertaining to Biharmonic Maps

Moser, Roger

doi:10.1080/03605300802224698

Cited by 31 publications

(56 citation statements)

References 24 publications

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“…Recent progress has been made, however, based on the observation that the lack of coercivity can be removed for one type of intrinsic biharmonic mappings (the type studied in the present paper), provided that one works in a geometrically motivated variant of Sobolev spaces [Moser 2008;Scheven 2009]. This approach permits methods analogous to what has been used for harmonic mappings.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…see [Scheven 2009;Moser 2008]. In [Scheven 2009] it is shown that a mapping u ∈ W 2,2 (S, N ) is biharmonic precisely if it satisfies…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Moreover, for each x ∈ 0 there exist M x ∈ ‫ގ‬ and biharmonic mappings v (ii) Moser [2008] showed that every biharmonic mapping v ∈ W 2,2 (S, N ) in fact satisfies v ∈ C ∞ (S, N ).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The existence of minimizers of E 2 under given boundary conditions on the mapping itself and on its first derivatives was established in [Moser 2008] using the direct method of the calculus of variations. For simplicity, from now on we will omit the adverb "intrinsically":…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Energy identity for intrinsically biharmonic maps in four dimensions

Hornung

Moser

2012

APDE

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Let u be a mapping from a bounded domain S ⊂ ‫ޒ‬ 4 into a compact Riemannian manifold N . Its intrinsic biharmonic energy E 2 (u) is given by the squared L 2 -norm of the intrinsic Hessian of u. We consider weakly converging sequences of critical points of E 2 . Our main result is that the energy dissipation along such a sequence is fully due to energy concentration on a finite set and that the dissipated energy equals a sum over the energies of finitely many entire critical points of E 2 .

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…see [Scheven 2009;Moser 2008]. In [Scheven 2009] it is shown that a mapping u ∈ W 2,2 (S, N ) is biharmonic precisely if it satisfies…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Energy identity for intrinsically biharmonic maps in four dimensions

Hornung

Moser

2012

APDE

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“…non-harmonic biharmonic, maps are studied. The theory of biharmonic maps has two main directions: one concerns the analytic properties (see [20,39,45,64]) and the other deals with geometric aspects. In the following we shall only refer to the latter.…”

Section: H) Being the Corresponding Structures If ϕ : M → N Is A Smomentioning

confidence: 99%

Harmonic and Biharmonic Maps at Iaşi

Balmus¹,

Fetcu²,

Oniciuc³

2010

Annals of the Alexandru Ioan Cuza University - Mathematics

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Abstract. We report on the achievements of the geometers from Iaşi in the field of harmonic and biharmonic maps.Mathematics Subject Classification 2000: 53C07, 53C40, 53C42, 53C43, 58E20. Key words: harmonic and biharmonic maps, minimal and biharmonic submanifolds.Nowadays, the theory of harmonic maps between Riemannian manifolds is a very important field of Riemannian geometry. The harmonic maps ϕ : (M, g) → (N, h) are critical points of the energy functional E which is defined on the infinit dimensional manifold of the smooth maps from M to N ,Here dϕ is the differential of the map ϕ and ∥·∥ denotes the Hilbert-Schmidt norm. The Euler-Lagrange equation associated to the energy E is given by the vanishing of the tension field τ (ϕ) = trace ∇dϕ and it is a second order elliptic equation, as one can see from its local expressionwhere Γ denotes the Christoffel symbols and ∆ is the Laplacian acting on functions. From here raises the deep link established by the harmonic maps *

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Biharmonic submanifolds of $${\mathbb{C}P^n}$$

et al. 2009

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We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifoldM in CP n and the bitension field of the inclusion of the corresponding Hopf-tube in S 2n+1 . Using this relation we produce new families of proper-biharmonic submanifolds of CP n . We study the geometry of biharmonic curves of CP n and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.

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A Variational Problem Pertaining to Biharmonic Maps

Cited by 31 publications

References 24 publications

Energy identity for intrinsically biharmonic maps in four dimensions

Energy identity for intrinsically biharmonic maps in four dimensions

Harmonic and Biharmonic Maps at Iaşi

Biharmonic submanifolds of $${\mathbb{C}P^n}$$

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