Heat Flow for Extrinsic Biharmonic Maps with Small Initial Energy

Lamm, Tobias

doi:10.1023/b:agag.0000047526.21237.04

Cited by 32 publications

(33 citation statements)

References 12 publications

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“…A weak solution to the extrinsic biharmonic map flow is a map u 2 W 2;2 .OE0; C1OE B 1 ; N / satisfying where u 0 2 W 2;2 .B 1 ; N /. Several existence results have been established for (1.3), see for instance [9] for small initial data or [2] and [18] for solution with finitely many singular times and arbitrary initial data. All these solutions satisfy the energy identity…”

Section: Introductionmentioning

confidence: 99%

Energy quantization for biharmonic maps

Laurain

Rivière

2013

Advances in Calculus of Variations

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In the present work we establish an energy quantization (or energy identity) result for solutions to scaling invariant variational problems in dimension 4 which includes biharmonic maps (extrinsic and intrinsic). To that end we first establish an angular energy quantization for solutions to critical linear 4th order elliptic systems with antisymmetric potentials. The method is inspired by the one introduced by the authors previously in "Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications" (2011) for 2nd order problems.

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Section: Introductionmentioning

confidence: 99%

Energy quantization for biharmonic maps

Laurain

Rivière

2013

Advances in Calculus of Variations

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“…Proof In[12] the author calculated the Euler-Lagrange equation of M | u| 2 . By combining this with the well-known Euler-Lagrange equation for the Dirichlet energy one gets the Euler-Lagrange equation for E ε .…”

mentioning

confidence: 99%

Fourth order approximation of harmonic maps from surfaces

Lamm

2006

Calc. Var.

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Let (M 2 , g) be a compact Riemannian surface and let (N n , h) be a compact Riemannian manifold, both without boundary, and assume that N is isometrically embedded into some R l . We consider a sequence u ε ∈ C ∞ (M, N ) (ε → 0) of critical points of the functional E ε (u) = M (|Du| 2 + ε| u| 2 ) with uniformly bounded energy. We show that this sequence converges weakly in W 1,2 (M, N ) and strongly away from finitely many points to a smooth harmonic map. One can perform a blow-up to show that there separate at most finitely many non-trivial harmonic two-spheres at these finitely many points. Finally we prove the so called energy identity for this approximation in the case that N = S l−1 → R l .

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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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Heat Flow for Extrinsic Biharmonic Maps with Small Initial Energy

Cited by 32 publications

References 12 publications

Energy quantization for biharmonic maps

Energy quantization for biharmonic maps

Fourth order approximation of harmonic maps from surfaces

Harmonic and Biharmonic Maps at Iaşi

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