Energy identity for intrinsically biharmonic maps in four dimensions

Abstract

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 .

A relaxation of the intrinsic biharmonic energy

“…For instance, in the critical domain dimension four, the energy concentration set is finite, and the energy loss due to concentration can be described by 'bubbling', cf. [7]. This is analogous to the situation for harmonic maps, see, e.g., [5].…”

Neck analysis for biharmonic maps

“…As a byproduct of this understanding, we give new proofs to other known results in the field of neck analysis. The first one is the following energy identity result, which was proved for extrinsic biharmonic maps by Wang and Zheng [21,22], for intrinsic Hessian biharmonic maps by Hornung and Moser [5] and for both cases by Laurain and Rivière [7].…”

“…It follows from Theorem A of [18] and Theorem 6.1 and Theorem 6.2 of [11]. An argument like Lemma 2.5 in [5] is needed to show that u is a weak biharmonic map to use the above mentioned theorems. Theorem 2.3.…”

“…In the case of harmonic maps, it was first introduced to the study of neck analysis by Lin and Wang [9] and it says the tangential part of the energy is the same as the radial part. The computation is generalized to biharmonic maps in [5], [22] and [7]. Because the biharmonic maps satisfy a fourth order PDE, boundary terms arise in the computation and the authors of [5] and [22] managed to show that the boundary terms are small so that they can still compare the tangential energy and the radial energy.…”

Analyse des problèmes conformément invariants

“…Dans ce cas on parle d'application biharmonique extrinsèque. Soit on considère l'énergie Ce résultat était déjà partiellement connu pour les applications intrinsèques, [26]. La méthode s'adaptant bien aux perturbations, on peut en déduire les conséquences suivante sur le comportement asymptotique du flot biharmonique.…”