Energy quantization for biharmonic maps

Abstract

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.

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].…”

“…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

“…Une autre question naturelle est de savoir si autour des points critiques de petites énergies la fonctionnelle est convexe (voir [12] et [30] pour le cas des applications harmoniques).…”

“…Ce résultat était déjà partiellement démontré par Changyou Wang [25] dans le cas des applications biharmoniques extrinsèques. Toutefois dans notre travail [30] nous avons bonnes espoir d'étendre le résultat au applications biharmonique intrinsèque et par la suite de pouvoir la notion de width obtenu par Colding et Minicozzi via les surfaces minimales dans [13] aux variétés de dimension 4.…”

Compactness results for sequences of approximate biharmonic maps

“…Recently, Hornung-Moser [8], Laurain-Rivière [12], and Wang-Zheng [22] determined the energy quantization result for sequences of intrinsic biharmonic maps, approximate intrinsic and extrinsic biharmonic maps, and approximate extrinsic biharmonic maps respectively. (In fact, the result of [12] applies to a broader class of solutions to scaling invariant variational problems in dimension four. )…”