Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications

Abstract

In the present work we establish a quantization result for the angular part of the energy of solutions to elliptic linear systems of Schrödinger type with antisymmetric potentials in two dimension. This quantization is a consequence of uniform Lorentz-Wente type estimates in degenerating annuli. We derive from this angular quantization the full energy quantization for general critical points to functionals which are conformally invariant or also for pseudo-holomorphic curves on degenerating Riemann surfaces.A.M.S. Classification :

Energy quantization for biharmonic maps

“…It can also be compared with the corresponding result established for second order problems in [11,Theorem 3.2]. Theorem 3.3.…”

“…The method we use goes first through the proof of an angular energy quantization result 2 for sequences of solutions to the general critical 4th order elliptic system with antisymmetric potentials introduced by Lamm and Rivière [10]. We follow in fact the approach that we originally introduced in [11] for second order problems. We have good reasons to think that the method could further be extended for proving a general energy quantization result for polyharmonic maps in critical dimension (see the "-regularity for polyharmonic maps in [4] and [3] for the general case, see also [14]).…”

Energy quantization of Willmore surfaces at the boundary of the moduli space

“…After deriving some equations relating H together with S and R, we prove that an appropriate combination of those quantities is bounded in L 2,1 . This will be done using some additional conservation laws combined with results from integrability by compensation theory from [21] or proved in the appendix. Finally combining those estimates with some algebraic fact, we prove that H converges to 0 in L 2 norm in the neck region.…”

“…L 2 −bounds on ∇S and ∇ R . Lemma 4.5 (lemma 2.4 [21]). Let a, b ∈ L 2 (B 1 ), 0 < ε < 1 4 , assume that ∇a ∈ L 2,∞ (B 1 ) and that ∇b ∈…”

“…Since both S and R are defined up to an additive constant, we can assume that S r = 0 and R r = 0, which gives |S ρ | ≤ C ∀ρ ∈ (r, 1). Now let us recall the following lemma from [21].…”