Neck analysis for biharmonic maps

Abstract

In this paper, we study the blow up of a sequence of (both extrinsic and intrinsic) biharmonic maps in dimension four with bounded energy and show that there is no neck in this process. Moreover, we apply the method to provide new proofs to the removable singularity theorem and energy identity theorem of biharmonic maps.1991 Mathematics Subject Classification. 58E20(35J50 53C43).

Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with applications to the Dirac-harmonic heat flow

“…Then using the (a) and (b) of Theorem 3.3, by iterating, we obtain 19) which implies (3.17) immediately. If J = ∅, without loss of generality, we may assume…”

“…The idea is from Qing-Tian's paper [22], which used a special case of the three circle theorem due to Simon [25] to show that the tangential energy of the sequence in the neck region decays exponentially. The second author in cooperation with H.Yin has extended this idea to some fourth order equations, see [19,20]. Let us first state the three circle theorem for harmonic functions (see [18,22,25]).…”

Neck analysis of extrinsic polyharmonic maps

“…The proof of Theorem 1.1 follows the general framework in [16], which was in turn motivated by the original idea of [22] and [20]. As is well known by now, the main challenge of the neck analysis is that the length of the neck, or equivalently, the ratio δ/(λ i R), is out of control.…”

“…The special case m = 2 is proved in [16]. However, the proof there is by brutal force and hence not instructive.…”

“…While many arguments in [16] generalizes without much effort to the case of polyharmonic maps, the two most important ingredients in the argument, namely, the three-circle lemma and the Pohozaev type argument needs completely different proofs. It turns out that the generalization from m = 2 (the biharmonic map case) to m > 2 exploits some fine structure of the operator ∆ m .…”