The Existence of Minimal Immersions of 2-Spheres

Dirac-harmonic maps from degenerating spin surfaces I: the Neveu–Schwarz case

“…Hence, by the standard covering argument (c.f. [17] and Theorem 2.3 in [18]), there exist finitely many points {x 1 , x 2 , . .…”

“…By employing the technique used by Sacks and Uhlenbeck in [17], we have the following lemma, which is part of Lemma 3.3 in [20].…”

Singularities of harmonic maps

“…A result of D. Preiss [90] (with a simple proof in [82] of this application) gives the rectifiability. If Z is nonempty, then an induction argument and a rescaling argument as in [95] produces a nonconstant harmonic map from S 2 to N . This is only possible if π 2 (N ) = 0.…”

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.…”