Geometric rigidity for sequences of $$ W^{2,2}$$ conformal immersions

Abstract

We analyse sequences of discs conformally immersed in RnRn with energy ∫D|Ak|2≤γn∫D|Ak|2≤γn , where γn=8πγn=8π if n=3n=3 and γn=4πγn=4π when n≥4n≥4 . We show that if such sequences do not weakly converge to a conformal immersion, then by a sequence of dilations we obtain a complete minimal surface with bounded total curvature, either Enneper’s minimal surface if n=3n=3 or Chen’s minimal graph if n≥4n≥4 . In the papers, (Kuwert and Li, Comm Anal Geom 20(2), 313–340, 2012; Rivière, Adv Calculus Variations 6(1), 1–31, 2013) it was shown that if a sequence of immersed tori diverges in moduli space then lim infk→∞W(fk)≥8πlim infk→∞W(fk)≥8π . We apply the above analysis to show that in R3R3 if the sequence diverges so that limk→∞W(fk)=8πlimk→∞W(fk)=8π then there exists a sequence of Möbius transforms σkσk such that σk∘fkσk∘fk converges weakly to a catenoid.