Geometric rigidity for sequences of W2,2 conformal immersions

We analyse sequences of discs conformally immersed in R-n with energy integral(D) vertical bar A(k)vertical bar(2) <= gamma(n), where gamma(n) = 8 pi if n = 3 and gamma(n) = 4 pi when n >= 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 = 3 or Chen's minimal graph if n >= 4. In the papers, (Kuwert and Li, Comm Anal Geom 20(2), 313-340, 2012; Riviere, Adv Calculus Variations 6(1), 1-31, 2013) it was shown that if a sequence of immersed tori diverges in moduli space then lim inf(k ->infinity) W(f(k)) >= 8 pi. We apply the above analysis to show that in R-3 if the sequence diverges so that lim(k ->infinity) W(f(k)) = 8 pi then there exists a sequence of Mobius transforms sigma(k) such that sigma(k) circle f(k) converges weakly to a catenoid.