On the convergence behavior of conformal immersion sequence from cylinders

Abstract

In this paper, we are concerned with the convergence behavior of a sequence of conformal immersions {fn} from long cylinders Pn with \(\int_{P_n } {|A_{f_n } |^2 } + \mu (f_n (P_n )) < \Lambda \). We show that if {fn} does not converge to a point, the total Gauss curvatures and the measures of the images of {fn} will not lose on the necks and each neck consists of a point.