On the convergence behavior of conformal immersion sequence from cylinders

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