Sequences of Smooth Global Isothermic Immersions

TL;DR

This paper investigates the convergence behavior of sequences of smooth global isothermic immersions of closed surfaces with bounded total curvature, revealing conditions under which they converge weakly and describing potential loss of compactness.

Contribution

It provides a detailed analysis of the weak convergence and compactness properties of isothermic immersions, including the description of defect measures and exceptional directions.

Findings

Weak convergence in W^{2,2} away from finitely many points.

Characterization of defect measures transported along holomorphic quadratic forms.

Examples demonstrating possible loss of compactness along exceptional directions.

Abstract

In the present work we study the behavior of sequences of smooth global isothermic immersions of a given closed surface and having a uniformly bounded total curvature. We prove that, if the conformal class of this sequence is bounded in the Moduli space of the surface, it weakly converges in W^{2,2} away from finitely many points, modulo extraction of a subsequence, to a possibly branched weak isothermic immersion of this surface. Moreover, if this limit happens to be smooth away from the branched points, we give an optimal description of the possible loss of strong compactness of such a subsequence by proving that, beside possibly finitely many atomic concentrations, the defect measure associated to the L^2 norm of the second fundamental form is "transported" along exceptional directions given by some holomorphic quadratic forms associated the limiting surface. We give examples where these losses of compactness, invariant along such exceptional directions, eventually happen.