Isothermal Coordinates of $W^{2,2}$ Immersions: A Counterexample

We study isothermal coordinates for the immersions of two-dimensional manifolds into Euclidean space and consider a class of immersions with square integrable second fundamental form, which are also called $W^{2,2}$ immersions. It is a widespread statement in the literature that such immersions have isothermal coordinates with uniformly bounded logarithm of the conformal factor. We show that this is not the case: We give an example of an immersion of the two-dimensional sphere into three-dimensional Euclidean space for which the logarithm of the conformal factor is unbounded. The reason is that immersions with square integrable second fundamental form do not admit a smooth approximation. In other words, they do not satisfy the hypotheses of the Toro theorem on bi-Lipschitz conformal coordinates.