Isothermal coordinates of $W^{2,2}$ immersions. Counterexample

We study isothermal coordinates for immersions of two-dimensional manifolds into Euclidean space. We consider a class of immersions with a square integrable second fundamental form, which are also called $W^{2,2}$ immersions. In the literature, it is widely elieved that such immersions have isothermal coordinates with a uniformly bounded logarithm of the conformal factor. In this paper, we show that this is not true. We give an example of an immersion of a two-dimensional sphere into three-dimensional Euclidean space, for which the logarithm of the conformal factor is unbounded. The reason for this phenomenon is that immersions with a square-integrable second quadratic form do not admit a smooth approximations. In other words, they do not satisfy the conditions of the Toro theorem on bi-Lipschitz conformal coordinates.