In geometric analysis both Sobolev functions on smooth Riemannian manifolds and models of possibly singular surfaces, such as varifolds and currents which generalise the concept of submanifold, are tools of basic importance. In this talk a theory of Sobolev functions on varifolds is presented which allows to combine these two tools.

We consider immersed Klein bottles in Euclidean four-space with low Willmore energy. It turns out that there are three distinct homotopy classes of immersions that are regularly homotopic to an embedding. One is characterized by the property that the immersions have Euler normal number zero. This class contains embedded Klein bottles with Willmore energy strictly less than $8\pi$. We prove that the infimum of the Willmore energy among all immersed Klein bottles in euclidean four-space is attained by a smooth embedding that is in this first homotopy class. In the other two homotopy classes we have that the Willmore energy is bounded from below by $8\pi$. We classify all immersed Klein bottles with Willmore energy $8\pi$ and Euler normal number $+4$ or $-4$. These surfaces are minimizers of the second or the third homotopy class.