In this talk we describe recent joint work with R. Schätzle in which we extend a rigidity result of DeLellis and Müller to arbitrary codimensions. More precisely, we show that every immersion of a two-dimensional surface into $\mathbb{R}^n$, whose tracefree second fundamental form is small in $L^2$ has to be close to a round sphere in the $W^{2,2}$-norm.

For the study of asymptotically flat manifolds in mathematical general relativity, surfaces of constant mean curvature (CMC) haven proved to be a useful tool. In 1996, Huisken-Yau showed that any asymptotically flat Riemannian manifold can be uniquely foliated by closed CMC surfaces. Furthermore, they interpreted this foliation as a definition of the center of mass. We prove that this definition is compatible with the definition of linear momentum by Arnowitt-Deser-Misner: The evolution of this foliation (asymptotically) corresponds to a translation with direction given by the quotient of (ADM) linear momentum and mass - equivalent to the center of mass in Newtonian systems.