Geometry and Analysis Seminar

Tobias Lamm (Insti. fur Math. Goethe-Univ., Frankfurt)
“Rigidity results for conformal immersions in $\ R^n$”

By a classical result of Codazzi every closed, totally umbilic surface is a round sphere. De Lellis and Mller proved a rigidity statement corresponding to this result. More precisely, they showed that for every closed surface in R3, whose traceless second fundamental form is “small” in L2, there exists a conformal parametrization whose distance to a standard parametrization of a round sphere is small in W2;2. In a recent joint work with H. Nguyen (War- wick) we were able to extend this result to arbitrary codimensions. Moreover, we obtained related rigidity results for inversions of the catenoid and Enneper`s minimal surface. In my talk I will review the analytic preliminaries (i.e. the results of Mller-Sverak and Kuwert-Li) and I will sketch the proof of the above mentioned results.