We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow with one additional dimension.
We present joint work with Mariel Sáez.

Given an immersed surface $\Sigma$ in the Euclidean space, the Willmore functional is defined as the $L^2$ norm of the mean curvature. If the surface is immeresd in a Riemannian manifold there are several natural generalizations: the $L^2$ curvature functionals. The topic as links with general realtivity (the Hawking mass), Biology (Hellfrich energy), string theory (Polyakov extrinsic action), etc. After introducing the various functionals and discussing their properties, I will give a panoramic view of the different techniques and the recent results obtained in various collaboration with Kuwert, Rivière and Schygulla.