Given an immersion f of the 2-sphere in a Riemannian manifold $(M,g)$ we study quadratic curvature functionals of the type: $$ \int _{f(S^2)}H^2,\quad \int _{f(S^2)}|A|^2,\quad \int _{f(S^2)}|Aº|^2, $$ where $H$ is the mean curvature, A is the second fundamental form, and $Aº$ is the tracefree second fundamental form. Minimizers, and more generally critical points of such functionals can be seen respectively as GENERALIZED minimal, totally geodesic and totally umbilical immersions. In the seminar I will review some results (obtained in collaboration with Kuwert, Rivière and Shygulla) regarding the existence and the regularity of minimizers of such functionals. An interesting observation regarding the results obtained with Rivière is that the theory of Willmore surfaces can be usesful to complete the theory of minimal surfaces (in particular in relation to the existence of canonical smooth representatives in homotopy classes, a classical program started by Sacks and Uhlembeck).

Given an immersion of a surface into the euclidean 3 space, the Willmore functional is defined as the $ L^2$ norm of the mean curvature. If we consider immersions in a Riemannian manifold there are many possible generalizations of the Willmore functional; in the seminar we will speak about these generalizations and study the existence of minimizers and critical points of the corresponding functionals under curvature conditions on the ambient manifold. The topic has links with general relativity, string theory, biology, nonlinear elasticity theory etc.