Compactness and semicontinuity theorems for minimizing sequences in geometric measure theory are usually achieved either exploiting some additional homological structure or suitably modifying the sequence. In some recent joint works with Ghiraldin, Maggi, de Philippis and De Rosa we have shown that it is possible to achieve compactness and semicontinuity directly. The methods are flexibile enough to handle higher codimension and anisotropic functionals and become particularly simple in codimension one for the area functional.

We discuss the existence of surfaces of type $S^2$ minimizing the Willmore functional with prescribed isoperimetric ratio, and some asymptotics as the ratio goes to zero.