We will discuss existence of minimal disks into a Riemannian manifold having a boundary lying on a specified embedded submanifold and that meet the submanifold orthogonally along the boundary. A general existence result has been obtained by A. Fraser. Her construction was inspired by Sacks-Uhlenbeck construction of minimal $2$-spheres : the existence is obtained by a limit procedure for a perturbed energy functional whose critical points are called $\alpha$-harmonic maps. We will explain how it is possible to adapt ideas of Colding-Minicozzi. These ideas go back to the replacement method of Birkhoff for the existence of geodesics. This approach gives general energy identities that include bubbles. This is a joint work with P. Laurain.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/3050