Abstract: One of the oldest problem in geometric analysis and in the calculus of variations consists in finding a surface of leat area surface spannend by a given smooth curve $\gamma$. The approach of Douglas and Rad ensures, for each prescribed genus, the existence of a surface minimizing the area among all the ones with smaller genus. On the other hand, after the seminal work of Federer and Fleming, it is always possible to find a mass minimizing current spanned by $\gamma$. When the Federer fleming solution is smooth (up to the boundary) , it naturally coincides with the Dougals Rado solution. In co-dimension one, thanks to the seminal work of De Giorgi and Hardt-Simon, this is indeed the case. In higher co-dimension the question is not yet completely understood and new phenomena might appear. In this talk, after surveying the known results, I will present the construction of a smooth curve in a 4 dimensional Riemannian manifold which spans an area minimizing surface with infinite genus. This is based on a joint work with C. De Lellis and J. Hirsch.
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