In this talk I will present the following regularity results in minimal surface theory: the singular points of a mass minimizing three dimensional cone in the Euclidean space are contained in at most finitely many half lines (joint with De Lellis and Spolaor).  

In this talk, we prove that the Min-Oo's conjecture holds if we consider a compact connected locally conformally flat manifold with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality, and the mean curvature of the boundary is controled bellow by the mean curvature of a geodesic ball in the standard unit-sphere. This is a joint work with E. Barbosa and M.P. Cavalcante.