Regularity of nonlocal minimal cones in dimension 2
  We show that the only nonlocal $s$-minimal cones in $\R^2$ are the trivial ones for all $s \in (0,1)$. As a consequence we obtain that the singular set of a nonlocal minimal surface has at most $n-3$ Hausdorff dimension. 
Ovidiu Savin and Enrico Valdinoci
math.AP