In 1990, Hoffman and Meeks proved the strong halfspace theorem for minimal surfaces, that is any connected, proper, possibly branched minimal surface in three-dimensional Euclidean space that is contained in a halfspace must be a plane. In a joint work with Matteo Cozzi, we prove the strong halfspace theorem for nonlocal minimal surfaces. Interestingly, our result holds for hypersurfaces of any dimension, in direct contrast to the classical case which does not hold for hypersurfaces of dimension three or higher.
In the first half of my talk, I will introduce and motivate nonlocal minimal surfaces and try to highlight some interesting similarities/differences with their classical counterparts. In the second half, I will discuss the nonlocal halfspace theorem including some ideas of the proof.