Weyl Law and convergence in the classical limit for min-max nonlocal minimal surfaces

In {CFS}, the author and collaborators construct infinitely many nonlocal s-minimal hypersurfaces (via min-max methods) on any closed n-dimensional Riemannian manifold M, obtaining an analogue of Yau's conjecture for s∈(0,1). The present article proves a Weyl Law for the fractional perimeters of these hypersurfaces, and it shows their convergence as s→ 1 (for $n=3$) to smooth classical minimal surfaces. Our results are used in particular to give a novel proof of the density and equidistribution of classical minimal surfaces for generic metrics in three dimensions, showcasing nonlocal minimal surfaces also as a new approximation theory for the area functional.