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

We study nonlocal minimal surfaces as a new approximation theory for the area functional, and more specifically in the context of Yau’s conjecture on the existence of minimal surfaces in closed three-dimensional manifolds. This programme offers an alternative to the Almgren–Pitts and Allen–Cahn approaches, with advantageous features both from the existence and regularity viewpoints. We build on recent work in which the author and collaborators constructed infinitely many nonlocal $s$-minimal hypersurfaces (via min-max methods) on any closed $n$-dimensional Riemannian manifold $M$, establishing a full analogue of Yau’s conjecture for $s\in (0,1)$. The present article first proves a Weyl-type Law for the fractional perimeters of these hypersurfaces. The rest – and main part – of the article is devoted to obtaining uniform estimates (in the classical limit $s\to 1$) for min-max $s$-minimal surfaces in closed three-manifolds, eventually establishing their convergence to smooth, classical minimal surfaces. We recover in particular recent results on existence, generic density and equidistribution of minimal surfaces, which are a strong form of Yau’s conjecture in this setting.