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Abstract

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. In recent work, 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 Law for the fractional perimeters of these hypersurfaces. The rest -- and main part -- of the article is devoted to obtaining effective estimates (as $s\to 1$) for min-max $s$-minimal surfaces in closed three-dimensional manifolds, including classical perimeter and mean curvature estimates, and eventually establishing their convergence to smooth, classical minimal surfaces. We recover in particular recent results on generic density and equidistribution of minimal surfaces, which are a strong form of Yau's conjecture in this setting.