Minimal surfaces are ubiquitous in geometry but they are very hard to find. In 1982, Yau conjectured that every closed Riemannian three-manifold contains infinitely many smooth, closed, immersed minimal surfaces. Until very recently, the best result was due to Almgren (1965) and Pitts (1981), who proved the existence of one minimal surface.

In this talk I will discuss my joint work with Irie and Neves (2017) in which we settle Yau's conjecture for generic metrics by proving a much stronger property holds true: the union of all closed, smooth, embedded, minimal surfaces is dense in the manifold. Even more, we prove (joint work with Neves and Song) that there is a sequence of minimal surfaces that is equidistributed in the manifold. The main tool used in the proof of these results is the Weyl Law for the Volume Spectrum conjectured by Gromov and proved by Liokumovich, myself and Neves (2016), in combination with min-max methods.

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