In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in R 3 is at least 2π 2 . We prove this conjecture using the min-max theory of minimal surfaces.

Abstract. In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min-max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces.In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.

In this paper, we prove compactness for the full set of solutions to the Yamabe Problem if n ≤ 24. After proving sharp pointwise estimates at a blowup point, we prove the Weyl Vanishing Theorem in those dimensions, and reduce the compactness question to showing positivity of a quadratic form. We also show that this quadratic form has negative eigenvalues if n ≥ 25.

The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the minmax minimal hypersurface.We advance the theory further and prove the first general Morse index bounds for minimal hypersurfaces produced by it. We also settle the multiplicity problem for the classical case of one-parameter sweepouts.

Given M a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum {ωp(M )} p∈N satisfies a Weyl law that was conjectured by Gromov.