We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold (M n , g), 3 ≤ n ≤ 7, can degenerate. Loosely speaking, our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces, up to controlled errors. Several compactness/finiteness theorems follow from our local picture.

We prove that any manifold diffeomorphic to S 3 and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal twosphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three manifolds. We apply our methods to solve a problem posed by S.T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered about the origin in R 4 . Finally, considering the example of degenerating ellipsoids we show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp.

We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such manifolds can be isotoped to be minimal and iii) the "doublings" of the Clifford torus by Kapouleas-Yang can be constructed variationally by an equivariant min-max procedure. In higher dimensions we also prove that the width of manifolds with positive Ricci curvature is achieved by an index 1 orientable minimal hypersurface.

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Entropy is a natural geometric quantity measuring the complexity of a surface embedded in R 3 . For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface is at least that of the self-shrinking two-sphere. We prove this conjecture for all closed embedded 2-spheres. Assuming a conjectural Morse index bound (announced recently by Marques-Neves), we can improve the result to apply to all closed embedded surfaces that are not tori. Our results can be thought of as the parabolic analog to the Willmore conjecture and our argument is analogous in many ways to that of Marques-Neves on the Willmore problem. The main tool is the min-max theory applied to the Gaussian area functional in R 3 which we also establish. To any closed surface in R 3 we associate a four parameter canonical family of surfaces and run a min-max procedure. The key step is ruling out the min-max sequence approaching a self-shrinking plane, and we accomplish this with a degree argument. To establish the min-max theory for R 3 with Gaussian weight, the crucial ingredient is a tightening map that decreases the mass of non-stationary varifolds (with respect to the Gaussian metric of R 3 ) in a continuous manner. Surprisingly, the resolution of the Willmore Conjecture by F.C. Marques and A. Neves [MN12] hinges on asking and answering the following question:In round S 3 , what is the non-equatorial embedded minimal surface with smallest area?Using min-max theory, [MN12] proved that the answer is the Clifford torus. This swiftly led to a proof of the Willmore conjecture. In this paper, we address the analogous question for singularity models for the mean curvature flow and cast the question in terms of min-max *