Daniel Ketover

Minimal hypersurfaces with bounded index

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

Genus bounds for min-max minimal surfaces

(M^{n},g)

Entropy of closed surfaces and min-max theory

(Mn,g),

Equivariant min-max theory

3\le n\le 7

Minimal two-spheres in three-spheres

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.

Degeneration of Min-Max Sequences in 3-manifolds

The long standing classification problem in the theory of Heegaard splittings of 3-manifolds is to exhibit for each closed 3-manifold a complete list, without duplication, of all its irreducible Heegaard surfaces, up to isotopy. We solve this problem for non Haken hyperbolic 3-manifolds.