Equivariant min-max theory

Abstract

We develop an equivariant min-max theory as proposed by Pitts-Rubinstein in 1988 and then show that it can produce many of the known minimal surfaces in ${\mathbb{S}}^3$ up to genus and symmetry group. We also produce several new infinite families of minimal surfaces in ${\mathbb{S}}^3$ proposed by Pitts-Rubinstein. These examples are doublings and desingularizations of stationary integral varifolds in ${\mathbb{S}}^3$.

Cited by in corpus (10)

• Free boundary minimal surfaces of unbounded genus
• Free boundary minimal surfaces in the unit ball : recent advances and open questions
• Free boundary minimal surfaces with connected boundary and arbitrary genus
• Free boundary minimal surfaces of any topological type in Euclidean balls via shape optimization
• Min-max theory for constant mean curvature hypersurfaces
• Self-shrinking Platonic solids
• Min-max theory for $G$-invariant minimal hypersurfaces
• Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces
• Monotonicity results for the first Steklov eigenvalue on compact surfaces
• On the intersection of minimal hypersurfaces of $S^k$