For each n at least 2, an equivariant min-max procedure yields a new embedded minimal surface Gamma_n in S^3 with genus 2n or 2n-2, area just above the sphere's, full symmetry group G_n for n at least 4, and Morse index at least 4n-1.
Genus one critical catenoid
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We use variational methods to construct a free boundary minimal surface in the three-dimensional unit ball with genus one, two boundary components and prismatic symmetry. Key ingredients are an extension of the equivariant min-max theory to include orientation-reversing isometries and the discovery of a nontrivial two-parameter sweepout.
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A new family of minimal surfaces of even genus in the three-dimensional sphere
For each n at least 2, an equivariant min-max procedure yields a new embedded minimal surface Gamma_n in S^3 with genus 2n or 2n-2, area just above the sphere's, full symmetry group G_n for n at least 4, and Morse index at least 4n-1.