Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I

Abstract

We construct closed embedded minimal surfaces in the round three-sphere $\mathbb{S}^3 (1)$, resembling two parallel copies of the equatorial two-sphere $\mathbb{S}^2_{\mathrm{eq}}$, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of $\mathbb{S}^2_{\mathrm{eq}}$, or along the equatorial circle and the poles. To carry out these constructions we refine and reorganize the doubling methodology in ways which we expect to apply also to further constructions. In particular, we introduce what we call “linearized doubling”, which is an intermediate step where singular solutions to the linearized equation are constructed subject to appropriate linear and nonlinear conditions. Linearized doubling provides a systematic approach for dealing with the obstructions involved and also understanding in detail the regions further away from the catenoidal bridges.