Nonorientable Minimal Surfaces Embedded in the Round $4$-sphere

Abstract

We show that every closed, nonorientable surface can be minimally embedded in ${\mathbb{S}}^4$, providing in particular the first known examples of embedded, nonorientable minimal surfaces in ${\mathbb{S}}^4$ with negative Euler characteristic. All of the surfaces we obtain have area below $8\pi$, which has applications to the existence of nonorientable surfaces minimizing the Willmore functional with prescribed topology in ${\mathbb{R}}^n$ for $n\ge 4$. Moreover, the number of geometrically distinct embeddings of each nonorientable genus is shown to grow at least exponentially with respect to the genus. Among these surfaces, we identify a distinguished, highly symmetric family which converges in the large-genus limit to a union of four half-spheres, meeting along a great circle, whose poles are vertices of a regular tetrahedron. Rescalings of this family converge to a new singly-periodic nonorientable minimal surface in ${\mathbb{R}}^4$, which seems to provide the first example of a complete, embedded, nonorientable minimal surface in ${\mathbb{R}}^4$ without continuous symmetry group. The proofs further develop the equivariant eigenvalue optimization methodology from our earlier work, applied to carefully chosen families of symmetry groups acting on nonorientable surfaces. Interestingly, we also find natural pairs of surfaces and group actions for which there is no metric maximizing the first normalized eigenvalue of the Laplacian.