Genus two embedded minimal surfaces in ${\mathbb{S}}^3$ with dihedral symmetry

Abstract

We prove that the Lawson surface ${\xi }_{2,1}$ is the unique closed embedded minimal surface of genus $2$ in ${\mathbb{S}}^3$ whose isometry group contains the dihedral group $D_4$ generated by two reflections across orthogonal totally geodesic two-spheres and a half-turn about a great circle. This weakens the full-symmetry hypotheses in previous characterizations of Lawson surfaces and leads to a substantially different geometric problem.

Withdrawn by the authors because Lemma 4.2 contains a gap: the moduli space of the relevant R_2-symmetric right-angled geodesic hexagons has an additional parameter omitted from the stated classification. This affects the reduction of the closing problem, so the main theorem is not proved by the present manuscript. A corrected version is in preparation