Minimal surfaces with low genus in lens spaces

Abstract Given a Riemannian RP 3 RP^3 with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or there exist one minimal real projective plane together with two distinct minimal 2-spheres. Our proof is based on a variant multiplicity one theorem for the Simon–Smith min-max theory under certain equivariant settings. In particular, we show under the positive Ricci assumption that RP 3 RP^3 contains at least four distinct minimal real projective planes and four distinct minimal tori. Additionally, the number of minimal tori can be improved to five for a generic positive Ricci metric on RP 3 RP^3 by the degree method. Moreover, using the same strategy, we show that, in the lens space L ⁢ (4 ⁢ m, 2 ⁢ m ± 1) L (4m, 2m 1), m ≥ 1 m 1, with a bumpy metric or a metric of positive Ricci curvature, there either exist N ⁢ (m) N (m) distinct minimal Klein bottles, or there exist one minimal Klein bottle and three distinct minimal 2-spheres, where N ⁢ (1) = 4 N (1) =4, N ⁢ (m) = 2 N (m) =2 for m ≥ 2 m 2, and the first case happens under the positive Ricci assumption.