Minimal surfaces with low genus in lens spaces

Given a Riemannian RP³ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or 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 RP³ 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 RP³ by the degree method. Moreover, using the same strategy, we show that in the lens space L (4m, 2m 1), m 1, with a bumpy metric or a metric of positive Ricci curvature, there either exist N (m) numbers of distinct minimal Klein bottles, or exist one minimal Klein bottle and three distinct minimal 2-spheres, where N (1) =4, N (m) =2 for m 2, and the first case happens under the positive Ricci assumption.