Minimal surfaces with low genus in lens spaces

TL;DR

This paper investigates the existence and multiplicity of minimal surfaces with low genus in lens spaces and real projective spaces under certain curvature conditions, using min-max theory and equivariant methods.

Contribution

It establishes new multiplicity results for minimal surfaces in lens spaces and real projective spaces with positive Ricci curvature or bumpy metrics, employing a variant of the Simon-Smith min-max theory.

Findings

Existence of at least four distinct minimal real projective planes in $\

In $\

In lens spaces $L(4m,2m\pm 1)$, either multiple minimal Klein bottles or a Klein bottle plus three minimal 2-spheres exist.

Abstract

Given a Riemannian $\mathbb{RP}^3$ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or 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 $\mathbb{RP}^3$ 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 $\mathbb{RP}^3$ by the degree method. Moreover, using the same strategy, we show that in the lens space $L(4m,2m\pm 1)$, $m\geq 1$, 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 $m\geq 2$, and the first case happens under the positive Ricci assumption.