Index one minimal surfaces in spherical space forms

Abstract

We prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two. This confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.

Index one minimal surfaces in positively curved $3$-manifolds

“…In the case of constant curvature, Viana [18] proved that there is a positive c such that, if Σ ⊂ M 3 is a closed 2-sided embedded minimal surface of index 1 and genus g = 3 in a spherical space form M , then the order of the fundamental group of M is bounded from above, |π 1 (M )| ≤ c.…”

“…As a consequence of the property of RP 3 and the result of [18], it follows that the question about the behaviour of index one minimal surfaces could be stated as follows:…”