Generic finiteness of minimal surfaces with bounded Morse index

2015/09/23 by Carlotto, Alessandro
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1509.07101

Abstract

Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains an embedded minimal RP². In particular, we derive a generic finiteness result whenever N does not contain a copy of RP³ in its prime decomposition. We discuss the obstructions to any further generalization of such a result. When the metric g is required to be (scalar positive and) strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by B. White) the same conclusion holds true for any closed 3-manifold.

Related

• Existence of minimal surfaces of arbitrary large Morse index
• Minimal spheres of arbitrarily high Morse index
• Finite topology minimal surfaces in homogeneous three-manifolds
• Metrics without Morse index bounds
• Singular limit laminations, Morse index, and positive scalar curvature