Generic finiteness of minimal surfaces with bounded Morse index

TL;DR

This paper proves that in certain 3-manifolds with positive scalar curvature, the set of minimal surfaces with bounded Morse index is finite unless the manifold contains a specific embedded surface, establishing a generic finiteness result.

Contribution

It establishes a generic finiteness theorem for minimal surfaces with bounded Morse index in 3-manifolds under positive scalar curvature, including cases with strongly bumpy metrics.

Findings

Finiteness of minimal surfaces with bounded Morse index in generic 3-manifolds.

Obstructions related to embedded RP^2 and RP^3 in the manifold.

Extension of results to strongly bumpy metrics for all closed 3-manifolds.

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 RP^2. In particular, we derive a generic finiteness result whenever N does not contain a copy of RP^3 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.