Deformation rigidity for Z/2 eigensections

TL;DR

This paper proves a deformation rigidity result for certain critical Z/2 eigensections on S^2, showing small deformations are equivalent to rotations, extending known symmetry-based rigidity phenomena.

Contribution

It establishes a new deformation rigidity theorem for critical eigensections associated with flat line bundles on S^2, generalizing previous symmetric case results.

Findings

Every minimal non-degenerate critical eigensection is deformation rigid.

Small deformations preserving the critical eigensection are induced by SO(3)-rotations.

The result extends rigidity phenomena beyond symmetric examples.

Abstract

We prove a rigidity result for certain critical Z/2 eigensections of the Laplacian on S^2 associated to a flat real line bundle determined by a branch-point configuration. More precisely, we show that every minimal non-degenerate critical eigensection is deformation rigid: any sufficiently small deformation of the configuration that still admits a critical eigensection must come from an SO(3)-rotation. This generalizes the rigidity phenomenon previously discovered in symmetric examples of Taubes-Wu.