Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three

The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton $N^2\times \mathbb{R}$, the product of Hamilton's cigar soliton $N^2$ and the real line $\mathbb{R}$ with the product metric. R. Hamilton has conjectured that there should exist a family of collapsed positively curved three-dimensional complete gradient steady solitons, with $\mathsf{S}^1$-symmetry, connecting the 3D cigar soliton. In this paper, we make the first initial progress and prove that the infinitesimal deformation at the 3D cigar soliton is non-essential. In Appendix A, we show that the 3D cigar soliton is the unique complete nonflat gradient steady Ricci soliton in dimension three that admits two commuting Killing vector fields.