Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three

Abstract

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.

17 pages

References in corpus (1)

• Gaussian densities and stability for some Ricci solitons

Cited by in corpus (3)

• Classification of expanding and steady Ricci solitons with integral curvature decay
• On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor
• Martin compactification of a complete surface with negative curvature