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# Communications in Analysis and Geometry

## Volume 26 (2018)

### Number 3

### Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three

Pages: 505 – 529

DOI: https://dx.doi.org/10.4310/CAG.2018.v26.n3.a2

#### Authors

#### 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 R. Hamilton’s cigar soliton $N^2$ and the real line $\mathbb{R}$ with the product metric. 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. Moreover, 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.

#### 2010 Mathematics Subject Classification

53C10, 53C21, 53C24, 53C25, 53C44

The research of the first author was partially supported by NSF Grant DMS-0909581.

Received 11 January 2015

Published 27 July 2018