Communications in Analysis and Geometry

Volume 26 (2018)

Number 3

Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three

Pages: 505 – 529



Huai-Dong Cao (Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania, U.S.A.)

Chenxu He (Department of Mathematics, University of Oklahoma, Norman, Ok., U.S.A.; and Department of Mathematics, University of California at Riverside)


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