Ricci soliton
In differential geometry, a complete Riemannian manifold is called a Ricci soliton if, and only if, there exists a smooth vector field such that
for some constant . Here is the Ricci curvature tensor and represents the Lie derivative. If there exists a function such that we call a gradient Ricci soliton and the soliton equation becomes
Note that when or the above equations reduce to the Einstein equation. For this reason Ricci solitons are a generalization of Einstein manifolds.
Self-similar solutions to Ricci flow
A Ricci soliton yields a self-similar solution to the Ricci flow equation
In particular, letting
and integrating the time-dependent vector field to give a family of diffeormorphisms , with the identity, yields a Ricci flow solution by taking
In this expression refers to the pullback of the metric by the diffeomorphism . Therefore, up to diffeomorphism and depending on the sign of , a Ricci soliton homothetically shrinks, remains steady or expands under Ricci flow.
Examples of Ricci solitons
Shrinking ()
Steady ()
- The 2d cigar soliton (a.k.a Witten's black hole)
- The 3d rotationally symmetric Bryant soliton and its generalization to higher dimensions [5]
- A family of non-Kahler steady Ricci solitons on the complex line bundles over . These solitons are asymptotic to the -dimensional Bryant soliton quotiented by . [6]
- Ricci flat manifolds
Expanding ()
- Expanding Kahler-Ricci solitons on the complex line bundles over . [7]
- Einstein manifolds of negative scalar curvature
Singularity models in Ricci flow
Shrinking and steady Ricci solitons are fundamental objects in the study of Ricci flow as they appear as blow-up limits of singularities. In particular, it is known that all Type I singularities are modeled on non-collapsed gradient shrinking Ricci solitons[8]. Type II singularities are expected to be modeled on steady Ricci solitons in general, however to date this has not been proven, even though all known examples are.
Notes
- Mikhail Feldman, Tom Ilmanen, and Dan Knopf, "Rotationally Symmetric Shrinking and Expanding Gradient Kähler-Ricci Solitons", J. Differential Geom. Volume 65, Number 2 (2003), 169-209.
- Koiso, N., "On rotationally symmmetric Hamilton’s equation for Kahler-Einstein metrics", Recent Topics in Diff. Anal. Geom., Adv. Studies Pure Math.,18-I,Academic Press, Boston, MA (1990), 327–337
- Cao, H.-D., Existence of gradient K¨ahler-Ricci solitons, Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, MA, (1996) 1-16
- Wang, X. J. and Zhu, X. H., Ka¨hler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188 (2004), no. 1, 87–103.
- R.L. Bryant, "Ricci flow solitons in dimension three with SO(3)-symmetries", available at
- Appleton, Alexander (2017). "A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four". arXiv:1708.00161.
- Mikhail Feldman, Tom Ilmanen, and Dan Knopf, "Rotationally Symmetric Shrinking and Expanding Gradient Kähler-Ricci Solitons", J. Differential Geom. Volume 65, Number 2 (2003), 169-209.
- J. Enders, R. Mueller, P. Topping, "On Type I Singularities in Ricci flow", Communications in Analysis and Geometry, 19 (2011) 905–922
References
- Cao, Huai-Dong (2010). "Recent Progress on Ricci solitons". arXiv:0908.2006.
- Topping, Peter (2006), Lectures on the Ricci flow, Cambridge University Press, ISBN 978-0521689472