Ricci soliton

In differential geometry, a complete Riemannian manifold $(M,g)$ is called a Ricci soliton if, and only if, there exists a smooth vector field $V$ such that

\operatorname {Ric} (g)=\lambda \,g-{\frac {1}{2}}{\mathcal {L}}_{V}g,

for some constant $\lambda \in \mathbb {R}$ . Here $\operatorname {Ric}$ is the Ricci curvature tensor and ${\mathcal {L}}$ represents the Lie derivative. If there exists a function $f:M\rightarrow \mathbb {R}$ such that $V=\nabla f$ we call $(M,g)$ a gradient Ricci soliton and the soliton equation becomes

\operatorname {Ric} (g)+\nabla ^{2}f=\lambda \,g.

Note that when $V=0$ or $f=0$ 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 $(M,g_{0})$ yields a self-similar solution to the Ricci flow equation

\partial _{t}g_{t}=-2\operatorname {Ric} (g_{t}).

In particular, letting

\sigma (t):=1-2\lambda t

and integrating the time-dependent vector field $X(t):={\frac {1}{\sigma (t)}}V$ to give a family of diffeormorphisms $\Psi _{t}$ , with $\Psi _{0}$ the identity, yields a Ricci flow solution $(M,g_{t})$ by taking

g_{t}=\sigma (t)\Psi _{t}^{\ast }(g_{0}).

In this expression $\Psi _{t}^{\ast }(g_{0})$ refers to the pullback of the metric $g_{0}$ by the diffeomorphism $\Psi _{t}$ . Therefore, up to diffeomorphism and depending on the sign of $\lambda$ , a Ricci soliton homothetically shrinks, remains steady or expands under Ricci flow.

Examples of Ricci solitons

Shrinking ( $\lambda >0$ )

Gaussian shrinking soliton $(\mathbb {R} ^{n},g_{eucl},f(x)={\frac {\lambda }{2}}|x|^{2})$
Shrinking round sphere $S^{n},n\geq 2$
Shrinking round cylinder $S^{n-1}\times \mathbb {R} ,n\geq 3$
The four dimensional FIK shrinker [1]
Compact gradient Kahler-Ricci shrinkers [2] [3] [4]
Einstein manifolds of positive scalar curvature

Steady ( $\lambda =0$ )

The 2d cigar soliton (a.k.a Witten's black hole) $\left(\mathbb {R} ^{2},g={\frac {dx^{2}+dy^{2}}{1+x^{2}+y^{2}}},V=-2(x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}})\right)$
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 $O(-k),k>n,$ over $\mathbb {C} P^{n},n\geq 1$ . These solitons are asymptotic to the $4n$ -dimensional Bryant soliton quotiented by $\mathbb {Z} _{k}$ . [6]
Ricci flat manifolds

Expanding ( $\lambda <0$ )

Expanding Kahler-Ricci solitons on the complex line bundles $O(-k),k>n$ over $\mathbb {C} P^{n},n\geq 1$ . [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

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[1] 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.

[2] 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

[3] 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

[4] 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.

[5] R.L. Bryant, "Ricci flow solitons in dimension three with SO(3)-symmetries", available at

[6] Appleton, Alexander (2017). "A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four". arXiv:1708.00161.

[7] 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.

[8] J. Enders, R. Mueller, P. Topping, "On Type I Singularities in Ricci flow", Communications in Analysis and Geometry, 19 (2011) 905–922