Ricci soliton

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

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

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

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

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

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

In particular, letting

${\displaystyle \sigma (t):=1-2\lambda t}$

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

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

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

Examples of Ricci solitons

Shrinking (${\displaystyle \lambda >0}$)

• Gaussian shrinking soliton ${\displaystyle (\mathbb {R} ^{n},g_{eucl},f(x)={\frac {\lambda }{2}}|x|^{2})}$
• Shrinking round sphere ${\displaystyle S^{n},n\geq 2}$
• Shrinking round cylinder ${\displaystyle S^{n-1}\times \mathbb {R} ,n\geq 3}$
• The four dimensional FIK shrinker ^[1]
• The four dimensional BCCD shrinker ^[2]
• Compact gradient Kahler-Ricci shrinkers ^[3][4][5]
• Einstein manifolds of positive scalar curvature

Steady (${\displaystyle \lambda =0}$)

• The 2d cigar soliton (a.k.a. Witten's black hole) ${\displaystyle \left(\mathbb {R} ^{2},g={\frac {dx^{2}+dy^{2}}{1+x^{2}+y^{2}}},V=-2(x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}})\right)}$
• The 3d rotationally symmetric Bryant soliton and its generalization to higher dimensions ^[6]
• Ricci flat manifolds

Expanding (${\displaystyle \lambda <0}$)

• Expanding Kahler-Ricci solitons on the complex line bundles ${\displaystyle O(-k),k>n}$ over ${\displaystyle \mathbb {C} P^{n},n\geq 1}$.^[1]
• 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.^[7] 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

References

• Cao, Huai-Dong (2010). "Recent Progress on Ricci solitons". arXiv:0908.2006 [math.DG].
• Topping, Peter (2006), Lectures on the Ricci flow, Cambridge University Press, ISBN 978-0521689472

This page was last edited on 2 January 2024, at 20:27