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 (discovered by M. Feldman, T. Ilmanen, D. Knopf) ^[1]
• The four dimensional BCCD shrinker (discovered by Richard Bamler, Charles Cifarelli, Ronan Conlon, and Alix Deruelle)^[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.

Soliton Identities

Taking the trace of the Ricci soliton equation ${\displaystyle Ric+{\frac {1}{2}}{\mathcal {L}}_{V}g=\lambda g}$ gives

${\displaystyle S+\operatorname {div} V=\lambda n}$,

1

where ${\displaystyle S}$ is the scalar curvature and ${\displaystyle n=\operatorname {dim} M}$. By taking the divergence of the Ricci soliton equation and invoking the contracted Bianchi identities and (1), it follows that

${\displaystyle \Delta V+\operatorname {Ric} \circ V=0\quad {\text{or equivalently, in components,}}\quad \Delta V_{i}+\operatorname {Ric} _{ij}V^{j}=0.}$

For gradient Ricci solitons ${\displaystyle V=\nabla f}$, similar arguments show

${\displaystyle S+\Delta f=\lambda n\quad {\text{and}}\quad \nabla (S+|\nabla f|^{2}-2\lambda f)=0.}$

In particular, if ${\displaystyle M}$ is connected, then there exists a constant ${\displaystyle C}$ such that

${\displaystyle S+|\nabla f|^{2}=2\lambda f+C.}$

Often, in the shrinking or expanding cases (${\displaystyle \lambda \neq 0}$), ${\displaystyle f}$ is replaced by ${\displaystyle f-{\frac {C}{2\lambda }}}$ to obtain a gradient Ricci soliton normalized such that ${\displaystyle S+|\nabla f|^{2}=2\lambda f}$.