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Ricci soliton

From Wikipedia, the free encyclopedia

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

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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 diffeomorphisms , 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

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Shrinking ()

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  • Gaussian shrinking soliton
  • Shrinking round sphere
  • Shrinking round cylinder
  • 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 ()

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  • The 2d cigar soliton (a.k.a. Witten's black hole)
  • The 3d rotationally symmetric Bryant soliton and its generalization to higher dimensions [6]
  • Ricci flat manifolds

Expanding ()

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  • Expanding Kahler-Ricci solitons on the complex line bundles over .[1]
  • Einstein manifolds of negative scalar curvature

Singularity models in Ricci flow

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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

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Taking the trace of the Ricci soliton equation gives

, (1)

where is the scalar curvature and . By taking the divergence of the Ricci soliton equation and invoking the contracted Bianchi identities and (1), it follows that


For gradient Ricci solitons , similar arguments show

In particular, if is connected, then there exists a constant such that

Often, in the shrinking or expanding cases (), is replaced by to obtain a gradient Ricci soliton normalized such that .

Notes

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  1. ^ a b Feldman, Mikhail; Ilmanen, Tom; Knopf, Dan (2003), "Rotationally Symmetric Shrinking and Expanding Gradient Kähler-Ricci Solitons", Journal of Differential Geometry, 65 (2): 169–209, doi:10.4310/jdg/1090511686
  2. ^ Bamler, R.; Cifarelli, C.; Conlon, R.; Deruelle, A. (2022). "A new complete two-dimensional shrinking gradient Kähler-Ricci soliton". arXiv:2206.10785 [math.DG].
  3. ^ Koiso, Norihito (1990), "On rotationally symmetric Hamilton's equation for Kahler-Einstein metrics", Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics, vol. 18-I, Academic Press, Boston, MA, pp. 327–337, doi:10.2969/aspm/01810327, ISBN 978-4-86497-076-1
  4. ^ Cao, Huai-Dong (1996), "Existence of gradient Kähler-Ricci solitons", Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, MA, pp. 1–16, arXiv:1203.4794
  5. ^ Wang, Xu-Jia; Zhu, Xiaohua (2004), "Kähler-Ricci solitons on toric manifolds with positive first Chern class", Advances in Mathematics, 188 (1): 87–103, doi:10.1016/j.aim.2003.09.009
  6. ^ Bryant, Robert L., Ricci flow solitons in dimension three with SO(3)-symmetries (PDF)
  7. ^ Enders, Joerg; Müller, Reto; Topping, Peter M. (2011), "On Type I Singularities in Ricci flow", Communications in Analysis and Geometry, 19 (5): 905–922, doi:10.4310/CAG.2011.v19.n5.a4, hdl:10044/1/10485

References

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