Advertisement

A Family of Expanding Ricci Solitons

  • Andreas Gastel
  • Manfred Kronz
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 59)

Abstract

The Ricci flow is a natural evolution equation for Riemannian metrics,introduced by Richard S. Hamilton in 1982. A family \({\left( {g\left( {t, \cdot } \right)} \right)_{t \in I}}\) of metrics on a Riemannian manifold M, depending on a time parameter \(t \in I \subseteq \mathbb{R} \) is a solution to the Ricci flow if it solves the equation
$$\frac{\partial }{{\partial t}}g\left( {t,\cdot} \right) = - 2Ricg\left( {t,\cdot} \right),$$
where Ric g(t, •) is the Ricci tensor associated with the evolving metric g(t, •). In general, a solution of the Ricci flow starting with smooth initial data will not possess a smooth continuation for all time. The formation of singularities has been discussed extensively in Hamilton’s article [H], and there is a particular type of solutions which is expected (and in some cases known) to appear as parabolic blowup limit of Ricci flows around a singularity, namely Ricci solitons. Ricci solitons are solutions to the Ricci flow for which there exist scalars σ(t) and diffeomorphisms Ψ t :MM such that
$$g\left( {t,\cdot} \right) = \sigma \left( t \right)\Psi _t^* g\left( {T,\cdot} \right){\text{ for all t}} \in {\text{I and some fixed T}} \in {\text{I}}{\text{.}}$$
(1)

Keywords

Riemannian Metrics Einstein Metrics Einstein Manifold Ricci Soliton Ricci Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AIV]
    S. Angenent, T. Ilmanen, J.J.L. Velazquez: Fattening from smooth initial data in mean curvature flow. In preparation, see [Il].Google Scholar
  2. [B1]
    C. Böhm: Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces, Invent. Math. 134 (1998), 145–176.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [B2]
    C. Böhm: Non-compact cohomogeneity one Einstein manifolds. Bull. Soc. Math. France 127 (1999), 135–177.MathSciNetzbMATHGoogle Scholar
  4. [EW]
    J. Eschenburg, M. Wang: The initial value problem for cohomogeneity one Einstein manifolds. J. Geom. Anal. 10 (2000), 109–137.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [FIK]
    M. Feldman, T. Ilmanen, D. Knopf: Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons. Preprint, 2002.Google Scholar
  6. [FK]
    D. Ferus, H. Karcher: Non-rotational minimal spheres and minimizing cones. Comm Math. Heiv. 60 (1985), 247–269.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [G]
    A. Castel: Nonuniqueness for the Yang-Mills heat flow. J. Diff. Eq. 187 (2003), 391–411.CrossRefGoogle Scholar
  8. [H]
    R.S. Hamilton: The formation of singularities in the Ricci flow. Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136, Internat. Press, Cambridge, MA, 1995.Google Scholar
  9. [Il]
    T. Ilmanen: Nonuniqueness in geometric heat flows. In Lectures on mean curvature flow and related equations, Trieste (1995), lecture 4, 38–56. http://www.math.ethz.ch/~ilmanen/notes.psGoogle Scholar
  10. [Iv]
    T. Ivey: New examples of complete Ricci solitons. Proc. Amer. Math. Soc. 122 (1994), 241–245.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Andreas Gastel
    • 1
  • Manfred Kronz
    • 2
  1. 1.Mathematisches Institut derHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany
  2. 2.Mathematisches Institut derUniversität Erlangen-NürnbergErlangenGermany

Personalised recommendations