The Journal of Geometric Analysis

, Volume 28, Issue 2, pp 983–1004 | Cite as

Examples of Ricci-Mean Curvature Flows

  • Hikaru Yamamoto


Let \(\pi :{\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\rightarrow {\mathbb {P}}^{n-1}\) be a projective bundle over \({\mathbb {P}}^{n-1}\) with \(1\le k \le n-1\). We denote \({\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\) by \(N_{k}^{n}\) and endow it with the U(n)-invariant gradient shrinking Kähler Ricci soliton structure constructed by Cao (Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, 1996) and Koiso (Recent topics in differential and analytic geometry. Advanced studies in pure mathematics, Boston, 1990). In this paper, we show that lens space \(L(k\, ;1)(r)\) with radius r embedded in \(N_{k}^{n}\) is a self-similar solution. We also prove that there exists a pair of critical radii \(r_{1}<r_{2}\), which satisfies the following. The lens space \(L(k\, ;1)(r)\) is a self-shrinker if \(r<r_{2}\) and self-expander if \(r_{2}<r\), and the Ricci-mean curvature flow emanating from \(L(k\, ;1)(r)\) collapses to the 0-section of \(\pi \) if \(r<r_{1}\) and to the \(\infty \)-section of \(\pi \) if \(r_{1}<r\). This paper gives explicit examples of Ricci-mean curvature flows.


Mean curvature flow Self-similar solution Ricci flow Ricci soliton 

Mathematics Subject Classification

53C42 53C44 



This work was supported by JSPS KAKENHI Grant Number 16H07229. The author would like to thank the reviewers for their careful reading and suggestions to help improve the paper.


  1. 1.
    Cao, H.-D.: Existence of gradient Kähler-Ricci solitons. Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), pp. 1–16. A K Peters, Wellesley (1996)Google Scholar
  2. 2.
    Cao, H.-D., Li, H.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. Partial Diff. Equ. 46(3–4), 879–889 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci flow: techniques and applications. Part I. Geometric aspects. Mathematical surveys and monographs, 135. American Mathematical Society, Providence (2007)Google Scholar
  4. 4.
    Han, X., Li, J.: The Lagrangian mean curvature flow along the Kähler–Ricci flow. In: Recent developments in geometry and analysis. Advanced Lectures in Mathematics (ALM), vol. 23, pp. 147–154. International Press, Somerville (2012)Google Scholar
  5. 5.
    Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Diff. Geom. 31(1), 285–299 (1990)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Koiso,N.: On rotationally symmetric Hamilton’s equation for Kähler-Einstein metrics. Recent topics in differential and analytic geometry. Advanced Studies in Pure Mathematics, vol. 18-I, pp. 327–337. Academic Press, Boston (1990)Google Scholar
  7. 7.
    Lotay, J. D., Pacini, T.: Coupled flows, convexity and calibrations: Lagrangian and totally real geometry. arXiv:1404.4227
  8. 8.
    Lott, J.: Mean curvature flow in a Ricci flow background. Commun. Math. Phys. 313(2), 517–533 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Magni, A., Mantegazza, C., Tsatis, E.: Flow by mean curvature inside a moving ambient space. J. Evol. Equ. 13(3), 561–576 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Smoczyk, K.: The Lagrangian mean curvature flow. Univ. Leipzig (Habil.-Schr.) (2000)Google Scholar
  11. 11.
    Yamamoto, H.: Ricci-mean curvature flows in gradient shrinking Ricci solitons. arXiv:1501.06256

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceTokyo University of ScienceTokyoJapan

Personalised recommendations