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The Journal of Geometric Analysis

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

Examples of Ricci-Mean Curvature Flows

  • Hikaru Yamamoto
Article
  • 138 Downloads

Abstract

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.

Keywords

Mean curvature flow Self-similar solution Ricci flow Ricci soliton 

Mathematics Subject Classification

53C42 53C44 

Notes

Acknowledgements

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.

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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

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

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