Examples of Ricci-mean curvature flows

Abstract

Let $\pi :\mathbb{P}(\mathcal{O}(0)\oplus \mathcal{O}(k))\to {\mathbb{P}}^{n-1}$ be a projective bundle over ${\mathbb{P}}^{n-1}$ with $1\le k\le n-1$. In this paper, we show that lens space $L(k;1)(r)$ with radius $r$ embedded in $\mathbb{P}(\mathcal{O}(0)\oplus \mathcal{O}(k))$ is a self-similar solution, where $\mathbb{P}(\mathcal{O}(0)\oplus \mathcal{O}(k))$ is endowed with the $U(n)$-invariant gradient shrinking Ricci soliton structure. 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 zero section of $\pi$ if $r<r_1$ and to the $\infty$-section of $\pi$ if $r_1<r$. This gives explicit examples of Ricci-mean curvature flows.

18 pages, 1 figure, 1 table

References in corpus (1)

• Ricci-mean curvature flows in gradient shrinking Ricci solitons