Examples of Ricci-mean curvature flows

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 $1\leq k \leq n-1$. 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.