An example of non-embeddability of the Ricci flow

Abstract

For an evolution of metrics $(M,g_{t})$ there is a t-smooth family of embeddings $e_{t}:M\rightarrow {\mathbb {R}}^{N}$ inducing $g_{t}$, but in general there is no family of embeddings extending a given initial embedding $e_{0}$. We give an example of this phenomenon when $g_{t}$ is the evolution of $g_{0}$ under the Ricci flow. We show that there are embeddings $e_{0}$ inducing $g_{0}$ which do not admit of t-smooth extensions to $e_{t}$ inducing $g_{t}$ for any $t>0$. We also find hypersurfaces of $\text {dim}>2$ that will not remain a hypersurface under Ricci flow for any positive time.