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 $e_t:M\to {\mathbb{R}}^N$ 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 dim>2 that will not remain a hypersurface under Ricci flow for any positive time.

arXiv comment5 pages