An example of non-embeddability of the Ricci flow

TL;DR

This paper demonstrates that under Ricci flow, certain initial embeddings of a manifold cannot be smoothly extended over time, revealing limitations in embedding evolution compatibility.

Contribution

It provides explicit examples showing non-embeddability of Ricci flow evolution, highlighting fundamental geometric constraints.

Findings

Existence of initial embeddings with no smooth extension under Ricci flow

Certain hypersurfaces cannot remain hypersurfaces during Ricci flow

Illustration of non-embeddability phenomena in geometric evolution

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.