Ricci Flow on Surfaces of Revolution: an Extrinsic View

An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in R^{n+k}, evolving as a function of time t such that the metrics induced on the submanifolds S(t) by the ambient Euclidean metric yield the Ricci flow on M. When does such a representation exist? We formulate this question precisely and describe a new, comprehensive way of addressing it for surfaces of revolution in R^3. Of special interest is the Ricci flow on a toroidal surface of revolution, that is, a surface of revolution whose profile curve is an immersed curve which does not intersect the axis of revolution. In In this case, the extrinsic representation of the Ricci flow on a Riemannian cover of S is eternal. This flow can also be realized as a compact family of non-smooth, but isometric, embeddings of the torus into R^3.