Mathematical, Computing & Information Sciences
An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in Rn+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 R3. Our approach is to build the desired embedded surfaces of revolution S(t) in R3 into the flow at the outset by rewriting the Ricci flow equations in terms of extrinsic geometric quantities in a natural way. This identifies an extrinsic representation with a particular solution of the scalar logarithmic diffusion equation in one space variable. The result is a single, unified framework to construct an extrinsic representation in R3 of a Ricci flow on a surface of revolution S initialized by a metric g0. Of special interest is the Ricci flow on the torus S1 ×S1 embedded in R3. 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 nonsmooth, but isometric, embeddings of the torus into R3.
Coll, V., Dodd, J., & Johnson, D. (2020). Ricci flow on surfaces of revolution: an extrinsic view. Geometriae Dedicata, 207(1), 81-94.