Ricci flow on surfaces of revolution: an extrinsic view

A Ricci flow ( M , g ( t )) on an n -dimensional Riemannian manifold M is an intrinsic geometric flow. A family of smoothly embedded submanifolds ( S ( t ) , g E ) of a fixed Euclidean space E = R n + k is called an extrinsic representation in R n + k of ( M , g ( t )) if there exists a smooth one-parameter family of isometries ( S ( t ) , g E ) → ( M , g ( t ) ) . When does such a representation exist? We formulate a new way of framing this question for Ricci flows on surfaces of revolution immersed in R 3 . This framework allows us to construct extrinsic representations for the Ricci flow initialized by any compact surface of revolution immersed in R 3 . In particular, we exhibit the first explicit extrinsic representations in R 4 of the Ricci flows initialized by toroidal surfaces of revolution immersed in R 3 .