Ricci flow on surfaces of revolution: an extrinsic view

Abstract

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 \(E = \mathbb {R}^{n+k}\) is called an extrinsic representation in \(\mathbb {R}^{n+k}\) of (M, g(t)) if there exists a smooth one-parameter family of isometries \((S(t), g_E) \rightarrow (M, g(t))\). When does such a representation exist? We formulate a new way of framing this question for Ricci flows on surfaces of revolution immersed in \(\mathbb {R}^3\). This framework allows us to construct extrinsic representations for the Ricci flow initialized by any compact surface of revolution immersed in \(\mathbb {R}^3\). In particular, we exhibit the first explicit extrinsic representations in \(\mathbb {R}^4\) of the Ricci flows initialized by toroidal surfaces of revolution immersed in \(\mathbb {R}^3\).