Infinitesimal Ricci flows of minimal surfaces in the three-dimensional Euclidean space

Abstract

where Ric(g) is the Ricci tensor of g. The Ricci flow, as a technical tool, was heavily used in works devoted to the proof of the Poincare conjecture, and many results concerning existence and properties of Ricci flows of metrics on compact manifolds have been obtained (see, e.g., [1, 2]). On the other hand, it is of interest to study the geometrical properties of metrics involved in the Ricci flow. One of the problems which can be considered in this connection is to study a one-parameter family of embeddings ft of a manifold Σ into a Riemannian manifold (M,G) such that the metrics gt = f∗ t G induced onΣ satisfy (1). It is worth noting that one-parameter families of submanifolds with the property that differential geometric objects naturally associated with these submanifolds (e.g., curvatures) satisfy certain evolution equations, have been considered in the literature many times (see, e.g., [3]). In the present paper we study a family Σt of surfaces in the three-dimensional Euclidean space E3 such that the metric gt induced on Σt by the standard metric of E3 satisfies the Ricci flow (1). The family Σt will be called a Ricci flow of surfaces. Our considerations are local, and all the manifolds and maps are assumed to be analytical. For a two-dimensional Riemannian manifold (M,g), we have Ric = Kg, where K is the sectional curvature, hence in our situation the Ricci-flow equation is written as follows: