Geom Dedicata (2006) 120:37–48 DOI 10.1007/s10711-006-9082-z ORIGINAL PAPER Construction of Lagrangian self-similar solutions to the mean curvature flow in C n Henri Anciaux Received: 22 January 2005 / Accepted: 18 May 2006 / Published online: 4 July 2006 © Springer Science+Business Media B.V. 2006 Abstract We give new examples of self-shrinking and self-expanding Lagrangian solutions to the Mean Curvature Flow (MCF). These are Lagrangian submanifolds in C n , which are foliated by (n - 1)-spheres (or more generally by minimal (n - 1)- Legendrian submanifolds of S 2n-1 ), and for which the study of the self-similar equa- tion reduces to solving a non-linear Ordinary Differential Equation (ODE). In the self-shrinking case, we get a family of submanifolds generalising in some sense the self-shrinking curves found by Abresch and Langer. Key words Mean curvature flow · Lagrangian submanifolds · Self-similar Mathematics Subject Classifications (2000) 35K55 · 58J35 · 53A07 0 Introduction Self-similar flows arise as special solutions of the mean curvature flow (which we abbreviate as MCF) that preserve the shape of the evolving surface. Analytically speaking, this amounts to making a particular Ansatz in the parabolic Partial Differ- ential Equation (PDE) describing the flow in order to eliminate the time variable and reduce the equation to an elliptic one. In geometric terms, it means looking for curvature properties of a submanifold that make it evolve in the required way. The study of such explicit, simple examples of flows is hoped to give a better under- standing of the behaviour of the flow at and after a singularity. The mean curvature flow is no longer expected to be unique after a singularity has appeared; there are numerical illustrations of this fact (cf. [6]) and current efforts to give a proof of it (cf. [7]). H. Anciaux (B ) PUC-Rio, Rua Marques de Sao Vicente, 225 Gavea 22453-900, Rio de Janeiro, RJ, Brasil e-mail: henri@mat.puc-rio.br