Minimal Surfaces Associated with Nonpolynomial Contact Symmetry Flows

Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the selfintersecting minimal surface that fills the trefoil knot. Introduction. In this paper we consider the minimal surface equation EminΣ = { (1 + u2y) uxx − 2uxuyuxy + (1 + u 2 x) uyy = 0 } (1) whose solutions describe two-dimensional minimal surfaces Σ ⊂ E in nonparametric form Σ = {z = u(x, y)}, here x, y, z are the Cartesian coordinates. We construct two infinite sequences of minimal surfaces related to nonpolynomial contact symmetries of Eq. (1). Remark 1. Although the graphs of solutions for Eq. (1) determine the minimal surfaces only locally such that the projections of their tangent planes to 0xy are nondegenerate, this is not restrictive for our reasonings. The minimal surfaces constructed in section 2 are self-intersecting, being in fact described by multi-valued solutions of Eq. (1) and admitting singular points. The paper is organized as follows. In section 1 we describe the generators and the commutator relations of the contact symmetry algebra for EminΣ. We provide examples of the contact non-point generators and indicate the recursion operators for the commutative Lie subalgebra of sym EminΣ . In section 2 we prove that any surface which is invariant w.r.t. a contact non-point symmetry flow is always a plane, although nonplanar minimal surfaces in space are assigned to the same symmetry generators by the inverse Legendre transformation. Thus we construct two sequences of the minimal surfaces associated with nonpolynomial contact symmetries of EminΣ; one of the sequences starts with the helicoid [4] in E. The recursions for the symmetries provide discrete transformations between the surfaces, while the generators themselves determine their continuous transformations. In particular, we obtain explicit formulas for the self-intersecting minimal surface Σ6 that fills the trefoil knot; this surface succeeds the helicoid Σ5 with respect to the recursion relations. Date: April 6, 2006. 2000 Mathematics Subject Classification. 49Q05, 53A10, 70S10.