On complex-tangential curves on the unit sphere on C^2 and homogeneous polynomials

Abstract

We show that a closed complex-tangential C-2-curve gamma of constant curvature on the unit sphere partial derivative B-2 of C-2 is unitarily equivalent to gamma l,m(t) = (root l/d e(it root m/l), root m/d e(-it root l/m)) where d = l + m, l,m greater than or equal to 1 integers. As an application, we propose a conjecture that if a homogeneous polynomial ir on C2 admits a complex-tangential analytic curve on partial derivative B-2 with pi(gamma(t)) = 1 then pi is unitarily equivalent to a monomial pi(l,m)(z,w) = root d(d)/l(l)m(m)z(l)w(m) where l, m greater than or equal to 1 integers and show that the conjecture is true for homogeneous polynomials of degree less than or equal to 5. A relevant conjecture and partial answer on the maximum modulus set of a homogeneous polynomial pi on C-2 is also given.