On complex-tangential curves on the unit sphere on $\mathbf {C}^2$ and homogeneous polynomials

Abstract

We show that a closed complex-tangential $C^2$-curve $\gamma$ of constannt curvature on the unit sphere $\partial B_2$ of $\mathbf{C}^2$ is unitarily equivalent to \[ \gamma_{l,m}(t) = \left( \sqrt{l/d} e^{it\sqrt{m/l}}, \sqrt{m/d} e^{-it\sqrt{l/m}} \right) \] where $d = l + m$, $l, m \geq 1$ integers. As an application, we propose a conjecture that if a homogeneous polynomial $\pi$ on $\mathbf{C}^2$ admits a complex-tangential analytic curve on $\partial B_2$ with $\pi(\gamma(t)) = 1$ then $\pi$ is unitarily equivalent to a monomial \[ \pi_{l,m}(z,w) = \sqrt{\frac{d^d}{l^l m^m}} z^l w^m \] where $l, m \geq 1$ integers and show that the conjecture is true for homogeneous polynomial of degree $\leq 5$. A relevant conjecture and partial answer on the maximum modulus set of a homogeneous polynomial $\pi$ on $\mathbf{C}^2$ is also given.