Lamination of the moduli space of circles and their length spectrum for a non-flat complex space form (Q1423469)

This note is structured in six paragraphs. In the first paragraph the author introduces different notions. A smooth curve \(\gamma:\mathbb{R}\to M\) parametrized by its arc length on a complete Riemannian manifold \(M\) is called a circle of geodesic curvature \(k\) if it satisfies the differential equation \(\nabla_{\dot\gamma}\nabla_{\dot\gamma}\dot\gamma(t)= -k^2\dot\gamma(t)\). In this case \(k\) is a nonnegative constant and \(\nabla_{\dot\gamma}\) denotes the covariant differentiation along with respect to the Riemannian connection on \(M\). When \(k=0\) as \(\gamma\) is parametrized by its arc length, this equation is equivalent to the equation of geodesics. In this note the author studies the set of congruence classes of circles on a non-flat complex space form. In preceding papers published in 1995 and 1998 the author studied the length spectrum of circles in a non-flat complex space form. The second half of this paper is consecrated to add some results on length functions of circles in a non-flat complex space form. The second paragraph of this paper is devoted to the moduli space of circles on a non-flat complex space form. The third one on laminations of circles on a complex projective space'' uses the author's paper published in 1995 relatively to the circles on a complex projective space \(\mathbb{C} P^n(c)\) of constant holomorphic sectional curvature \(c\) and of complex dimension \(n\geq 2\) and shows the following: (i) Every Kähler circle of geodesic curvature \(k\) is a closed curve with length \(2\pi/\sqrt{k^2+c}\). (ii) Every totally real circle of geodesic curvature \(k\) is a closed curve with length \(4\pi/\sqrt{4k^2+ c}\). (iii) For \(K(> 0)\) and \(\tau\) \((0<\tau< 1)\) the solutions of the cubic equation \[ c\theta^2- 4(k^2+ c)\theta+ 2\sqrt{c},\;\tau= 0 \] are denoted by \(a_{k,\tau}\), \(b_{k,\tau}\), \(d_{k,\tau}\) \((a_{k,\tau}< b_{k,\tau}< d_{k,\tau})\). The central part of this paragraph is theorem 1. The fourth paragraph treats laminations of bounded circles on a complex hyperbolic space and shows theorem 2. Paragraph five is entitled ``Length functions on a complex projective space'' and is devoted to the study of the length functions for circles on a non-flat complex space form. In the final section, the author mentions briefly corresponding results on length functions for circles on a complex hyperbolic space. For other details see the authors references.