On the functional logdet and related flows on the space of closed embedded curves on S²

For any two-dimensional Riemannian manifold (M, g) we introduce a new functional, h_g, on the space of closed simple nonparametrized curves on M. This functional associates to any simple curve Γ the regularize determinant of the Laplace operator on the manifold obtained by cutting M along Γ and imposing Dirichlet boundary conditions. When M is of genus zero we derive a formula for the variation of h_g, we prove that the critical points are conformal circles (i.e., the curves which, with respect to the unique metric of constant curvature 1 in the conformal class (e^2αg:α ∈ C^∞(S², R)) of g, have constant geodesic curvature), and that the hessian of the functional at a critical point is nondegenerate in directions normal the critical submanifold (Theorem 1.1). We also construct smooth flows on the space of nonparametrized curves retracting the space onto the critical sub-manifold and show that they are gradient-like for our function. These flows deform a given closed embedded curve on S² to a conformal circle keeping the area of the domain bounded by each curve of the deformation constant (Theorem 1.3).