On the Functional logdet and Related Flows on the Space of Closed Embedded Curves on S2

Abstract

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 2α g:α ∈ C ∞( S 2, 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 S 2 to a conformal circle keeping the area of the domain bounded by each curve of the deformation constant (Theorem 1.3).