On the functional logdet and related flows on the space of closed embedded curves on \(S^ 2\)
The paper is devoted to the study of the critical points of a new functional defined on the space of closed, simple, nonparametrized curves on a two-dimensional Riemannian manifold. The definition of the new functional goes as follows. Any oriented nonparametrized embedded curve on the two dimensional sphere \(S^ 2\) divides it into two parts. Take the manifold with boundary which has as interior the points of the sphere not in the curve, and as boundary, the disjoint union of two copies of the curve with opposite orientation. The resulting compact manifold has two connected components, denoted \(D^{+}_{\Gamma}\) and \(D^ -_ \Gamma\) according to the orientation of \(\Gamma\). Moreover, any Riemannian structure on the sphere can be lifted to it. Thus, given a fixed Riemannian structure on the sphere, the authors consider the Laplacian operators for the lifted Riemannian metric on the component \(D^ +_ \Gamma\), and, following \textit{D. B. Ray} and \textit{I. Singer} [Adv. Math. 7, 145-210 (1971; Zbl 0239.58014)], they define the zeta- regularized determinant, \(\text{logdet}(\Delta_ g,D^ +_ \Gamma)\) of the Laplace-Beltrami operator. The new functional, \(h_ g\), is then the sum \(\text{logdet}(\Delta_ g,D^ +_ \Gamma) + \text{logdet}(\Delta_ g,D^ +_{\iota(\Gamma)})\), where \(\iota(\Gamma)\) denotes the same underlying curve, but with reversed orientation. Considered separately, the logdet functional of these zeta functionals have no critical points and are not bounded neither below nor above. But the sum, \(h_ g\), of both logdet functionals has interesting properties. The paper is devoted to the study of the critical points of this sum of logdet functionals. It is well known that in the conformal class of any Riemannian metric \(g\) on the two-dimensional sphere there is a unique metric of constant curvature \(1\). One of the main results of the paper is the characterization of the critical points of the new functional as conformal circles; i.e., curves of constant geodesic curvature for the constant curvature metric in the same conformal class. Another main result is the existence of smooth vector fields on the space of closed embedded curves which are gradient-like for this functional and generate flows which retract the whole space onto the submanifold of conformal circles.
scientific article; zbMATH DE number 599771