### 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).

Original language | English (US) |
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Pages (from-to) | 440-466 |

Number of pages | 27 |

Journal | Journal of Functional Analysis |

Volume | 120 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1994 |

### ASJC Scopus subject areas

- Analysis

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## Cite this

^{2}.

*Journal of Functional Analysis*,

*120*(2), 440-466. https://doi.org/10.1006/jfan.1994.1038