### 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) |
---|---|

Pages (from-to) | 440-466 |

Number of pages | 27 |

Journal | Journal of Functional Analysis |

Volume | 120 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1994 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

^{2}.

*Journal of Functional Analysis*,

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

**On the Functional logdet and Related Flows on the Space of Closed Embedded Curves on S ^{2}.** / Burghelea, D.; Kappeler, T.; Mcdonald, P.; Friedlander, Leonid.

Research output: Contribution to journal › Article

^{2}',

*Journal of Functional Analysis*, vol. 120, no. 2, pp. 440-466. https://doi.org/10.1006/jfan.1994.1038

^{2}. Journal of Functional Analysis. 1994 Mar;120(2):440-466. https://doi.org/10.1006/jfan.1994.1038

}

TY - JOUR

T1 - On the Functional logdet and Related Flows on the Space of Closed Embedded Curves on S2

AU - Burghelea, D.

AU - Kappeler, T.

AU - Mcdonald, P.

AU - Friedlander, Leonid

PY - 1994/3

Y1 - 1994/3

N2 - For any two-dimensional Riemannian manifold (M, g) we introduce a new functional, hg, 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 hg, 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 {e2αg: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).

AB - For any two-dimensional Riemannian manifold (M, g) we introduce a new functional, hg, 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 hg, 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 {e2αg: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).

UR - http://www.scopus.com/inward/record.url?scp=25144524629&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=25144524629&partnerID=8YFLogxK

U2 - 10.1006/jfan.1994.1038

DO - 10.1006/jfan.1994.1038

M3 - Article

VL - 120

SP - 440

EP - 466

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -