TY - JOUR

T1 - On the functional logdet and related flows on the space of closed embedded curves on S2

AU - Burghelea, Dan

AU - Kappeler, Thomas

AU - McDonald, Patrick

AU - Friedlander, Leonid

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

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).

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U2 - 10.1006/jfan.1994.1038

DO - 10.1006/jfan.1994.1038

M3 - Article

AN - SCOPUS:25144524629

VL - 120

SP - 440

EP - 466

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -