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

D. Burghelea, T. Kappeler, P. Mcdonald, Leonid Friedlander

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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 {eg: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).

Original languageEnglish (US)
Pages (from-to)440-466
Number of pages27
JournalJournal of Functional Analysis
Volume120
Issue number2
DOIs
StatePublished - Mar 1994

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Closed
Curve
Submanifolds
Critical point
Circle
Curvature
Simple Closed Curve
Laplace Operator
Theorem
Dirichlet Boundary Conditions
Geodesic
Riemannian Manifold
Bounded Domain
Genus
Determinant
Gradient
Metric
Zero

ASJC Scopus subject areas

  • Analysis

Cite this

On the Functional logdet and Related Flows on the Space of Closed Embedded Curves on S2. / Burghelea, D.; Kappeler, T.; Mcdonald, P.; Friedlander, Leonid.

In: Journal of Functional Analysis, Vol. 120, No. 2, 03.1994, p. 440-466.

Research output: Contribution to journalArticle

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