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

Dan Burghelea, Thomas Kappeler, Patrick McDonald, Leonid Friedlander

Research output: Contribution to journalArticle

3 Scopus citations

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

ASJC Scopus subject areas

  • Analysis

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