Möbius invariant metrics on the space of knots

Unitarily Invariant Metrics on the Grassmann Space

Let Gm,n be the Grassmann space of m-dimensional subspaces of F. Denote by θ1(X ,Y), . . . , θm(X ,Y) the canonical angles between subspaces X ,Y ∈ Gm,n. It is shown that Φ(θ1(X ,Y), . . . , θm(X ,Y)) defines a unitarily invariant metric on Gm,n for every symmetric gauge function Φ. This provides a wide class of new metrics on Gm,n. Some related results on perturbation and approximation of subs...

Torus Knots Extremizing the Möbius Energy

This work was accomplished while the authors were at the Institute for Advanced Study, and was partly supported by an NSF Postdoctoral Fellowship. Using the principle of symmetric criticality [Palais 1979], we construct torus knots and links that extremize the Möbiusinvariant energy introduced by O’Hara [1991] and Freedman, He and Wang [1993]. The critical energies are explicitly computable usi...

Möbius-invariant natural neighbor interpolation

On the Invariant Spectrum of S1−invariant Metrics on S

A theorem of J. Hersch (1970) states that for any smooth metric on S, with total area equal to 4π, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the standard round metric). For metrics invariant under the standard S-action on S, one can restrict the Laplace operator to the subspace of S-invariant functions and consi...

On the Invariant Spectrum of S−invariant Metrics on S

A theorem of J. Hersch (1970) states that for any smooth metric on S2, with total area equal to 4π, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the standard round metric). For metrics invariant under the standard S1-action on S2, one can restrict the Laplace operator to the subspace of S1-invariant functions and c...