Isoperimetric inequalities for immersed closed spherical curves

Abstract

Abstract.Let a: Sl ->S2 be a C2 immersion with length L and total curvature K . If a is regularly homotopic to a circle traversed once then L2 + K2 > 4n2 with equality if and only if a is a circle traversed once. If a has nonnegative geodesic curvature and multiple points then L + K > An with equality if and only if a is a great circle traversed twice.

Isoperimetric inequalities for wave fronts and a generalization of Menzin's conjecture for bicycle monodromy on surfaces of constant curvature

“…Weiner generalized this result to smooth immersions of a circle into the 2-sphere, in [14] (which we use to prove some later results). Using the Gauss-Bonnet Theorem, the classical isoperimetric inequality can be written as…”

“…We have that γ * is smooth because γ has no inflection points and is also homotopic to a circle traversed once because γ has an even number of double-tangent points. By [14] we then have that…”

“…This inequality makes sense for immersions as well as embeddings. In [14], Weiner shows that (1) holds for smooth immersions regularly homotopic to a circle traversed once. E. Teufel made a similar generalization for smooth immersions of a circle into the hyperbolic plane, in [13].…”