Li-Yau type inequality for curves in any codimension

Abstract

For immersed curves in Euclidean space of any codimension we establish a Li--Yau type inequality that gives a lower bound of the (normalized) bending energy in terms of multiplicity. The obtained inequality is optimal for any codimension and any multiplicity except for the case of planar closed curves with odd multiplicity; in this remaining case we discover a hidden algebraic obstruction and indeed prove an exhaustive non-optimality result. The proof is mainly variational and involves Langer--Singer's classification of elasticae and André's algebraic-independence theorem for certain hypergeometric functions. We also discuss applications to elastic flows, networks, and knots.

28 pages, 6 figures, final version

References in corpus (1)

• On the linear independence of values of $G$-functions

Cited by in corpus (7)

• Migrating elastic flows
• Migrating elastic flows II
• Complete classification of planar p-elasticae
• Variational stabilization of degenerate p-elasticae
• Regularity and structure of non-planar $p$-elasticae
• Elastic curves and self-intersections
• The critical points of the elastic energy among curves pinned at endpoints