Li-Yau type inequality for curves in any codimension

TL;DR

This paper proves a Li-Yau type inequality for immersed curves in Euclidean space of any codimension, relating bending energy to multiplicity, with optimal bounds except for a specific planar case.

Contribution

It establishes a universal lower bound on normalized bending energy for curves of any codimension and multiplicity, revealing a hidden algebraic obstruction in certain cases.

Findings

Inequality is optimal for all cases except planar closed curves with odd multiplicity.

Identifies a hidden algebraic obstruction in the non-optimal case.

Discusses applications to elastic flows, networks, and knots.

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 Andr\'{e}'s algebraic-independence theorem for certain hypergeometric functions. We also discuss applications to elastic flows, networks, and knots.