Elastic curves and self-intersections

TL;DR

This paper reviews classical elastica theory and provides a detailed proof of a Li--Yau type inequality for self-intersecting curves, also discussing open problems in the field.

Contribution

It offers a detailed proof of a Li--Yau type inequality for self-intersecting elastica and reviews classical elastica theory with insights into open problems.

Findings

Proof of Li--Yau type inequality for self-intersecting curves

Review of classical elastica theory

Discussion of open problems in elastica and related topics

Abstract

This is an expository note to give a brief review of classical elastica theory, mainly prepared for giving a more detailed proof of the author's Li--Yau type inequality for self-intersecting curves in Euclidean space. We also discuss some open problems in related topics.