A Li-Yau inequality for the 1-dimensional Willmore energy

TL;DR

This paper explores Li-Yau type inequalities for curves in the plane related to elastic energy, extending classical surface results and applying findings to gradient flow dynamics.

Contribution

It introduces Li-Yau inequalities for planar curves involving elastic energy and discusses their implications for gradient flows.

Findings

Li-Yau inequalities for planar elastic curves

Applications to gradient flow analysis

Extensions of classical surface inequalities

Abstract

By the classical Li-Yau inequality, an immersion of a closed surface in $\mathbb{R}^n$ with Willmore energy below $8\pi$ has to be embedded. We discuss analogous results for curves in $\mathbb{R}^2$, involving Euler's elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.