Embeddedness and graphicality of the elastic flow for complete curves

TL;DR

This paper investigates the elastic flow of complete curves in Euclidean space, establishing conditions for preserving embeddedness and graphicality, and deriving a new inequality for planar curves.

Contribution

It extends elastic energy methods to non-compact curves, providing optimal thresholds for embeddedness and graphicality, and introduces a novel Li--Yau type inequality.

Findings

Established positivity-preserving properties for elastic flow

Derived optimal thresholds for embeddedness and graphicality

Proved a new Li--Yau type inequality for complete planar curves

Abstract

We study positivity-preserving properties for the elastic flow of non-compact, complete curves in Euclidean space. Despite the fact that the canonical elastic energy is infinite in this context, we extend our recent work based on the adapted elastic energy to derive nontrivial optimal thresholds for maintaining planar embeddedness and graphicality, respectively. We also obtain a new Li--Yau type inequality for complete planar curves.