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

Abstract

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

Optimal thresholds for preserving embeddedness of elastic flows

“…Proposition 2.2 (Planar elasticae, see e.g. [31,Proposition B.8]). Let I ⊂ R be an interval and let γ ∈ C ∞ (I; R 2 ) be an elastica with signed scalar curvature k[γ].…”

“…Proof. By [31,Lemma 5.4] the only closed elasticae with a self-intersection are (up to scaling and isometries) given by ω-fold circles (ω ≥ 2) and ω-fold figure-eight elasticae (ω ≥ 1). For an ω-fold covering of the circle one readily checks that…”

“…Definition 2.1) are known to hold in general, see e.g. [8,9,11,27,31,34] and also a survey [26]. However, since elastic flows are of higher order, the global behavior of solutions is less understood.…”

“…Our focus will be on embeddedness along elastic flows. In previous studies the authors found the following optimal energy threshold for all-time embeddedness in [31] (n = 2) and [29] (n ≥ 2): Let C 8 = B[γ 8 ] > 0 denote the energy B of a figureeight elastica γ 8 , cf. Definition 2.3 and Figure 1b.…”

“…Since we are interested in minimization problems for B, from now on we specify the natural H 2 -Sobolev regularity for curves. We first recall the following general estimate for nonembedded closed curves, which is recently obtained by the last two authors for n = 2 [31] and by the first author for n ≥ 2 [29].…”