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

Müller, Marius; Rupp, Fabian

doi:10.1515/acv-2021-0014

Cited by 9 publications

(2 citation statements)

References 23 publications

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“…By [25,Lemma 4.3 and Remark 4.4], γ is non-embedded. As at the beginning of the proof of Theorem 3.14, one deduces from the bound on the elastic energy that the g H 2 -length n of each γ n is bounded from below uniformly away from 0.…”

Section: A Li-yau Inequality For Open Curves In Hmentioning

confidence: 95%

On the convergence of the Willmore flow with Dirichlet boundary conditions

Schlierf¹

2023

Preprint

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Very little is yet known regarding the Willmore flow of surfaces with Dirichlet boundary conditions. We consider surfaces with a rotational symmetry as initial data and prove a convergence result for initial data below an energy threshold which depends on the prescribed boundary conditions. Moreover, we show optimality by constructing singular examples for the Willmore flow above this energy threshold. Finally, a Li-Yau inequality for open curves in H 2 is proved.

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Section: A Li-yau Inequality For Open Curves In Hmentioning

confidence: 95%

On the convergence of the Willmore flow with Dirichlet boundary conditions

Schlierf¹

2023

Preprint

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“…It is well known that for p = 2 all planar elasticae are smooth (by bootstrap), classified into some classes (essentially due to Euler), and have explicit parameterizations in terms of Jacobian elliptic integrals and functions [26] (see also [15,32]).…”

Section: Introductionmentioning

confidence: 99%

Complete classification of planar p-elasticae

Miura¹,

Yoshizawa²

2022

Preprint

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Euler's elastica is defined by a critical point of the total squared curvature under the fixed length constraint, and its L p -counterpart is called p-elastica. In this paper we completely classify all p-elasticae in the plane and obtain their explicit formulae as well as optimal regularity. To this end we introduce new types of p-elliptic functions which streamline the whole argument and result. As an application we also classify all closed planar p-elasticae. Contents 1. Introduction 1 2. The Euler-Lagrange equations 9 3. p-Elliptic functions 11 4. Classification, representation, and regularity of p-elasticae 25 5. Closed p-elasticae 39 Appendix A. First variation of the p-bending energy 45 References 47

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Li–Yau type inequality for curves in any codimension

Miura

2023

Calc. Var.

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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 André’s algebraic-independence theorem for certain hypergeometric functions. We also discuss applications to elastic flows, networks, and knots.

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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 {\mathbb{R}^{n}} with Willmore energy below 8 … Show more

Cited by 9 publications

References 23 publications

On the convergence of the Willmore flow with Dirichlet boundary conditions

On the convergence of the Willmore flow with Dirichlet boundary conditions

Complete classification of planar p-elasticae

Li–Yau type inequality for curves in any codimension

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