Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2208.12678

Stationary curves under the Möbius-Plateau energy

Max Lipton·Gokul Nair
View original source ↗Read source document ↗Export metadata ↓

Abstract

Plateau problems with elastic boundary energies have been of recent theoretical and applied interest. However, strong assumptions have to be made to avoid self-intersections of the boundary curve during energy minimization. We introduce a class of Plateau problems for boundaries with self-repulsive energies that obviates self-contact in energy minimization problems. For the self-repulsive energy, we choose the Möbius Energy introduced by O'Hara due to its myriad regularity properties shown by Freedman et al. We first prove an existence theorem for this Möbius-Plateau problem in the class of closed Lipschitz curves of a given irreducible knot-type spanned by immersed discs. We then turn our attention to Möbius-Plateau variations of helicoidal strips, which are classified as "screw-like" or "ribbon-like" based on the signs of the radii of the boundary helices. By analyzing the Euler-Lagrange equations, we show that screw-like solutions are plentiful, whilst ribbon-like solutions impose strong constraints on their parameters: they must have high frequency (equivalently, low pitch), thin width in comparison to the frequency, and remain close to the axis.

Explore related subjects

math.DGmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Max Lipton, Gokul Nair. 2025-03-09. Stationary curves under the Möbius-Plateau energy. https://arxiv.org/abs/2208.12678

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the topology of manifolds with nonnegative Ricci curvature and linear volume growth

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces.

math.DG↗
arXiv2026

K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

Recently, Sun-Zhang have developed an algebraic theory for Kähler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration $(π: X \to Y, ξ)$. In particular, they conjecture that existence of a Kähler-Ricci shrinker metric is equivalent to a notion of K-stability. We prove one direction of this conjecture, namely that existence of a Kähler-Ricci shrinker metric $g$ implies K-polystability of $(π: X \to Y, ξ)$, in the case that the Ricci curvature of $g$ decays at infinity. As an application, we give a non-existence result: if $M$ is the blowup of a six-dimensional quadric along a two-dimensional subquadric, then the total space $X$ of the cube root of $K_M$ is a polarized Fano fibration not admitting a Kähler-Ricci shrinker.

math.DG↗
arXiv2026

Observações sobre funções potenciais de variedades quase-Einstein não compactas

Neste artigo, estudamos o conjunto de funções potenciais em variedades quase Einstein não compactas. Mostramos que o espaço de todas as funções potenciais positivas em uma variedade tridimensional não compacta quase-Einstein tem dimensão no máximo dois, e que a igualdade vale se e somente se a variedade for isométrica a um produto $B\times\mathbb{R}$, onde $B$ é uma superfície $λ$-Einstein ou um dos exemplos obtidos por L. Berard Bergery e descritos no livro de Besse. Além disso, provamos que qualquer variedade quase-Einstein assintoticamente plana $n$-dimensional com $λ=0$ é necessariamente Ricci-plana.

math.DG↗