Topic
Sphere eversion
About: Sphere eversion is a research topic. Over the lifetime, 22 publications have been published within this topic receiving 151 citations. The topic is also known as: Sphere eversion & Turning a sphere inside-out.
Papers
1 Mar 1997
TL;DR: In this paper, an eversion of a sphere driven by a gradient flow for elastic bending energy is considered, and the resulting eversion is isotopic to one of Morin's classical eversions.
Abstract: We consider an eversion of a sphere driven by a gradient flow for elastic bending energy. We start with a halfway model which is an unstable Willmore sphere with 4-fold orientation-reversing rotational symmetry. The regular homotopy is automatically generated by flowing down the gradient of the energy from the halfway model to a round sphere, using the Surface Evolver. This flow is not yet fully understood; however, our numerical simulations give evidence that the resulting eversion is isotopic to one of Morin’s classical sphere eversions. These simulations were presented as real-time interactive animations in the CAVE TM automatic virtual environment at Supercomputing’95, as part of an experiment in distributed, parallel computing and broad-band, asynchronous networking.
32 citations
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TL;DR: In this article, it was shown that the stable Hopf invariant is not realizable in R 2n, i.e., the composition of f with an embedding M ⊂ R 2 n is C 0 -approximable by embeddings.
Abstract: P. M. Akhmetiev used a controlled version of the stable Hopf invariant to show that any (continuous) map N → M between stably parallelizable compact n-manifolds, n 6 1, 2, 3, 7, is realizable in R 2n , i.e. the composition of f with an embedding M ⊂ R 2n is C 0 -approximable by embeddings. It has been long believed that any degree 2 map S 3 → S 3 , obtained by capping off at infinity a time-symmetric (e.g. Shapiro’s) sphere eversion S 2 ×I → R 3 , was non-realizable in R 6 . We show that there
27 citations
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TL;DR: In this article, the authors extend the classification of Bryant of variational branched Willmore spheres in dimension $S^3$ and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in $\mathbb{R}^3] and vanishing flux.
Abstract: We extend the classification of Robert Bryant of Willmore spheres in $S^3$ to variational branched Willmore spheres $S^3$ and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in $\mathbb{R}^3$ and vanishing flux. We also obtain a classification of variational branched Willmore spheres in $S^4$, generalising a theorem of Sebastian Montiel. As a result of our asymptotic analysis at branch points, we obtain an improved $C^{1,1}$ regularity of the unit normal of variational branched Willmore surfaces in arbitrary codimension. We also prove that the width of Willmore sphere min-max procedures in dimension $3$ and $4$, such as the sphere eversion, is an integer multiple of $4\pi$.
20 citations
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TL;DR: In this paper, a general Minmax procedure in Euclidian spaces for constructing Willmore surfaces of non zero indices was developed. And they implemented this procedure to the Willmore Minmax Sphere Eversion in the 3 dimensional euclidian space.
Abstract: We develop a general Minmax procedure in Euclidian spaces for constructing Willmore surfaces of non zero indices. We implement this procedure to the Willmore Minmax Sphere Eversion in the 3 dimensional euclidian space. We compute the cost of the Sphere eversion in terms of Willmore energies of Willmore Spheres in ${\R}^3$
20 citations
TL;DR: In this paper, a general Minmax procedure in Euclidian spaces for constructing Willmore surfaces of non zero indices was developed. And they implemented this procedure to the Willmore Minmax Sphere Eversion in the 3 dimensional euclidian space.
Abstract: We develop a general Minmax procedure in Euclidian spaces for constructing Willmore surfaces of non zero indices. We implement this procedure to the Willmore Minmax Sphere Eversion in the 3 dimensional euclidian space. We compute the cost of the Sphere eversion in terms of Willmore energies of Willmore Spheres in ${\R}^3$
14 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2019 | 1 |
| 2018 | 1 |
| 2017 | 4 |
| 2016 | 1 |
| 2015 | 1 |