“最优宇宙”和其他球体外翻

Willmore Minmax Surfaces and the Cost of the Sphere Eversion

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 ${\R}^3$

$2$ -纽结图与球面外翻的正则等价性

Regular-equivalence of $2$-knot diagrams and sphere eversions

对于$2$ -结的每个图$D$，我们提供了一种方法来构造相同结的新图$D'$，使得在$D$和$D'$之间的任何Roseman移动序列必然涉及分支点。证明是通过发展的观察，没有球外翻可以提升到一个合痕在$4$ -空间。

又一个球体外翻

Yet another sphere eversion

我们提出了一个（可能的）新的领域外翻的基础上的收缩性 * 的某个子集的沉浸的圆在平面上的空间。（*：通过强形变收缩）

“Holiverse”：$\R^3$中2-球的整体外翻

The `Holiverse': holistic eversion of the 2-sphere in $\R^3$

我们给出了一个简短的，简单的和概念性的证明，基于自旋结构，球的外翻：一个嵌入的2-球面$R^3$可以翻转内到外的正则同伦。这种外翻的成分是无缝连接的。我们也给出了证明的数学起源：Hopf纤维化，和实射影三维空间的拓扑结构。

Analytic sphere eversion using ruled surfaces

Sphere eversions have been described so far by either pictures with minimal topological complexity, numerical evolution or complex equations. We write down relatively simple explicit formulas for the whole eversion, both analytic and topologically simpler, including also Boy surface (real projective plane), using a family of ruled surfaces. We show their usefulness in visualizing the process using commonly available modeling software.

Formalising the $h$-principle and sphere eversion

In differential topology and geometry, the h-principle is a property enjoyed by certain construction problems. Roughly speaking, it states that the only obstructions to the existence of a solution come from algebraic topology. We describe a formalisation in Lean of the local h-principle for first-order, open, ample partial differential relations. This is a significant result in differential topology, originally proven by Gromov in 1973 as part of his sweeping effort which greatly generalised many previous flexibility results in topology and geometry. In particular it reproves Smale's celebrate…