On the total absolute curvature of sphere eversions

A simple eversion of the 2-sphere with a unique quadruple point

We give an example of an eversion of the 2-sphere in the Euclidean 3-space, inspired by Morse theory, with a unique quadruple point. No homotopical tool is used.

"The Optiverse" and other sphere eversions

For decades, the sphere eversion has been a classic subject for mathematical visualization. The 1998 video "The Optiverse" shows geometrically optimal eversions created by minimizing elastic bending energy. We contrast these minimax eversions with earlier ones, including those by Morin, Phillips, Max, and Thurston. The minimax eversions were automatically generated by flowing downhill in energy using Brakke's Evolver.

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 ${\R}^3$

Regular homotopy and total curvature

We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvature functional on the space of 2-sphere immersions into 3-space. We show that the infimum over all sphere eversions of the maximum of the total curvature during an eversion is at most 8πand we establish a non-injectivity result for local minima.

关于$\mathbb{R}^3$中曲率有界球体问题

On the curvature bounded sphere problem in $\mathbb{R}^3$

该研究证明了在$\mathbb{R}^3$中，满足法曲率有界条件的拓扑球体若被包含在半径为2的开球内，则其包围区域必须包含单位球，并探讨了相关体积问题。

Volume bounds of conic 2-spheres

We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian curvature bound. We also give the geometric models realizing the extremal volume. In particular, when the curvature is bounded in absolute value by $1$, we compute the minimal volume of a conic sphere in the sense of Gromov. In order to apply the level set analysis and iso-perimetric inequality as in our previous works, we develop some new analytical tools to treat regions with vanishing curvature.