Bi-quartic parametric polynomial minimal surfaces

四次极小超曲面的Morse指标

The Morse index of quartic minimal hypersurfaces

针对$g=4$或$6$的$S^n$齐性极小超曲面Morse指标与零化度的未解问题，论文计算了两个$g=4$的此类超曲面的对应量，还发现其Laplace谱含不可根式表示的无理特征值。

Holomorphic differentials and Laguerre deformation of surfaces

A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their Laguerre Gauss maps, the T-transforms of L-minimal isothermic surfaces have constant mean curvature in some translate of hyperbolic 3-space or de Sitter 3-space in Minkowski 4-space, or have mean curvature zero in some translate of a time-oriented lightcone in Minkowski 4-space. As an application, we show that various instances of the La…

Segre quartic surfaces and minitwistor spaces

Segre surfaces in the title mean quartic surfaces in $\mathbb{CP}^4$ which are the images of weak del Pezzo surfaces of degree four under the anti-canonical map. We first show that minimal minitwistor spaces with genus one are exactly Segre quartic surfaces. By a kind of Penrose correspondence, Zariski open subsets of the projective dual varieties of these surfaces admit Einstein-Weyl structure. We investigate structures of these dual varieties in detail. In particular, we determine the degrees of these varieties (namely the classes of the Segre surfaces), as well as structure of several compo…

Parametric polynomial minimal surfaces of arbitrary degree

Weierstrass representation is a classical parameterization of minimal surfaces. However, two functions should be specified to construct the parametric form in Weierestrass representation. In this paper, we propose an explicit parametric form for a class of parametric polynomial minimal surfaces of arbitrary degree. It includes the classical Enneper surface for cubic case. The proposed minimal surfaces also have some interesting properties such as symmetry, containing straight lines and self-intersections. According to the shape properties, the proposed minimal surface can be classified into fo…

完备极小曲面的一个强参数 h-原理

A strong parametric h-principle for complete minimal surfaces

本文证明开 Riemann 曲面到 $\mathbb R^n$ 的完备非平坦共形极小浸入满足强参数 h-原理，并建立其与连续映射空间的同伦型关系。

Laplacian 代数与极小超锥体的子校准

Laplacian Algebras and Subcalibrations for Minimal Hypercones

本文提出基于 Laplacian 代数的代数机制，通过闭包恒等式和代数不等式构造极小超锥体并判定其面积极小性、严格稳定性，应用于等参超锥体及复行列式实部零集，并给出奇次刚性与回拉判据。