ON CURVES AND TUBES IN $\mathbb{R}^n$

子黎曼几何中的管状领域以及海森堡群中曲线的韦伊不变性结果

Tubes in sub-Riemannian geometry and a Weyl's invariance result for curves in the Heisenberg groups

本文研究了子黎曼结构中子流形的距离正则性，管状邻域体积渐进行为，以及海森堡群中曲线的韦伊不变性，证明了管体积仅依赖于Reeb角。

Geometrically induced discrete spectrum in curved tubes

The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating with respect to the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincides with the spectrum of the straight tube of the same cross-section and that the discrete spectrum is not empty.

Semi-Riemannian hypersurfaces in $\mathbb{L}^{n+1}$ with a totally geodesic foliation of codimension one

We classify hypersurfaces of the Minkowski space $Ł^{n+1}$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or three dimensional strip. Moreover, if the hypersurface is embedded then it is a partial tube over a curve.

齐次空间中不变常平均曲率管

Invariant constant mean curvature tubes in homogeneous spaces

研究齐次$\mathbb{E}(κ,τ)$-空间中螺旋运动下不变常平均曲率管族的整体几何，证明嵌入性与叶化结果，并数值分析紧致情形的等周剖面。

D'Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders

B. Csikós and M. Horváth proved that if a connected Riemannian manifold of dimension at least $4$ is harmonic, then the total scalar curvatures of tubes of small radius about a regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about geodesic segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D'Atri sp…

关于具有整体Grauert管的Zoll流形的一个注记

A Note on Zoll Manifolds with Entire Grauert Tubes

本文计算了具有整体Grauert管的Zoll流形切丛的代数指标，并证明了此类流形中$\u211d\mathbb{HP}^2$型必与标准$\u211d\mathbb{HP}^2$等距。

非平坦四元数空间形式中的2型曲率适应超曲面

Curvature-adapted hypersurfaces of 2-type in non-flat quaternionic space forms

本文分类了非平坦四元数空间形式中在四元-Hermitian矩阵空间内的Chen 2型曲率适应实超曲面，确定了射影与双曲空间中的具体几何类型，并判别了其质量对称性与极小性。

管状域上正曲率凯勒度量及其在最优传输中的应用

Positively curved Kähler metrics on tube domains and their applications to optimal transport

本文研究管状域上具有 $O(n) \times \mathbb{R}^n$ 对称性的凯勒流形，证明非负双截曲率条件下的唯一性结果，刻画非负正交反双截曲率的充要条件，并探讨其在最优传输中合成曲率界的应用。

Minimal Surfaces in $\mathbb{H}_2\times \mathbb{R}$: Nonfillable Curves

We study the asymptotic Plateau problem in $\mathbb{H}_2\times \mathbb{R}$. We give the first examples of non-fillable finite curves with no thin tail in the asymptotic cylinder. Furthermore, we study the fillability question for infinite curves in the asymptotic boundary.

A characterization of involutes and evolutes of a given curve in $\mathbb{E}^{n}$

The orthogonal trajectories of the first tangents of the curve are called the involutes of $x$. The hyperspheres which have higher order contact with a curve $x$ are known osculating hyperspheres of $x$. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve $x$ in $n$-dimensional Euclidean space $\mathbb{E}^{n}$. In the present study, we give a characterization of involute curves of order $k$ (resp. evolute curves) of the given curve $x$ in $n$-dimensional Euclidean space $\mathbb{E}^{n}$. Further, we obtain some results on these type of cur…

Legendrian curve shortening flow in $\mathbb{R}^{3}$

Motivated by Legendrian curve shortening flows in $\mathbb{R}^{3}$, we study the curve shortening flow of figure-eight curves in the plane. We show that, under some symmetry and curvature conditions, a figure-eight curve will shrink to a point at the first singular time.

Area-minimizing properties of Pansu spheres in the sub-riemannian $3$-sphere

We consider the sub-Riemannian $3$-sphere $(\mathbb{S}^3,g_h)$ obtained by restriction of the Riemannian metric of constant curvature $1$ to the planar distribution orthogonal to the vertical Hopf vector field. It is known that $(\mathbb{S}^3,g_h)$ contains a family of spherical surfaces $\{\mathcal{S}_λ\}_{λ\geq 0}$ with constant mean curvature $λ$. In this work we first prove that the two closed half-spheres of $\mathcal{S}_0$ with boundary $C_0=\{0\}\times\mathbb{S}^1$ minimize the sub-Riemannian area among compact $C^1$ surfaces with the same boundary. We also see that the only $C^2$ solut…

Nonvanishing of Cartan CR curvature on boundaries of Grauert tubes around hyperbolic surfaces

We show that the boundaries of thin strongly pseudoconvex Grauert tubes, with respect to the Guillemin-Stenzel Kähler metric canonically associated with the Poincaré metric on closed hyperbolic real-analytic surfaces, has nowhere vanishing Cartan CR-curvature. This result provides a wealth of examples of compact $3$-dimensional Levi nondegenerate CR manifolds having no CR-umbilical point. We provide two proofs utilizing two recent formulas for determining the Cartan CR-curvature of any local $\mathcal{C}^6$-smooth hypersurfaces in $\mathbb{C}^2$. One was obtained in 2012 by the second named au…

实解析映射芽的 $\rho$-管和 $\rho$-球上的 Milnor 纤维化

Milnor fibrations on rho-tubes and rho-spheres for real analytic map germs

本文研究实解析映射芽在替换欧氏距离为解析控制函数后的Milnor纤维化，给出管和球纤维化的判据，并通过例子表明控制函数的选择可决定纤维化是否存在。

A simple closed curve in $\mathbb{R}^3$ whose convex hull equals the half-sum of the curve with itself

If $Γ$ is the range of a Jordan curve that bounds a convex set in $\mathbb{R}^2,$ then $\frac{1}{2}(Γ+Γ)=\mathsf{co}(Γ),$ where $+$ is the Minkowski sum and $\mathsf{co}$ is the convex hull. Answering a question of V.N. Ushakov, we construct a simple closed curve in $\mathbb{R}^3$ with range $Γ$ such that $\frac{1}{2}(Γ+Γ)=[0,1]^3=\mathsf{co}(Γ).$ Also we show that such simple closed curve cannot be rectifiable.

Complete embedded complex curves in the ball of $\mathbb{C}^2$ can have any topology

In this paper we prove that the unit ball $\mathbb{B}$ of $\mathbb{C}^2$ admits complete properly embedded complex curves of any given topological type. Moreover, we provide examples containing any given closed discrete subset of $\mathbb{B}$.

Injectivity Criteria for Holomorphic Curves in $\mathbb{C}^n$

Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves $ϕ:\mathbb{D}\to\mathbb{C}^n$. The result can be considered a generalization of a classical condition for univalence of Nehari.