The Curvatures of Regular Curves and Euclidean Invariants of their Derivatives

在Ricci曲率和Lie导数条件下的时空局部正则性

Local regularity of spacetimes under Ricci curvature and Lie derivative conditions

本文研究了4维时空的局部正则性，通过Ricci曲率和Lorentzian度量关于任意类时向量场的Lie导数的先验界，推导出一般的正则性估计。

Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds

Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an almost-Euclidean upper bound on volumes of geodesic balls, then unit balls in such a space are Gromov-Hausdorff close, and in fact bi-Hölder and bi-$W^{1,p}$ homeomorphic, to Euclidean balls. We prove a compactness and limit space structure theorem under the same assumptions.

Evolution of area-decreasing maps between two-dimensional Euclidean spaces

We consider the mean curvature flow of the graph of a smooth map $f:\mathbb{R}^2\to\mathbb{R}^2$ between two-dimensional Euclidean spaces. If $f$ satisfies an area-decreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map $f_t$. Further, we prove uniform decay estimates for the mean curvature vector of the graph and all higher-order derivatives of the corresponding map $f_t$.

关于Carnot群之间映射的导数等价性

On the equivalence of derivatives for maps between Carnot groups

本文用初等欧几里得方法证明了Carnot群间保持水平曲线且水平方向连续可微的映射必为连续Pansu可微，并重证了Magnani的均值估计。

Polynomial Weingarten Tubular Surfaces in Euclidean, Hyperbolic and Lorentzian 3-spaces

In this article we fully classify regular tubular surfaces in Euclidean, Lorentzian and hyperbolic 3-spaces whose Gaussian and mean curvatures $K$ and $H$ verify a polynomial relation. More precisely, we determine the set $S(Q)$ of all regular tubular surfaces whose curvatures verify a given polynomial relation $Q(K,H)=0$, and the set $Q(S)$ of all polynomial relations vanished by the curvatures of a given tubular surface $S$.

On the Invariant Theory of Weingarten Surfaces in Euclidean Space

We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential equation. We apply these results to the special Weingarten surfaces: minimal surfaces, surfaces of constant mean curvature and surfaces of constant Gauss curvature.

Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space

A Bertrand curve in the 4-dimensional Euclidean space is a space curve whose first normal line is the same as the first normal line of another curve. On the other hand, a Mannheim curve in the 4-dimensional Euclidean space is a space curve whose first normal line is the same as the second or third normal line of another curve. By definitions, another curve is a parallel curve with respect to the direction of the first normal vector. As smooth curves with singular points, we consider framed curves in the Euclidean space. Then we define and investigate Bertrand and Mannheim curves of framed curv…

On regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb R^{n+1}$, $n\geq 2$, defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb R^{n+1}$, $n\geq 2$, defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

投影几何中的照明体

Illumination Bodies in Projective Geometries

本文将照明体的概念推广到常曲率黎曼空间和投影芬斯勒几何，证明其体积导数定义了这些背景下凸体的表面积，推广了欧氏空间中的仿射表面积。

Weierstrass-type Representation of Weakly Regular Pseudospherical Surfaces in Euclidean Space

The paper presents a generalized Weierstrass representation for pseudospherical surfaces in terms of 3x3 matrices, using moving frames and loop group decompositions. The construction of all such surfaces, starting from a given representation, constitutes the subject of a separate, ulterior report. The appendix of the present work gives an elementary proof of the Birkhoff decomposition theorem for "loop-groups" of spectral parameter based on the real line, instead of the unit circle.

An equivalence between pseudo-holomorphic embeddings into almost-complex Euclidean space and CR regular embeddings into complex space

We show that a pseudo-holomorphic embedding of an almost-complex $2n$-manifold into almost-complex $(2n + 2)$-Euclidean space exists if and only if there is a CR regular embedding of the $2n$-manifold into complex $(n + 1)$-space. We remark that the fundamental group does not place any restriction on the existence of either kind of embedding when $n$ is at least three. We give necessary and sufficient conditions in terms of characteristic classes for a closed almost-complex 6-manifold to admit a pseudo-holomorphic embedding into $\R^8$ equipped with an almost-complex structure that need not be…

具有近欧几里得边界数据的新渐近平直静态真空度量

New asymptotically flat static vacuum metrics with near Euclidean boundary data

针对Bartnik近欧几里得边界数据的静态真空延拓猜想，研究改进了此前的静态正则超曲面结果，证明一类光滑单侧超曲面族的开稠子族中任意超曲面均为静态正则，采用了受相关猜想研究启发的新论证。

Minimal Surfaces in the Four-Dimensional Euclidean Space

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for strongly regular minimal surfaces of general type in terms of their invariants. We introduce canonical parameters on strongly regular minimal surfaces of general type and prove that any such a surface is determined up to a motion by two invariant functions satisfying a system of tw…

On the symplectic volume of the moduli space of Spherical and Euclidean polygons

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the Spherical and Euclidean polygons with fixed perimeter the regular one is the most flexible and that among all the spherical polygons the regular one with side-length $π/2$ is the most flexible.