An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)

Mapped Null Hypersurfaces and Legendrian Maps

For an $(m+1)$-dimensional space-time $(X^{m+1}, g),$ define a mapped null hypersurface to be a smooth map $ν:N^{m}\to X^{m+1}$ (that is not necessarily an immersion) such that there exists a smooth field of null lines along $ν$ that are both tangent and $g$-orthogonal to $ν.$ We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle $ST^*M$ of an immersed spacelike hypersurface $μ:M^m\to X^{m+1}.$ We show that a Legendrian map $\wt λ: L^{m-1}\to (ST^*M)^{2m-1}$ defines a mapped null hypersurface in $X.$ On the other hand, the intersection of a …

Linking, Legendrian linking and causality

The set N of all null geodesics of a globally hyperbolic (d+1)-dimensional spacetime (M,g) is naturally a smooth (2d-1)-dimensional contact manifold. The sky of an event is the subset of N defined by all null geodesics through that event, and is an embedded Legendrian submanifold of N diffeomorphic to a (d-1)-dimensional sphere. It was conjectured by Low that for d=2 two events are causally related iff their skies are linked (in an appropriate sense). We use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases, and suggest that for d=3 smo…

三维反德西特空间中伪球类空标架曲线的零曲面

Null surfaces of pseudo-spherical spacelike framed curves in the anti-de Sitter 3-space

本研究引入三维反德西特空间中伪球类空标架曲线的零曲面，分类其零锥波前奇点并关联标架曲线奇点，通过定义反德西特距离平方函数从勒让德奇点理论视角解释零锥波前，辅以示例验证结果。

Compact special Legendrian surfaces in $S^5$

A surface $Σ\subset S^5 \subset \mathbb{C}^3$ is called \emph{special Legendrian} if the cone $0 \times Σ\subset \mathbb{C}^3$ is special Lagrangian. The purpose of this paper is to propose a general method toward constructing compact special Legendrian surfaces of high genus. It is proved \emph{there exists a compact, orientable, Hamiltonian stationary Lagrangian surface of genus $1+\frac{k(k-3)}{2}$ in $ \mathbb{C}P^2$ for each integer $ k \geq 3$, which is a smooth branched surface except at most finitely many conical singularities.} If this surface is smooth, it is minimal and the Legendri…

三维反德西特空间中伪球面标架浸入的焦集奇点

Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space

本文在三维反德西特时空中引入伪球面非零标架曲线，利用活动标架定义渐屈线和焦曲面，揭示渐屈线恰为焦曲面的奇点集，并通过勒让德奇点理论将其解释为波前。

Complete minimal surfaces densely lying in arbitrary domains of $\mathbb{R}^n$

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to\mathbb{R}^n$ with $\overline{X(M)}=\mathbb{R}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open topology. Moreover, we show that every domain in $\mathbb{R}^n$ contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface. Our method of p…