极长距离的非凸性

Fundamental groups, slalom curves and extremal length

We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for this invariant. The bounds differ by a multiplicative constant. The main motivation comes from $3$-braid invariants and their application.

关于双曲度量的极值长度

On the extremal length of the hyperbolic metric

本文证明了双曲黎曼面的极值长度仅由其拓扑决定，并给出了面积为一的极值度量直径的上界。

Convexity of balls in the outer space

In this paper we study the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop $α$, the length of $α$ along a balanced folding path is not larger than the maximum of its lengths at the end points. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counterexamples.

Extremal length functions are log-plurisubharmonic

In this paper, we show that the extremal length functions on Teichmüller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined by the Teichmüller distance from a reference point is plurisubharmonic, and the Teichmüller space is hyperconvex. To show the log-plurisubharmonicity, we give an explicit formula of the Levi form of the extremal length functions in generic case.

Bounded Outdegree and Extremal Length on Discrete Riemann Surfaces

Let $T$ be a triangulation of a Riemann surface. We show that the 1-skeleton of $T$ may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed from $T$ by attaching new edges and vertices and subdividing its faces. Such refinements provide a mechanism of convergence of the discrete triangulation to the classical surface. We will prove a bound on the distortion of the discrete extremal lengths of path families on $T$ under the refinement process. Our bound will depend only on the refinement and not …

Teichmüller 空间中极值长度的渐进行为

The asymptoticity of extremal length in Teichmüller space

研究Teichmüller空间中极值长度的渐进行为，推导极限表达式及Teichmüller距离公式，并确定渐近射线的条件。

比较最短曲线的双曲长度与极值长度

Comparing hyperbolic and extremal lengths for shortest curves

本文通过给出短部分裤子分解的领口宽度下界，进而利用可压缩曲线的双曲长度推导出 Schottky 流形重整化体积的上界。

Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube

Given a triangulation of a closed topological cube, we show that (under some technical condition) there is an essentially unique tiling of a rectangular parallelepiped by cubes, indexed by the vertices of the triangulation. Moreover, i - the combinatorics is preserved, and ii- the boundary is preserved: vertices corresponding to the cubes at the corners of the rectangular parallelepiped are at the corners of the topological cube. Also, the sizes of the cubes are obtained as a solution of a variational problem which is a discrete version of the notion of extremal length in three dimensional Euc…

Teichmüller测地线沿能量函数的拟凸性

Quasi-convexity of energy functions along Teichmüller geodesics

本文将双曲长度函数沿Teichmüller测地线的拟凸性推广到调和映射能量函数的两种情形，证明了相关拟凸性并推广了Masur的渐近增长结果，推导了能量函数的变分公式。

顶点穿孔立方体的极值长度短程线

The extremal length systole of the cube punctured at its vertices

研究顶点穿孔立方体的极值长度短程线，通过分支覆盖、椭圆积分等方法证明其由围绕边的12条曲线实现，并给出二次微分特征以高精度估计其值。

Extremal length boundary of Teichmüller space contains non-Busemann points

In this paper, we shall show that the metric boundary of the Teichmueller space with respect to the Teichmueller distance contains non-Busemann points when the complex dimension of the Teichmueller space is at least two.

Unification of extremal length geometry on Teichmuller space via intersection number

In this paper, we give a framework for the study of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur. There is a natural compactification using extremal length geometry introduced by Gardiner and Masur. The compactification is realized in a certain projective space. We develop the extremal length geometry in the cone which is defined as the inverse image of the compactification via the quotient mapping. The compactification is identified with a subset of the cone by taking an appropriate lift. The cone contains canonically the space of measured fol…

Length of a curve is quasi-convex along a Teichmuller geodesic

We show that for every simple closed curve α, the extremal length and the hyperbolic length of αare quasi-convex functions along any Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space equipped with the Teichmuller metric, balls are quasi- convex.

Turn graphs and extremal surfaces in free groups

This note provides an alternate account of Calegari's rationality theorem for stable commutator length in free groups.

幽灵多边形、泊松括号与凸性

Ghost polygons, Poisson bracket and convexity

本文研究半单群中曲面群阿诺索夫表示模空间的长度与关联函数，利用幽灵代数的幽灵多边形和括号推导泊松括号公式，证明该函数集在泊松括号下稳定，应用于证明长度函数凸性及展示交换函数子代数。

曲线复形上的有向图 I：双曲长度与极值长度

Oriented graphs on curve complex I: hyperbolic and extremal length

本文研究闭曲面曲线复形上由函数诱导的有向图，引入 Dehn 拟位似函数，证明不同正函数诱导不同有向图，并由此得到仅凭不相交简单闭曲线的长度比较即可确定双曲度量的新刚性结果。