Joel Hass, Peter Scott
We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of minimal configurations not realized by geodesics in any hyperbolic metric.
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We prove by an algebraic method that the embedding of the Teichmuller space in the space of geodesic currents is totally linearly independent. We prove a similar result for all negatively curved surfaces using an ergodic argument.
We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curves along the two geodesics are not identical.
We obtain simple characterizations of the connected components of the space of closed curves on the 2-sphere whose geodesic curvatures are constrained to lie in an open interval $(κ_1,κ_2)$, in terms of $κ_1$ and $κ_2$. Many results concerning the topology of these spaces are established. In particular, we determine the homeomorphism class of some of these components.
Totally Geodesic Submanifolds in Products of Non-Positively Curved Manifolds
研究非正曲率闭 manifold M 和 M×M 中满足正交条件的 n 维完全测地子流形，证明在 M 满足一定不可约性条件下，若 M×M 存在无限多个此类子流形或单个致密子流形，则 M 是局部对称空间。
Let $S$ be a closed Riemann surface of genus $p>1$ with one point removed. In this paper, we identify those point-pushing pseudo-Anosov maps on $S$ that preserve at least one bi-infinite geodesic in the curve complex.
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.
Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus $n$ with $k$ punctures for any $n$ and $k$.
This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.
We provide a combinatorial condition characterizing curves that are short along a Teichmueller geodesic. This condition is closely related to the condition provided by Minsky for curves in a hyperbolic 3-manifold to be short. We show that short curves in a hyperbolic manifold homeomorphic to S x R are also short in the corresponding Teichmueller geodesic, and we provide examples demonstrating that the converse is not true.
Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichmüller spaces, and hyperbolic 3-manifolds. In the same paper, they showed that there are at least one and at most finitely many tight geodesics between any two vertices in the curve complex. Bowditch found a uniform finiteness property on tight geodesics [9], and this property has given further important applications in some of the above studies. In this paper, we introduce classes of geodesics which are not tight but still …
We give an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, our approach is new, simple, and more effective for all distances accessible by computer. Our method gives a new preferred finite set of geodesics between any two vertices of the complex, called efficient geodesics, which are different from the tight geodesics introduced by Masur and Minsky.
For the complex of curves of a closed orientable surface of genus $g$, $\mathcal{C}(S_{g>1})$, the notion of efficient geodesic in was introduced in arXiv:1408.4133. There it was established that there always exists (finitely many) efficient geodesics between any two vertices, $ v_α , v_β \in \mathcal{C}(S_g)$, representing homotopy classes of simple closed curves, $α, β\subset S_g$. The main tool for used in establishing the existence of efficient geodesic was a dot graph, a booking scheme for recording the intersection pattern of a reference arc, $γ\subset S_g$, with the simple closed curves…
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting of geodesics. Furthermore, the method gives inequalities with a uniform constant depending only on the topology of the surface.
The Restriction of Efficient Geodesics to the Non-separating Complex of Curves
本文证明高效测地线算法可限制于非分离曲线复形，并通过点图与手术分析说明任意测地线都能转化为内部顶点均为非分离曲线的高效测地线。
We parametrize the commensurability classes of curves on Shimura surfaces that are totally geodesic, i.e., the commensurability classes of so-called $\mathbb{C}$-Fuchsian subgroups. In particular, if a Shimura surface contains one commensurability class of totally geodesic curves, it contains infinitely many.
We introduce a method for constructing Weil-Petersson (WP) geodesics with certain behavior in the Teichmüller space. This allows us to study the itinerary of geodesics among the strata of the WP completion and its relation to subsurface projection coefficients of their end invariants. As an application we demonstrate the disparity between short curves in the universal curve over a WP geodesic and those of the associated hyperbolic $3$-manifold.
Let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group $F_n$. It acts properly on the outer space $X_n$ of marked metric graphs, which is a finite-dimensional infinite simplicial complex with some simplicial faces missing. In this paper, we construct complete geodesic metrics and complete piecewise smooth Riemannian metrics on $X_n$ which are invariant under $\mathrm{Out}(F_n)$. One key ingredient is the identification of metric graphs with tropical curves and the use of the tropical Jacobian map from the moduli space of tropical curves to the moduli space of principally pol…
We give a uniform bound of the bounded geodesic image theorem for the closed oriented surfaces. The proof utilizes the bicorn curves introduced by Przytycki and Sisto (see arXiv:1502.02176). With the uniformly bounded Hausdorff distance of the bicorn paths and 1-slimness of the bicorn curve triangles, we are able to show the bound is 44 for both non-annular and annular subsurfaces. In a particular case when the distance between a geodesic and an essential boundary component of subsurface (or core if it is annular) is $\geq 18$, then the bound can be as small as 3, which is comparable to the bo…
The minima of the geodesic length functions of uniform filling curves
本研究利用填充闭定向曲面的曲线与儿童画的联系，明确了均匀填充曲线测地长度函数的极小值，证明其在相关儿童画确定的Grothendieck-Belyi曲面上达到，且均为数域上的黎曼曲面。
It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting Poisson line process.
Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two different metrics on a surface have no geodesic arcs in common, but in special cases the surface is decomposed into polygons geodesic for both the flat and the hyperbolic metric. This is the case for certain surfaces which are translation and half-turn tiled by an Euclidean rectangle. We explore them in this paper. Their multi-geodesic tessellation provides a…
Let $g_t$ be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold $M$ starting at the hyperbolic metric. We construct foliations of the Grassmann bundle $Gr_2(M)$ of tangent 2-planes whose leaves are (lifts of) minimal surfaces in $(M,g_t)$. These foliations are deformations of the foliation of $Gr_2(M)$ by (lifts of) totally geodesic planes projected down from the universal cover $\mathbb{H}^3$. Our construction continues to work as long as the sum of the squares of the principal curvatures of the (projections to $M$) of the leaves remains pointwise small…
We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length at most $L$ grows exponentially in $L$. We get exponentially tighter bounds given weak conditions on self-intersection number.
On the spectrum of the number of geodesics and tight geodesics in the curve complex
本文研究曲面曲线复形中测地线与紧测地线数量的谱，证明了两者在一般情况下的包含关系及长度为2时的相等性，并证明该谱完全由曲面类型$(g,n)$确定。
In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely determines the curve. For an orientable surface $S$ of negative Euler characteristic, we extend the known result that simple curves have this property to curves with self-intersection number one (with one exceptional case on closed surfaces of genus two that we describe completely), while for hyperbolizable 3-manifolds, we show that curves freely homotopic to…
The curve graphs are not locally finite. In this paper, we show that the curve graphs satisfy a property which is equivalent to graphs being uniformly locally finite via Masur--Minsky's subsurface projections. As a direct application of this study, we show that there exist computable bounds for Bowditch's slices on tight geodesics, which depend only on the surface. As an extension of this application, we define a new class of geodesics, weak tight geodesics, and we also obtain a computable finiteness statement on the cardinalities of the slices on weak tight geodesics.
We give a combinatorial proof, using the hyperbolicity of the curve graphs, of the bounded geodesic image theorem of Masur and Minsky. Recently it has been shown that curve graphs are uniformly hyperbolic, thus a universal bound can be given for the diameter of the geodesic image. We also generalize the theorem for projections to markings of the whole surface.
The mapping class group of a surface $§$ acts on the set of closed geodesics on $§$. This action preserves self-intersection number. In this paper, we count the orbits of curves with at most $K$ self-intersections, for each $K \geq 1$. (The case when $K=0$ is already known.) We also restrict our count to those orbits that contain geodesics of length at most $L$, for each $L >0$. This result complements a recent result of Mirzakhani, which gives the asymptotic growth of the number of closed geodesics of length at most $L$ in a single mapping class group orbit. Furthermore, we develop a new, com…
We study the geodesic complexity of the ordered and unordered configuration spaces of graphs in both the $\ell_1$ and $\ell_2$ metrics. We determine the geodesic complexity of the ordered two-point $\varepsilon$-configuration space of any star graph in both the $\ell_1$ and $\ell_2$ metrics and of the unordered two-point configuration space of any tree in the $\ell_1$ metric, by finding explicit geodesics from any pair to any other pair, and arranging them into a minimal number of continuously-varying families. In each case the geodesic complexity matches the known value of the topological com…
Let $Σ$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $Σ$. A geodesic curve $γ$ in $Σ$ is filling, if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word in some generators of $π_1(Σ)$, is filling. In the process, we find an explicit bound for the combinatorial length of a curve given by its Dehn-Thurston coordinate, in terms of the hyperbolic length. This gives us an efficient method for producing a collection which is guaranteed…
Given a filling primitive geodesic curve in a closed hyperbolic surface one obtains a hyperbolic three-manifold as the complement of the curve's canonical lift to the projective tangent bundle. In this paper we give the first known lower bound for the volume of these manifolds in terms of the length of generic curves. We show that estimating the volume from below can be reduced to a counting problem in the unit tangent bundle and solve it by applying an exponential multiple mixing result for the geodesic flow.
Super efficiency of efficient geodesics in the complex of curves
本文证明曲线复形中高效测地线具有“超高效性”，即顶点候选列表的界独立于距离且仅依赖于亏格，其证明依赖于新的相交增长不等式与点图分析。
Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.
Veering branched surfaces, surgeries, and geodesic flows
本文引入转向分支曲面作为研究转向三角剖分的对偶方法，通过手术操作显式构造了对应于负曲率曲面测地流的转向分支曲面，并给出了测地流的显式Markov分割及钻孔单位切丛的分类。
Let $Υ$ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $Υ$ of length $\leq L$, choose one, say $γ_{L}$, at random and let $N (γ_{L})$ be the number of its self-intersections. It is known that there is a positive constant $κ$ depending on the metric such that $N (γ_{L})/L^{2} \rightarrow κ$ in probability as $L\rightarrow \infty$. The main results of this paper concern the size of typical fluctuations of $N (γ_{L})$ about $κL^{2}$. It is proved that if the metric has constant curvature -1 then typical fluctuations are of order $L$, in particular, $(N (…
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