Geodesic Disk Research Articles

Teichmüller curves, Galois actions and $ \widehat {GT} $‐relations

AbstractTeichmüller curves are geodesic discs in Teichmüller space that project to algebraic curves C in the moduli space Mg. Some Teichmüller curves can be considered as components of Hurwitz spaces. We show that the absolute Galois group Gℚ acts faithfully on the set of these embedded curves.We also compare the action of Gℚ on π1(C) with the one on π1(Mg) and obtain a relation in the Grothendieck–Teichmüller group, seemingly independent of the known ones. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Schiffer’s Problem and an Isoperimetric Inequality for the First Buckling Eigenvalue of Domains on $$\mathbb{S}^{2}$$

We consider the overdetermined eigenvalue problem on a sufficiently regular connected open domain Ω on the 2-sphere \(\mathbb{S}^2\): $$ \begin{array}{@{}l@{}} \Delta u+\alpha u = 0\quad {\rm in}\quad \Omega,\\ u = {\rm constant},\ \displaystyle\frac{\partial u}{\partial \nu} = {\rm constant\quad on}\quad \partial \Omega, \end{array}$$ where α ≠ 0. We show that if α = 2 and Ω is simply connected then the problem admits a (nonzero) solution if and only if Ω is a geodesic disk. We furthermore extend to domains on \(\mathbb{S}^{2}\) the isoperimetric inequality of Payne–Weinberger for the first buckling eigenvalue of compact planar domains. As a corollary we prove that Ω is a geodesic disk if the above overdetermined eigenvalue problem admits a (nonzero) solution with ∂u/∂ν = 0 on ∂Ω and α = λ2 the second eigenvalue of the Laplacian with Dirichlet boundary condition. This extends a result proved in the case of the Euclidean plane by C. Berenstein.

Geodesic discs in Teichmüller space

LetT(S) be the Teichmuller space of a Riemann surfaceS. By definition, a geodesic disc inT(S) is the image of an isometric embedding of the Poincare disc intoT(S). It is shown in this paper that for any non-Strebel pointτ ∈ T(S), there are infinitely many geodesic discs containing [0] and τ.

Piecewise-smooth surfaces as the union of geodesic disks

We consider a question of G. Fejes Tóth, whether the boundary of a convex body can be the union of three geodesic disks with disjoint interior.

A new shape descriptor for surfaces in 3D images

We introduce a linear shape descriptor for (open) surfaces in 3D images. To extract the shape descriptor, the border of the surface is first identified. Then, the distance transform of the surface is computed, where each voxel in the surface is labelled with the minimum distance to its closest border voxel. On the distance transform, the centres of the maximal geodesic discs (CMGDs) are detected. These voxels are suitably linked to each other by growing paths in the direction of the steepest gradient, to finally obtain the linear shape descriptor of the surface. The shape descriptor can be extracted from any open surface-like object, i.e., an object with thickness at most two-voxel.

On Teichmüller Geometry *

Much work has been done on the geodesic segments and geodesic disks passing two points in infinite dimensional Teichmüller spaces. This paper deals with the uniqueness question of geodesic segments joining two points in the tangent space of an infinite dimensional Teichmüller space and the uniqueness question of geodesic disks in an infinite dimensional Teichmüller space with given tangent vectors.

The isoperimetric inequality for minimal surfaces in a Riemannian manifold

Abstract It is proved that every minimal surface with one or two boundary components in a simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant K satisfies the sharp isoperimetric inequality 4π A ≦ L2 + K A2. Here equality holds if and only if the minimal surface is a geodesic disk in a surface of constant Gaussian curvature K.

Asymptotic estimates for best and stepwise approximation of convex bodies IV

In this article we first prove a stability theorem for coverings in 𝔼 2 by congruent solid circles: if the density of such a covering is close to its lower bound , then most of the centers of the circles are arranged in almost regular hexagonal patterns. A version of this result then is extended to coverings by geodesic discs in two-dimensional Riemannian manifolds. Given a sufficiently differentiable convex body C in 𝔼 3 , the following two problems are closely related: (i) Approximation of C with respect to the Hausdorff metric, the Banach-Mazur distance and a notion of distance due to Schneider by inscribed or circumscribed convex polytopes. (ii) Covering of the boundary of C by geodesic discs with respect to suitable Riemannian metrics. The stability result for Riemannian manifolds and the relation between approximation and covering yield rather precise information on the form of best approximating inscribed convex polytopes P n of C with respect to the Hausdorff metric: if the number n of vertices is large, then most of the vertices are arranged in almost regular hexagonal patterns. Consequently, the majority of facets of P n are almost regular triangles. Here ‘regular’ is meant with respect to the Riemannian metric of the second fundamental form. Similar results hold for circumscribed polytopes and also for the Banach-Mazur distance and Schneider's notion of distance.

Weakly disk-homogeneous spaces

Naturally reductive Riemannian homogeneous spaces and more generally, g.o. spaces, have the property that the volume of a geodesic disk normal to a geodesic and with center on that geodesic remains constant when the center moves along that central geodesic. Riemannian manifolds having that property for arbitrary geodesics and all sufficiently small geodesic disks are called weakly disk-homogeneous. Since, up to local isometries, there are no other examples known for dimension > 2, we investigate whether a possible converse holds or not. Besides some general results, we give a positive answer for three-dimensional and for several classes of four-dimensional manifolds. Some related results are discussed, in particular about four-dimensional Einstein C-spaces.

On the geometry of infinite dimensional Teichmüller spaces

We prove that in any infinite dimensional Teichmuller space, there exists a minimal geodesic lying in two distinct geodesic disks.

Holomorphic families of geodesic discs in infinite dimensional Teichmüller spaces

The theory of quasiconformal mappings plays an important role in Teichmüller theory. The Teichmüller spaces of Riemann surfaces are defined as quotient spaces of the spaces of Beltrami differentials, and the Teichmüller distances are defined to measure quasiconformal deformations between the Riemann surfaces representing points in the Teichmüller spaces. The Teichmüller spaces are complex Banach manifolds equipped with natural complex structures such that the canonical projections are holomorphic. It is known (see Gardinar [4]) that the Teichmüller distance, defined independently of the complex structures, coincides with the Kobayashi distance.In spite of the naturality of the definition of a Teichmüller space as a quotient of Beltrami differentials, for given two Beltrami differentials it is hard to determine whether they are equivalent or not. For this reason, it is not trivial to describe geodesic lines with respect to the Teichmüller-Kobayashi metric.

Positive scalar curvature and periodic fundamental groups

is formed by double contraction of the riemannian curvature tensor of g (compare [He, pages 74-75]). Geometrically speaking, the scalar curvature function measures the difference between the volumes of the riemannian and euclidean geodesic disks. The existence of a riemannian metric with positive scalar curvature on a smooth manifold turns out to be of interest in many contexts. For example, by results of J. Kazdan and F. Warner [KW] the entire question of realizing a smooth function as a scalar curvature reduces to the existence of such a metric, and results of R. Schoen [Schn] on the Yamabe problem show that a metric with positive scalar curvature can be conformally deformed to one with

On the area growth of a hyperbolic surface

We conjecture that if the rate of area growth of a geodesic disc of radius r on a smooth simply-connected complete surface with non-positive Gaussian curvature is faster than r2(logr)1+e for some ε ≥ 0, then the surface is hyperbolic. We prove this under an additional assumption that the surface is rotationally symmetric.

Stochastic Determination of Moduli of Annular Regions and Tori

Let $A = A(r, 1)$ be an annulus $\{z: r < |z| < 1\}$ with the Poincare metric $g$ on $A$. Let $\mathbf{Z} = (Z_t, P_a)$ be a Brownian motion on $A$ corresponding to $g$. If we take a geodesic disc $D$ centered at $c$ in $A$, then the probability $P_a(\exists t, Z_t \in \partial D$ such that $Z_s, 0 < s < t$, winds around the origin in the positive direction) is a function of $r, |c|$, and the radius $\rho$ of $D$. In the present paper we shall calculate the value $S$ of the supremum of these winding probabilities. Then it will turn out that there exists a 1 to 1 correspondence between $S$ and $r$. Noting that $r$ is called the modulus of $A$, we have an explicit formula of moduli of annular regions. Further we shall give an explicit formula of moduli of tori in a similar way.

The volume of geodesic disks in a Riemannian manifold

The volume of geodesic disks in a Riemannian manifold

Bonnesen-Style Isoperimetric Inequalities

(1979). Bonnesen-Style Isoperimetric Inequalities. The American Mathematical Monthly: Vol. 86, No. 1, pp. 1-29.