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Publication# Quasiconformal Embeddings of Y-Pieces

Abstract

In this paper we construct quasiconformal embeddings from Y-pieces that contain a short boundary geodesic into degenerate ones. These results are used in a companion paper to study the Jacobian tori of Riemann surfaces that contain small simple closed geodesics.

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Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Rieman

Paper

Paper is a thin sheet material produced by mechanically or chemically processing cellulose fibres derived from wood, rags, grasses, or other vegetable sources in water, draining the water through a fi

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a gr

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To any compact Riemann surface of genus g one may assign a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus and we call a Gram matrix of the lattice of a Jacobian a period Gram matrix. The aim of this thesis is to contribute to the Schottky problem, which is to discern the Jacobians among the PPAVs. Buser and Sarnak approached this problem by means of a geometric invariant, the first successive minimum. They showed that the square of the first successive minimum, the squared norm of the shortest non-zero vector, in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. The main goal of this work was to improve this result and to get insight into the connection between the geometry of a compact Riemann surface that is given in hyperbolic geometric terms, and the geometry of its Jacobian. We show the following general findings: For a hyperelliptic surface the first successive minimum is bounded from above by a universal constant. The square of the second successive minimum of the Jacobian of a Riemann surface of genus g is equally of order log(g). We provide refined upper bounds on the consecutive successive minima if the surface contains several disjoint small simple closed geodesics and a lower bound for the norm of certain lattice vectors of the Jacobian, if the surface contains small non-separating simple closed geodesics. If the concrete geometry of the Riemann surface is known, more precise statements can be made. In this case we obtain theoretical and practical estimates on all entries of the period Gram matrix. Here we establish upper and lower bounds based on the geometry of the cut locus of simple closed geodesics and also on the geometry of Q-pieces. In addition the following two results have been obtained: First, an improved lower bound for the maximum value of the norm of the shortest non-zero lattice vector among all PPAVs in even dimensions. This follows from an averaging method from the geometry of numbers applied to a family of symmetric PPAVs. Second, a new proof for a lower bound on the number of homotopically distinct geodesic loops, whose length is smaller than a fixed constant. This lower bound applies not only to geodesic loops on Riemann surfaces, but on arbitrary manifolds of non-positive curvature.