On the number of minimal surfaces with a given boundary

We generalize the following result of White: Suppose $N$ is a compact, strictly convex domain in \RR^3 with smooth boundary. Let $\Sigma$ be a compact 2-manifold with boundary. Then a generic smooth curve $\Gamma\cong \partial\Sigma$ in $\partial N$ bounds an odd or even number of embedded minimal surfaces diffeomorphic to $\Sigma$ according to whether $\Sigma$ is or is not a union of disks. First, we prove that the parity theorem holds for any compact riemannian 3-manifold $N$ such that $N$ is strictly mean convex, $N$ is homeomorphic to a ball, $\partial N$ is smooth, and $N$ contains no closed minimal surfaces. We then further relax the hypotheses by allowing $N$ to be weakly mean convex and to have piecewise smooth boundary. We extend the parity theorem yet further by showing that, under an additional hypothesis, it remains true for minimal surfaces with prescribed symmetries. The parity theorems are used in an essential way to prove the existence of embedded genus-$g$ helicoids in \SS^2\times \RR. We give a very brief outline of this application. (The full argument will appear elsewhere.)Comment: 13 pages Dedicated to Jean Pierre Bourguignon on the occasion of his 60th birthday. One tex 'newcommand' revised because arxiv version had an error. Two illustrations and one proof have been added. May 2009: Abstract, key words, MSC codes added. One typo fixed. Paper has been published in Asterisqu

Discrete conformal maps and ideal hyperbolic polyhedra

We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring M\"obius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to addresses the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and updated, minor changes in exposition. v3, final version: typos corrected, improved exposition, some material moved to appendice

Packings induced by piecewise isometries cannot contain the Arbelos

Copyright © American Institute of Mathematical SciencesPlanar piecewise isometries with convex polygonal atoms that are piecewise irrational rotations can naturally generate a packing of phase space given by periodic cells that are discs. We show that such packings cannot contain certain subpackings of Apollonian packings, namely those belonging to a family of Arbelos subpackings. We do this by showing that the unit complex numbers giving the directions of tangency within such an isometric-generated packing lie in a finitely generated subgroup of the circle group, whereas this is not the case for the Arbelos subpackings. In the opposite direction, we show that, given an arbitrary disc packing of a polygonal region, there is a piecewise isometry whose regular cells approximate the given packing to any specified precision

On conformally flat circle bundles over surfaces

We study surface groups $\Gamma$ in $SO(4,1)$, which is the group of Mobius tranformations of $S^3$, and also the group of isometries of $\mathbb{H}^4$. We consider such $\Gamma$ so that its limit set $\Lambda_\Gamma$ is a quasi-circle in $S^3$, and so that the quotient $(S^3 - \Lambda_\Gamma) / \Gamma$ is a circle bundle over a surface. This circle bundle is said to be conformally flat, and our main goal is to discover how twisted such bundle may be by establishing a bound on its Euler number. By combinatorial approaches, we have two soft bounds in this direction on certain types of nice structures. In this article we also construct new examples, a "grafting" type path in the space of surface group representations into $SO(4,1)$: starting inside the quasi-Fuschsian locus, going through non-discrete territory and back.Comment: 28 pages, 7 figures. Updated from Thesis version: more correct bound of (3/2)n^2, updated exposition in section 3.

Statistics of torus piecewise isometries

By now, we have learned quite well how to study hyperbolic (locally expanding/contracting or both) chaotic dynamical systems, thanks to a large extent to the development of the so called operator approach. Contrary to this almost nothing is known about piecewise isometries, except for a special case of one-dimensional interval exchange mappings. The last case is fundamentally different from the general situation in the presence of an invariant measure (Lebesgue measure), which helps a lot in the analysis. We start by showing that already the restriction of the rotation of the plane to a torus demonstrates a number of rather unexpected properties. Our main results describe sufficient conditions for the existence/absence of invariant measures of torus piecewise isometries. Technically these results are based on the approximation of the maps under study by weakly periodic ones.Comment: 17 page

On the Moduli Space of Singular Euclidean Surfaces

The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces.Comment: To appear in the Handbook of Teichmuller Theory, vol. 1, ed. A. Papadopoulos, European Math. Society Series, 200

Detecting Similarity of Rational Plane Curves

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.Comment: 22 page