arXiv research

Surprising circles found in Coxeter group boundaries.

Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.

The paper solves circle packings on surfaces with boundaries.

The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.

The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.

Fix two parallel circles in $\mathbb{R}^3$ centered about a common axis. Among surfaces of revolution immersed in $\mathbb{R}^3$ whose boundary is given by these circles, there is one which maximizes the first Dirichlet eigenvalue. If the circles are sufficiently close together, then this surface is unique.

The paper explores universal circles for Anosov foliations and their uniqueness.

Study shows universal circle isomorphic to flow space ideal boundary.

Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to t…

We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a byproduct, we get that any circle decomposition of a surface is upper semicontinuou…

Slim curves on 3-sphere help spherical CR uniformizations.

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if $T$ is a Riemannian 2-torus with boundary in $\mathbb R ^n$, such that the boundary curve is a standard unit circle, then the length o…

We proof that having boundary of standard 3-dimensional simplex $\partial Δ^3$ as a base of triangulation one can triangulate only trivial and Hopf circle bundles.

Constructs minimal surfaces near the boundary of a ball.

Study shows how a curve shortens to a half-circle under specific flow.

Classifies ancient flows in a disc with boundary.

In this paper we investigate free boundary minimal surfaces in the unit ball in Euclidean 3-space, and by using holomorphic techniques we prove that intersection curves of free boundary minimal surfaces with the unit sphere are all circles.

The paper finds circle packings with specific curvatures in hyperbolic geometry.

Formula calculates surface torsion via pants decompositions.

The boundary of a Möbius manifold carries a canonical Möbius structure. This enables one to define the cobordism group of $n$-dimensional (closed) Möbius manifolds. The purpose of this note is to show that the cobordism group of Möbius circles is zero, i.e., every Möbius circle bounds a Möbius surface. We also complete…

The Riemannian hemisphere has a lower bound for its mass.

An alternate proof shows how foliation extensions work in 3D spaces.

A ``hyperideal circle pattern'' in $S^2$ is a finite family of oriented circles, similar to the ``usual'' circle patterns but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patterns are directly related to hyperideal hyperbolic polyhedra, and also to circle packings. …

Study shows universal circles for taut foliations in 3-manifolds.

We specify exactly which groups can act geometrically on CAT(0) spaces whose visual boundary is homeomorphic to either a circle or a suspension of a Cantor set.

Study of harmonic maps to the circle with applications to hyperbolic 3-manifolds.

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary cir…

Two optimization problems for Loewner energy curves and their symmetries.

R-circles in general three dimensional CR manifolds (of contact type) are the analogues to traces of Lagrangian totally geodesic planes on the sphere viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain legendrian curves on the manifold. We prove that a diffeomorphism betwee…

Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.

Reciprocity laws for line bundles on circle fibrations over complex manifolds.

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain latti…

A $T$-Schottky group is a discrete group of Möbius transformations whose generators identify pairs of, possibly-tangent, Jordan curves on the complex sphere, ${\hat{\IC}}$. If the curves are Euclidean circles then the group is termed classical $T$-Schottky. We describe the boundary of the space of classical $T$-Schottk…

Study infinite circle patterns in the Weil-Petersson class using discrete harmonic functions.

Study angle structures on 3-manifolds, linking to representation theory.

This paper computes the quantization of the moduli space of flat SO(3)-bundles over an oriented surface with boundary, with prescribed holonomies around the boundary circles. The result agrees with the generalized Verlinde formula conjectured by Fuchs and Schweigert.

Geometrically constructs Virasoro-Bott group from circle diffeomorphisms.

We study of the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of {\it a priori} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of the area and the height in terms of the boundary. In case that the boundary is a …

Method constructs spirals with given tangents and curvatures.

Let t_a be the Dehn twist about a circle a on an orientable surface. It is well known that for each circle b and an integer n, I(t_a^n(b),b)=|n|I(a,b)^2, where I(,) is the geometric intersection number. We prove a similar formula for circles on nonorientable surfaces. As a corollary we prove some algebraic properties o…

Study complex deformations of the circle using group cohomology and Virasoro algebra.

Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.

Study proves existence and nonexistence for annular surfaces with specific curvature and boundary.

We will prove a Moser-type theorem for self-dual harmonic 2-forms on closed 4-manifolds, and use it to classify local forms on neighborhoods of singular circles on which the 2-form vanishes. Removing neighborhoods of the circles, we obtain a symplectic manifold with contact boundary - we show that the contact form on e…

Stable actions of hyperbolic groups on their boundaries.

Study of laminations for pseudo-Anosov flows on three-manifolds.

Let $\cT$ be Teichmüller space of a closed surface of genus at least 2. For any point $c\in \cT$, we describe an action of the circle on $\cT\times \cT$, which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of $\cT$. This circle action shares some of the …

We prove that a H-surface M in H^2xR, |H| <= 1/2, inherits the symmetries of its boundary when the boundary is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to πFurthermore we prove that the asymptotic boun…