Related papers: On surfaces in three dimensional contact manifolds

We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author.…

A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal…

This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology…

It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totally geodesic…

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…

The geometry of the Heisenberg group acting on the plane arises naturally in geometric topology as a degeneration of the familiar spaces $\mathbb{S}^2,\mathbb{H}^2$ and $\mathbb{E}^2$ via conjugacy limit as defined by Cooper, Danciger, and…

There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…

This note studies the Burnside problem for homeomorphism groups of compact connected manifolds. For surfaces, we prove that the identity component of the homeomorphism group is torsion-free precisely when the surface is not the sphere,…

Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any…

A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…

For the quantum Heisenberg manifolds, using the action of Heisenberg group we construct a family of spectral triples. It is shown that associated Kasparov module is same for all these spectral triples. Then we show that element is…

In this thesis we analyze a very simple model of two dimensional quantum gravity based on causal dynamical triangulations (CDT). We present an exactly solvable model which indicates that it is possible to incorporate spatial topology…

We classify all surfaces with constant Gaussian curvature $K$ in Euclidean $3$-space that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are real functions of one variable. If $K=0$, we prove…

The Null Surface Formulation of General Relativity is developed for 2+1 dimensional gravity. The geometrical meaning of the metricity condition is analyzed and two approaches to the derivation of the field equations are presented. One…

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's…

We prove a compactness theorem for holomorphic curves in 4-dimensional symplectizations that have embedded projections to the underlying 3-manifold. It strengthens the cylindrical case of the SFT compactness theorem by using intersection…

This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2…

We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a (straight) ruled surface, i.e. it consists of straight lines tangent to a horizontal vector field along a…

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(\kappa,\tau)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg…

Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.