arXiv subjects
Publications and source records attributed to Charalampos Charitos.
A new direct proof of the Virtual Haken Conjecture, which asserts that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group has a finite cover that is Haken, will be given.
In this work we describe a class of subsets of the Euclidean plane which, with the induced length metric, are locally $CAT(0)$ spaces and we show that the gluing of two such subsets along a piece of their boundary is again a locally $CAT(0)$ space provided that the sum of the signed curvatures at every gluing point is non-positive. A generalization to subsets of smooth Riemannian surfaces of curvature $k\leq 0$ is given.
For a family $\mathcal{C}$ of properly embedded curves in the 2-dimensional disk $\mathbb{D}^{2}$ satisfying certain uniqueness properties, we consider convex polygons $P\subset \mathbb{D}^{2}$ and define a metric $d$ on $P$ such that $(P,d)$ is a geodesically complete metric space whose geodesics are precisely the curves $\left\{ c\cap P\bigm\vert c\in \mathcal{C}\right\}.$ Moreover, in the special case $\mathcal{C}$ consists of all Euclidean lines, it is shown that $P$ with this new metric is not isometric to any convex domain in $\mathbb{R} ^{2}$ equipped with its Hilbert metric. We generalize this construction to certain classes of uniquely geodesic metric spaces homeomorphic to $\mathbb{R}^{2}.$
A new metric on the open 2-dimensional unit disk is defined making it a geodesically complete metric space whose geodesic lines are precisely the Euclidean straight lines. Moreover, it is shown that the unit disk with this new metric is not isometric to any hyperbolic model of constant negative curvature, nor to any convex domain in R2 equipped with its Hilbert metric.
Joseph-Nicolas Delisle was one of the most important scientists at the Saint Petersburg Academy of Sciences during the first period when Euler was working there. Euler was helping him in his work on astronomy and in geography. In this paper, Delisle's geographical projection is presented and Euler's study of this projection isexplained, highlighting some important mathematical points, in particular on the metric geometry of surfaces.The final version of this paper will appear in the book \emph{Mathematical Geography in the Eighteenth Century: Euler, Lagrange and Lambert}, ed. Renzo Caddeo and Athanase Papadopoulos, Springer International Publishing, 2022.
Morse foliations of codimension one on the sphere S^3 are studied and the existence of special components for these foliations is derived. As a corollary the instability of Morse foliations can be proven in almost all cases.
The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.
For a handlebody of genus $g\geq6$ it is shown that every automorphism of the complex of separating meridians can be extended to an automorphism on the complex of all meridians and, in consequence, it is geometric.
If $Ω$ is the interior of a convex polygon in $\mathbb{R}^{2}$ and $f,g$ two asymptotic geodesics, we show that the distance function $d\left(f\left(t\right),g\left(t\right)\right)$ is convex for $t$ sufficiently large. The same result is obtained in the case $\partial Ω$ is of class $C^{2}$ and the curvature of $\partial Ω$ at the point $f\left(\infty\right)=g\left(\infty\right) $ does not vanish. An example is provided for the necessity of the curvature assumption.
Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen-Margulis' measure finiteness assumption used in recent work of Ricks is removed. We also construct examples of CAT(0) spaces which do not admit finite Bowen-Margulis measure.
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points. Dynamical properties for the space of geodesics are also proved.
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.
We prove that for any orientable connected surface of finite type which is not a a sphere with at most four punctures or a torus with at most two punctures, any homeomorphism of the space of geodesic laminations of this surface, equipped with the Thurston topology, is induced by a homeomorphism of the surface.
For a genus g handlebody H a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H, is constructed and several properties are established. In particular, this complex naturally contains, as a subcomplex, the complex of curves of the boundary surface of H. As in the classical theory, the group of automorphisms of this complex is identified with the mapping class group of the handlebody.
Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a space $\mathcal{T}$ of complete hyperbolic metrics on $N$ with cone singularities along the edges of the tetrahedra. We prove that $\mathcal{T}$ is homeomorphic to a Euclidean space and we compute its dimension. By means of examples, we examine if the elements of $% \mathcal{T}$ are uniquely determined by the angles around the edges of $N.$
If M is a manifold with compressible boundary, we analyze essential disks in M, as well as incompressible, but not necessarily boundary incompressible, surfaces in M. We are most interested in the case where M is a handlebody or compression body. The analysis depends on a new normal surface theory. We hope the normal surface theory will be used in other papers to describe objects representing limits of essential disks in a handlebody or a 3-manifold with compressible boundary. For certain automorphisms of handlebodies, these disk limits should serve as invariant objects akin to laminations and analogous to the invariant laminations for pseudo-Anosov automorphisms of surfaces.
A complex of incompressible surfaces in a handlebody is constructed so that it contains, as a subcomplex, the complex of curves of the boundary of the handlebody. For genus 2 handlebodies, the group of automorphisms of this complex is used to characterize the mapping class group of the handlebody. In particular, it is shown that all automorphisms of the complex of incompressible surfaces are geometric, that is, induced by a homeomorphism of the handlebody.