scholarly journals EMBEDDING OF METRIC GRAPHS ON HYPERBOLIC SURFACES

2019 ◽  
Vol 99 (03) ◽  
pp. 508-520
Author(s):  
BIDYUT SANKI
Keyword(s):  
Euler Characteristic ◽  
Isometric Embedding ◽  
Hyperbolic Surface ◽  
Hyperbolic Surfaces ◽  
Metric Graphs ◽  
Metric Graph ◽  
Complementary Region

An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$ , there is an embedding of $(G,d)$ (possibly after a rescaling of $d$ ) on a surface of genus $g$ . Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$ . The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$ , where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.

scholarly journals A metric graph satisfying w41=1$w_4^1 = 1$ that cannot be lifted to a curve satisfying dim⁡ (W41)=1$\dim \;(W_4^1 ) = 1$

2016 ◽  
Vol 14 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Marc Coppens
Keyword(s):  
Linear Systems ◽  
Harmonic Morphism ◽  
Metric Graphs ◽  
Clifford Index ◽  
Special Linear ◽  
Metric Graph ◽  
Index 2

AbstractFor all integers g ≥ 6 we prove the existence of a metric graph G with $w_4^1 = 1$ such that G has Clifford index 2 and there is no tropical modification G′ of G such that there exists a finite harmonic morphism of degree 2 from G′ to a metric graph of genus 1. Those examples show that not all dimension theorems on the space classifying special linear systems for curves have immediate translation to the theory of divisors on metric graphs.

scholarly journals Graphs of systoles on hyperbolic surfaces

2019 ◽  
Vol 11 (01) ◽  
pp. 1-20 ◽  
Author(s):  
Bidyut Sanki ◽  
Siddhartha Gadgil
Keyword(s):  
Planar Graphs ◽  
Classical Result ◽  
Hyperbolic Surface ◽  
Necessary Condition ◽  
Major Result ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Fat Graphs ◽  
Admissible Graph

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal forms a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible).There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first main result is that this condition is also sufficient.It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

scholarly journals Density of half-horocycles on geometrically infinite hyperbolic surfaces

2012 ◽  
Vol 33 (4) ◽  
pp. 1162-1177
Author(s):  
BARBARA SCHAPIRA
Keyword(s):  
Tangent Bundle ◽  
Hyperbolic Surface ◽  
Hyperbolic Surfaces ◽  
Weak Assumption ◽  
Unit Tangent Bundle

AbstractOn the unit tangent bundle of a hyperbolic surface, we study the density of positive orbits $(h^s v)_{s\ge 0}$ under the horocyclic flow. More precisely, given a full orbit $(h^sv)_{s\in {\mathbb R}}$, we prove that under a weak assumption on the vector $v$, both half-orbits $(h^sv)_{s\ge 0}$ and $(h^s v)_{s\le 0}$ are simultaneously dense or not in the non-wandering set $\mathcal {E}$of the horocyclic flow. We give also a counterexample to this result when this assumption is not satisfied.

Harmonic manifolds and embedded surfaces arising from a super regular tesselation

2017 ◽  
Vol 26 (09) ◽  
pp. 1743003
Author(s):  
G. Brumfiel ◽  
H. Hilden ◽  
M. T. Lozano ◽  
J. M. Montesinos ◽  
E. Ramirez ◽  
...  
Keyword(s):  
Euler Characteristic ◽  
Quotient Space ◽  
Hyperbolic Manifolds ◽  
Covering Space ◽  
Hyperbolic Surfaces ◽  
Harmonic Manifolds ◽  
Embedded Surfaces

The main result of this paper is the construction of two Hyperbolic manifolds, [Formula: see text] and [Formula: see text], with several remarkable properties: (1) Every closed orientable [Formula: see text]-manifold is homeomorphic to the quotient space of the action of a group of order [Formula: see text] on some covering space of [Formula: see text] or [Formula: see text]. (2) [Formula: see text] and [Formula: see text] are tesselated by 16 dodecahedra such that the pentagonal faces of the dodecahedra fit together in a certain way. (3) There are 12 closed non-orientable hyperbolic surfaces of Euler characteristic [Formula: see text] each of which is tesselated by regular right angled pentagons and embedded in [Formula: see text] or [Formula: see text]. The union of the pentagonal faces of the tesselating dodecahedra equals the union of the 12 images of the embedded surfaces of Euler characteristic [Formula: see text].

scholarly journals Short closed geodesics with self-intersections

2020 ◽  
Vol 169 (3) ◽  
pp. 623-638
Author(s):  
VIVEKA ERLANDSSON ◽  
HUGO PARLIER
Keyword(s):  
Linear Function ◽  
Hyperbolic Surface ◽  
Minimal Length ◽  
The Self ◽  
Closed Geodesics ◽  
Intersection Numbers ◽  
Hyperbolic Surfaces ◽  
Fixed Integer

AbstractOur main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

SYSTOLIC FILLINGS OF SURFACES

2018 ◽  
Vol 98 (3) ◽  
pp. 502-511
Author(s):  
BIDYUT SANKI
Keyword(s):  
Closed Geodesic ◽  
Disjoint Union ◽  
Hyperbolic Surface ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Simple Closed Geodesics

A filling of a closed hyperbolic surface is a set of simple closed geodesics whose complement is a disjoint union of hyperbolic polygons. The systolic length is the length of a shortest essential closed geodesic on the surface. A geodesic is called systolic, if the systolic length is realised by its length. For every $g\geq 2$, we construct closed hyperbolic surfaces of genus $g$ whose systolic geodesics fill the surfaces with complements consisting of only two components. Finally, we remark that one can deform the surfaces obtained to increase the systole.

scholarly journals The nodal count {0,1,2,3,…} implies the graph is a tree

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120504 ◽  
Author(s):  
Ram Band
Keyword(s):  
Metric Graphs ◽  
Oscillation Theorem ◽  
Nodal Count ◽  
Metric Graph ◽  
Converse Theorems ◽  
Link Type ◽  
Tree Graphs ◽  
Metric Tree ◽  
Magnetic Stability ◽  
Almost All

Sturm's oscillation theorem states that the n th eigenfunction of a Sturm–Liouville operator on the interval has n −1 zeros (nodes) (Sturm 1836 J. Math. Pures Appl. 1 , 106–186; 373–444). This result was generalized for all metric tree graphs (Pokornyĭ et al. 1996 Mat. Zametki 60 , 468–470 ( doi:10.1007/BF02320380 ); Schapotschnikow 2006 Waves Random Complex Media 16 , 167–178 ( doi:10.1080/1745530600702535 )) and an analogous theorem was proved for discrete tree graphs (Berkolaiko 2007 Commun. Math. Phys. 278 , 803–819 ( doi:10.1007/S00220-007-0391-3 ); Dhar & Ramaswamy 1985 Phys. Rev. Lett. 54 , 1346–1349 ( doi:10.1103/PhysRevLett.54.1346 ); Fiedler 1975 Czechoslovak Math. J. 25 , 607–618). We prove the converse theorems for both discrete and metric graphs. Namely if for all n , the n th eigenfunction of the graph has n −1 zeros, then the graph is a tree. Our proofs use a recently obtained connection between the graph's nodal count and the magnetic stability of its eigenvalues (Berkolaiko 2013 Anal. PDE 6 , 1213–1233 ( doi:10.2140/apde.2013.6.1213 ); Berkolaiko & Weyand 2014 Phil. Trans. R. Soc. A 372 , 20120522 ( doi:10.1098/rsta.2012.0522 ); Colin de Verdière 2013 Anal. PDE 6 , 1235–1242 ( doi:10.2140/apde.2013.6.1235 )). In the course of the proof, we show that it is not possible for all (or even almost all, in the metric case) the eigenvalues to exhibit a diamagnetic behaviour. In addition, we develop a notion of ‘discretized’ versions of a metric graph and prove that their nodal counts are related to those of the metric graph.

scholarly journals A new spectral invariant for quantum graphs

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Michał Ławniczak ◽  
Pavel Kurasov ◽  
Szymon Bauch ◽  
Małgorzata Białous ◽  
Afshin Akhshani ◽  
...  
Keyword(s):  
Differential Equations ◽  
Euler Characteristic ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Quantum Graphs ◽  
Metric Graphs ◽  
Physical Systems ◽  
The Difference ◽  
Microwave Networks ◽  
Spectral Invariant

AbstractThe Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic $$\chi _G:= |V|-|V_D|-|E|$$ χ G : = | V | - | V D | - | E | , with $$|V_D|$$ | V D | denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic $$\chi _G$$ χ G of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic $$\chi _G$$ χ G can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic $$\chi _G$$ χ G a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.

scholarly journals METRIC GRAPH RECONSTRUCTION FROM NOISY DATA

2012 ◽  
Vol 22 (04) ◽  
pp. 305-325 ◽  
Author(s):  
MRIDUL AANJANEYA ◽  
FREDERIC CHAZAL ◽  
DANIEL CHEN ◽  
MARC GLISSE ◽  
LEONIDAS GUIBAS ◽  
...  
Keyword(s):  
Real World ◽  
Metric Spaces ◽  
Data Sets ◽  
Metric Geometry ◽  
Metric Graphs ◽  
Real World Data ◽  
Data Set ◽  
World Data ◽  
Metric Graph ◽  
Graph Reconstruction

Many real-world data sets can be viewed of as noisy samples of special types of metric spaces called metric graphs.19 Building on the notions of correspondence and Gromov-Hausdorff distance in metric geometry, we describe a model for such data sets as an approximation of an underlying metric graph. We present a novel algorithm that takes as an input such a data set, and outputs a metric graph that is homeomorphic to the underlying metric graph and has bounded distortion of distances. We also implement the algorithm, and evaluate its performance on a variety of real world data sets.

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