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Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces [article]

Éric Colin de Verdière, Alfredo Hubard, Arnaud de Mesmay
2015 arXiv   pre-print
This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated  ...  surfaces and Gromov's systolic inequality for surfaces are essentially equivalent.  ...  Acknowledgements We would like to thank Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh for pointing out and discussing with us their results on Voronoi diagrams of Riemannian surfaces [16] and  ... 
arXiv:1408.4036v2 fatcat:kjyhkuh6kjc4ndbo4llubmkhhi

Guest Editors' Foreword

Siu-Wing Cheng, Olivier Devillers
2015 Discrete & Computational Geometry  
This special issue of Discrete & Computational Geometry contains a selection of seven papers whose preliminary versions appeared in the Proceedings of the Annual Symposium on Computational Geometry, Kyoto  ...  There is a known lower bound of three, and there has been a series of results that gradually reduce the maximum degree to six.  ...  Colin de Verdière, Hubard, and de Mesmay use Riemannian systolic inequalities to study cuts on a triangulated surface.  ... 
doi:10.1007/s00454-015-9680-3 fatcat:3j56wwx2j5fe7emkodkxjc4wcm

Page 6700 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
a Harnack-type inequality and a volume growth condi- tion.  ...  Petersburg) Extremal decompositions of a Riemann surface, and quasi- conformal mappings of a special form. II. (Russian. Russian summary) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.  ... 

Metaphors in systolic geometry [article]

Larry Guth
2010 arXiv   pre-print
The metaphors connect the systolic inequality to minimal surfaces, topological dimension, scalar curvature, and hyperbolic geometry.  ...  This is an expository essay about systolic geometry. It describes a central theorem in the subject and why the proof is difficult.  ...  Buser constructed a pants decomposition of any genus G hyperbolic surface using curves of length G. On the other hand, the curves in a pants decomposition must be larger than the systole ∼ log G.  ... 
arXiv:1003.4247v1 fatcat:qgssaer44vds3gor7jqisorakq

Combinatorial systolic inequalities [article]

Ryan Kowalick, Jean-François Lafont, Barry Minemyer
2015 arXiv   pre-print
We show that a class of smooth manifolds satisfies a systolic inequality for all Riemannian metrics if and only if it satisfies a corresponding combinatorial systolic inequality for all smooth triangulations  ...  We establish combinatorial versions of various classical systolic inequalities.  ...  The work of the second author was partially supported by the NSF, under grants DMS-1207782 and DMS-1510640.  ... 
arXiv:1506.07121v1 fatcat:cowkh2ecdzg6bdjabrf72i7dqi

Metaphors in Systolic Geometry

Larry Guth
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
When we put ridges in the surface of the torus, the systole only depends on the thinnest part and the thick parts contribute heavily to the area.  ...  It has systoleand area around 60, and so it obeys the systolic inequality.  ...  Buser constructed a pants decomposition of any genus G hyperbolic surface using curves of length G. On the other hand, the curves in a pants decomposition must be larger than the systole ∼ log G.  ... 
doi:10.1142/9789814324359_0072 fatcat:uixoc3gx2zhgbfk7oqoujomybm

Shortest Path Embeddings of Graphs on Surfaces

Alfredo Hubard, Vojtěch Kaluža, Arnaud de Mesmay, Martin Tancer
2017 Discrete & Computational Geometry  
We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics.  ...  Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest  ...  Acknowledgements We are grateful toÉric Colin de Verdière for his involvement in the early stages of this research.  ... 
doi:10.1007/s00454-017-9898-3 fatcat:pnan2b4mqfamlcisrcknjm62ne

The maximum number of systoles for genus two Riemann surfaces with abelian differentials

Chris Judge, Hugo Parlier
2019 Commentarii Mathematici Helvetici  
This article explores the length and number of systoles associated to holomorphic 1-forms on surfaces.  ...  In particular, we show that up to homotopy, there are at most 10 systolic loops on such a genus two surface and that the bound is realized by a unique translation surface up to homothety.  ...  If one of the triangles has a smaller ratio, then the above inequality is strict and thus equality only occurs if the Delaunay triangulation consisted of equilateral triangles.  ... 
doi:10.4171/cmh/463 fatcat:yuawh7fiungqra6ro5q7atjmiy

Minimal Delaunay triangulations of hyperbolic surfaces [article]

Matthijs Ebbens, Hugo Parlier, Gert Vegter
2020 arXiv   pre-print
Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations.  ...  Finally, to give a general lower bound, we will show that the Ω(√(g)) lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus g is tight for hyperbolic surfaces  ...  Hence the longest curve of any pants decomposition of a random surface is not convex. The lengths of edges in a given triangulation are another parameter set for M g .  ... 
arXiv:2011.09847v1 fatcat:vajxjygs6vekjnxfqs6ernejty

Diastolic and isoperimetric inequalities on surfaces

Florent Balacheff, Stéphane Sabourau
2010 Annales Scientifiques de l'Ecole Normale Supérieure  
-We prove a universal inequality between the diastole, defined using a minimax process on the one-cycle space, and the area of closed Riemannian surfaces.  ...  4 e série, t. 43, 2010, p. 579 à 605 DIASTOLIC AND ISOPERIMETRIC INEQUALITIES ON SURFACES ʙʏ Fʟʀɴ BALACHEFF ɴ Sʜɴ SABOURAU Aʙʀ.  ...  Every closed Riemannian surface M of genus g ≥ 1 satisfies the following asymptotically optimal systolic inequality (1.4) sys(M ) ≤ C log(g + 1) √ g » area(M ) where sys(M ) is the systole of M and C  ... 
doi:10.24033/asens.2128 fatcat:2qapdvuzvfbabaeewgxyqtn6eq

The maximum number of systoles for genus two Riemann surfaces with abelian differentials [article]

Chris Judge, Hugo Parlier
2019 arXiv   pre-print
For general genus g and a holomorphic 1-form ω with one zero, we provide the optimal upper bound, 6g-3, on the number of homotopy classes of systoles.  ...  In particular, we show that if X has genus two, then, up to homotopy, there are at most 10 systolic loops on (X,ω) and, moreover, that this bound is realized by a unique translation surface up to homothety  ...  The deck group of the universal covering map p permutes the cells of the Delaunay decomposition, and so we obtain a decomposition of X.  ... 
arXiv:1703.01809v6 fatcat:maegx53zzbcbtpmbrjzz5nhcoe

Homological Error Correcting Codes and Systolic Geometry [article]

Ethan Fetaya
2011 arXiv   pre-print
In my masters thesis I prove a square root bound on the distance of homological codes that come from two dimensional surfaces, as a result of the systolic inequality.  ...  I also give a detailed version of M.H. Freedman's proof that due to systolic freedom, this bound does not hold in higher dimensions.  ...  Now the fact that d is smaller then the order of τ tells us that τ d p = p (since τ d has no fixed points). Both point p and τ d p in Σ g are projected to the same point in H 2 /H g , this  ... 
arXiv:1108.2886v1 fatcat:wbxkxximtre2jisbijhtdppkla

Well-rounded equivariant deformation retracts of Teichmüller spaces [article]

Lizhen Ji
2014 arXiv   pre-print
., _g-equivariant deformation retracts, of the Teichmüller space _g of compact Riemann surfaces of genus g.  ...  We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichmüller space.  ...  I would like to thank Scott Wolpert for very helpful conversations, references and encouragement and Hugo Parlier for helpful correspondence, for example, the arguments in the proof of Proposition 4.3  ... 
arXiv:1302.0877v2 fatcat:g3u5hvh5pnf35ocpnaepbjkwju

On the 2-systole of stretched enough positive scalar curvature metrics on S 2 x S 2 [article]

Thomas Richard
2020 arXiv   pre-print
We use recent developments by Gromov and Zhu to derive an upper bound for the 2-systole of the homology class of S 2 x * in a S 2 x S 2 with a positive scalar curvature metric such that the set of spheres  ...  Besson and S. Sabourau for helpful discussions. He also thanks Jintian Zhu for pointing an inaccuracy in a previous version of this work.  ...  The author is supported by the grant ANR-17-CE40-0034 of the French National Research Agency ANR (Project CCEM).  ... 
arXiv:2007.02705v3 fatcat:5jjpvtas5zefhmseep65eogrx4

Complexity of PL-manifolds [article]

Bruno Martelli
2009 arXiv   pre-print
We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties.  ...  On the other hand, there are many closed 4-manifolds of complexity zero (manifolds without 3-handles, doubles of 2-handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary  ...  We would like to thank Katya Pervova for suggesting improvements on an earlier version of the manuscript, and Roberto Frigerio for the many discussions on bounded cohomology and Gromov norm.  ... 
arXiv:0810.5478v2 fatcat:tmeoytbzd5csvd5dny6uimspxu
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