Metaphors in Systolic Geometry
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Complete Hypersurfaces in Euclidean Spaces with Finite Strong Total
Complete hypersurfaces in Euclidean spaces with finite strong total curvature Manfredo do Carmo and Maria Fernanda Elbert Abstract We prove that finite strong total curvature (see definition in Section 2) complete hypersurfaces of (n + 1)-euclidean space are proper and diffeomorphic to a com- pact manifold minus finitely many points. With an additional condition, we also prove that the Gauss map of such hypersurfaces extends continuously to the punc- tures. This is related to results of White [22] and and M¨uller-Sver´ak[18].ˇ Further properties of these hypersurfaces are presented, including a gap theorem for the total curvature. 1 Introduction Let φ: M n ! Rn+1 be a hypersurface of the euclidean space Rn+1. We assume that n n n+1 M = M is orientable and we fix an orientation for M. Let g : M ! S1 ⊂ R be the n Gauss map in the given orientation, where S1 is the unit n-sphere. Recall that the linear operator A: TpM ! TpM, p 2 M, associated to the second fundamental form, is given by hA(X);Y i = −hrX N; Y i; X; Y 2 TpM; where r is the covariant derivative of the ambient space and N is the unit normal vector in the given orientation. The map A = −dg is self-adjoint and its eigenvalues are the principal curvatures k1; k2; : : : ; kn. Key words and sentences: Complete hypersurface, total curvature, topological properties, Gauss- Kronecker curvature 2000 Mathematics Subject Classification. 57R42, 53C42 Both authors are partially supported by CNPq and Faperj. 1 R n We say that the total curvature of the immersion is finite if M jAj dM < 1, where P 21=2 n n n+1 jAj = i ki , i.e., if jAj belongs to the space L (M). -
February 19, 2016
February 19, 2016 Professor HELGE HOLDEN SECRETARY OF THE INTERNATIONAL MATHEMATICAL UNION Dear Professor Holden, The Turkish Mathematical Society (TMD) as the Adhering Organization, applies to promote Turkey from Group I to Group II as a member of IMU. We attach an overview of the developments of Mathematics in Turkey during the last 10 years (2005- 2015) preceded by a historical account. With best regards, Betül Tanbay President of the Turkish Mathematical Society Report on the state of mathematics in Turkey (2005-2015) This is an overview of the status of Mathematics in Turkey, prepared for the IMU for promotion from Group I to Group II by the adhering organization, the Turkish Mathematical Society (TMD). 1-HISTORICAL BACKGROUND 2-SOCIETIES AND CENTERS RELATED TO MATHEMATICAL SCIENCES 3-NUMBER OF PUBLICATIONS BY SUBJECT CATEGORIES 4-NATIONAL CONFERENCES AND WORKSHOPS HELD IN TURKEY BETWEEN 2013-2015 5-INTERNATIONAL CONFERENCES AND WORKSHOPS HELD IN TURKEY BETWEEN 2013-2015 6-NUMBERS OF STUDENTS AND TEACHING STAFF IN MATHEMATICAL SCIENCES IN TURKEY FOR THE 2013-2014 ACADEMIC YEAR AND THE 2014-2015 ACADEMIC YEAR 7- RANKING AND DOCUMENTS OF TURKEY IN MATHEMATICAL SCIENCES 8- PERIODICALS AND PUBLICATIONS 1-HISTORICAL BACKGROUND Two universities, the Istanbul University and the Istanbul Technical University have been influential in creating a mathematical community in Turkey. The Royal School of Naval Engineering, "Muhendishane-i Bahr-i Humayun", was established in 1773 with the responsibility to educate chart masters and ship builders. Gaining university status in 1928, the Engineering Academy continued to provide education in the fields of engineering and architecture and, in 1946, Istanbul Technical University became an autonomous university which included the Faculties of Architecture, Civil Engineering, Mechanical Engineering, and Electrical and Electronic Engineering. -
Systolic Inequalities, Discrete Or Not
Systolic inequalities, discrete or not Arnaud de Mesmay Gipsa-lab, CNRS, Université Grenoble Alpes Based on joint work with Éric Colin de Verdière and Alfredo Hubard. 1 / 53 A primer on surfaces We deal with connected, compact and orientable surfaces of genus g without boundary. Discrete metric Riemannian metric Triangulation G. Scalar product m on the Length of a curve jγjG : tangent space. Number of edges. Riemannian length jγjm. 2 / 53 Systoles and pants decompositions We study the length of topologically interesting curves for discrete and continuous metrics. Non-contractible curves Pants decompositions 3 / 53 Motivations Why should we care ? Topological graph theory: If the shortest non-contractible cycle is long, the surface is planar-like. ) Uniqueness of embeddings, colourability, spanning trees. Riemannian geometry: René Thom: “Mais c’est fondamental !”. Links with isoperimetry, topological dimension theory, number theory. Algorithms for surface-embedded graphs: Cookie-cutter algorithm for surface-embedded graphs: Decompose the surface, solve the planar case, recover the solution. More practical sides: texture mapping, parameterization, meshing ... 4 / 53 Part 1: Length of shortest curves 5 / 53 Intuition p p It should have length O( A) or O( n), but what is the dependency on g ? On shortest noncontractible curves Discrete setting Continuous setting What is the length of the red curve? 6 / 53 On shortest noncontractible curves Discrete setting Continuous setting What is the length of the red curve? Intuition p p It should have length O( A) or O( n), but what is the dependency on g ? 7 / 53 Discrete Setting: Topological graph theory The edgewidth of a triangulated surface is the length of the shortest noncontractible cycle. -
Linear Subspaces of Hypersurfaces
LINEAR SUBSPACES OF HYPERSURFACES ROYA BEHESHTI AND ERIC RIEDL Abstract. Let X be an arbitrary smooth hypersurface in CPn of degree d. We prove the de Jong-Debarre Conjecture for n ≥ 2d−4: the space of lines in X has dimension 2n−d−3. d+k−1 We also prove an analogous result for k-planes: if n ≥ 2 k + k, then the space of k- planes on X will be irreducible of the expected dimension. As applications, we prove that an arbitrary smooth hypersurface satisfying n ≥ 2d! is unirational, and we prove that the space of degree e curves on X will be irreducible of the expected dimension provided that e+n d ≤ e+1 . 1. Introduction We work throughout over an algebraically closed field of characteristic 0. Let X ⊂ Pn be an arbitrary smooth hypersurface of degree d. Let Fk(X) ⊂ G(k; n) be the Hilbert scheme of k-planes contained in X. Question 1.1. What is the dimension of Fk(X)? In particular, we would like to know if there are triples (n; d; k) for which the answer depends only on (n; d; k) and not on the specific smooth hypersurface X. It is known d+k classically that Fk(X) is locally cut out by k equations. Therefore, one might expect the d+k answer to Question 1.1 to be that the dimension is (k + 1)(n − k) − k , where negative dimensions mean that Fk(X) is empty. This is indeed the case when the hypersurface X is general [10, 17]. Standard examples (see Proposition 3.1) show that dim Fk(X) must depend on the particular smooth hypersurface X for d large relative to n and k, but there remains hope that Question 1.1 might be answered positively for n large relative to d and k. -
Solutions to the Maximal Spacelike Hypersurface Equation in Generalized Robertson-Walker Spacetimes
Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 80, pp. 1{14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOLUTIONS TO THE MAXIMAL SPACELIKE HYPERSURFACE EQUATION IN GENERALIZED ROBERTSON-WALKER SPACETIMES HENRIQUE F. DE LIMA, FABIO´ R. DOS SANTOS, JOGLI G. ARAUJO´ Communicated by Giovanni Molica Bisci Abstract. We apply some generalized maximum principles for establishing uniqueness and nonexistence results concerning maximal spacelike hypersur- faces immersed in a generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so-called timelike convergence condition (TCC). As application, we study the uniqueness and nonexistence of entire solutions of a suitable maximal spacelike hypersurface equation in GRW spacetimes obeying the TCC. 1. Introduction In the previous decades, the study of spacelike hypersurfaces immersed in a Lorentz manifold has been of substantial interest from both physical and mathe- matical points of view. For instance, it was pointed out by Marsden and Tipler [24] and Stumbles [37] that spacelike hypersurfaces with constant mean curvature in a spacetime play an important role in General Relativity, since they can be used as initial hypersurfaces where the constraint equations can be split into a linear system and a nonlinear elliptic equation. From a mathematical point of view, spacelike hypersurfaces are also interesting because of their Bernstein-type properties. One can truly say that the first remark- able results in this branch were the rigidity theorems of Calabi [13] and Cheng and Yau [15], who showed (the former for n ≤ 4, and the latter for general n) that the only maximal (that is, with zero mean curvature) complete spacelike hypersurfaces of the Lorentz-Minkowski space Ln+1 are the spacelike hyperplanes. -
Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces
Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces Éric Colin de Verdière 1 Alfredo Hubard 2 Arnaud de Mesmay 3 1École normale supérieure, CNRS 2INRIA, Laboratoire d’Informatique Gaspard Monge, Université Paris-Est Marne-la-Vallée 3IST Austria, Autriche 1 / 62 A primer on surfaces We deal with connected, compact and orientable surfaces of genus g without boundary. Discrete metric Riemannian metric Triangulation G. Scalar product m on the Length of a curve jγjG : tangent space. Number of edges. Riemannian length jγjm. 2 / 62 Systoles and surface decompositions We study the length of topologically interesting curves and graphs, for discrete and continuous metrics. 1.Non-contractible curves 2.Pants decompositions 3.Cut-graphs 3 / 62 Part 0: Why should we care.. 4 / 62 .. about graphs embedded on surfaces ? The easy answer: because they are “natural”. They occur in multiple settings: Graphics, computer-aided design, network design. The algorithmic answer: because they are “general”. Every graph is embeddable on some surface, therefore the genus of this surface is a natural parameter of a graph (similarly as tree-width, etc.). The hard answer: because of Robertson-Seymour theory. Theorem (Graph structure theorem, roughly) Every minor-closed family of graphs can be obtained from graphs k-nearly embedded on a surface S, for some constant k. 5 / 62 ) We need algorithms to do this cutting efficiently. ) We need good bounds on the lengths of the cuttings. ... about cutting surfaces along cycles/graphs ? Algorithms for surface-embedded graphs: Cookie-cutter algorithm for surface-embedded graphs: Cut the surface into the plane. Solve the planar case. -
Rotational Hypersurfaces Satisfying ∆ = in the Four-Dimensional
POLİTEKNİK DERGİSİ JOURNAL of POLYTECHNIC ISSN: 1302-0900 (PRINT), ISSN: 2147-9429 (ONLINE) URL: http://dergipark.org.tr/politeknik Rotational hypersurfaces satisfying ∆푰퐑 = 퐀퐑 in the four-dimensional Euclidean space Dört-boyutlu Öklid uzayında ∆퐼푹 = 푨푹 koşulunu sağlayan dönel hiperyüzeyler Yazar(lar) (Author(s)): Erhan GÜLER ORCID : 0000-0003-3264-6239 Bu makaleye şu şekilde atıfta bulunabilirsiniz (To cite to this article): Güler E., “Rotational hypersurfaces satisfying ∆퐼퐑 = 퐀퐑 in the four-dimensional Euclidean space”, Politeknik Dergisi, 24(2): 517-520, (2021). Erişim linki (To link to this article): http://dergipark.org.tr/politeknik/archive DOI: 10.2339/politeknik.670333 Rotational Hypersurfaces Satisfying 횫퐈퐑 = 퐀퐑 in the Four-Dimensional Euclidean Space Highlights ❖ Rotational hypersurface has zero mean curvature iff its Laplace-Beltrami operator vanishing ❖ Each element of the 4×4 order matrix A, which satisfies the condition 훥퐼푅 = 퐴푅, is zero ❖ Laplace-Beltrami operator of the rotational hypersurface depends on its mean curvature and the Gauss map Graphical Abstract Rotational hypersurfaces in the 4-dimensional Euclidean space are discussed. Some relations of curvatures of hypersurfaces are given, such as the mean, Gaussian, and their minimality and flatness. The Laplace-Beltrami operator has been defined for 4-dimensional hypersurfaces depending on the first fundamental form. In addition, it is indicated that each element of the 4×4 order matrix A, which satisfies the condition 훥퐼푅 = 퐴푅, is zero, that is, the rotational hypersurface R is minimal. Aim We consider the rotational hypersurfaces in 피4 to find its Laplace-Beltrami operator. Design & Methodology We indicate fundamental notions of 피4. Considering differential geometry formulas in 3-space, we transform them in 4-space. -
Filling Area I
Filling Area I M. Heuer ([email protected]) June 14, 2011 In this lecture, we will be learning about the filling volume of a Rieman- nian manifold. For a compact surface embedded in R3, this can easily be imagined as filling up the inside of it with water and hence calculating the volume of the resulting 3-dimensional manifold (but with a different met- ric than the usual one). However, we will use a more general approach by considering n-dimensional manifolds and a so-called filling by a manifold of higher dimension having the surface as its boundary. In particular, we will learn how to calculate the filling volume for spheres. Moreover, we will get to know an estimate of the displacement of a point in relation to the area of the hyperelliptic surface. 1 Preliminaries { As we have seen in the preceding lectures, for a Riemann surface Σ and a hyper- elliptic involution J, we have a conformal branched 2-fold covering Q :Σ ! S2 of the sphere S2 2 π { Pu's inequality: systπ1(G) ≤ 2 area(G) { A closed Riemannian surface Σ is called ovalless real if it possesses a fixed point free, antiholomorphic involution τ. 2 Orbifolds Definition 2.1. { An action of a group G on a topological space X is called faithful if there exists an x 2 X for all g 2 G with g 6= e, e being the neutral element, such that g · x 6= x. { A group action of a group G on a topological space X is cocompact if X=G is compact. -
Rotation Hypersurfaces in Sn × R and Hn × R
Note di Matematica Note Mat. 29 (2009), n. 1, 41-54 ISSN 1123-2536, e-ISSN 1590-0932 DOI 10.1285/i15900932v29n1p41 Notehttp://siba-ese.unisalento.it, di Matematica 29, ©n. 2009 1, 2009, Università 41–54. del Salento __________________________________________________________________ Rotation hypersurfaces in Sn × R and Hn × R Franki Dillen i [email protected] Johan Fastenakels ii [email protected] Joeri Van der Veken iii Katholieke Universiteit Leuven, Department of Mathematics, Celestijnenlaan 200B Box 2400, BE-3001 Leuven, Belgium. [email protected] Received: 14/11/2007; accepted: 07/04/2008. Abstract. We introduce the notion of rotation hypersurfaces of Sn × R and Hn × R and we prove a criterium for a hypersurface of one of these spaces to be a rotation hypersurface. Moreover, we classify minimal rotation hypersurfaces, flat rotation hypersurfaces and rotation hypersurfaces which are normally flat in the Euclidean resp. Lorentzian space containing Sn ×R resp. Hn × R. Keywords: rotation hypersurface, flat, minimal MSC 2000 classification: 53B25 1 Introduction In[3]theclassicalnotionofarotationsurfaceinE3 was extended to rota- tion hypersurfaces of real space forms of arbitrary dimension. Motivated by the recent study of hypersurfaces of the Riemannian products Sn × R and Hn × R, see for example [2], [4], [5] and [6], we will extend the notion of rotation hyper- surfaces to these spaces. Starting with a curve α on a totally geodesic cylinder S1 × R resp. H1 × R and a plane containing the axis of the cylinder, we will construct such a hypersurface and we will compute its principal curvatures. Moreover, we will prove a criterium for a hypersurface of Sn × R resp. -
The Missing Boundaries of the Santaló Diagrams for the Cases (D
Discrete Comput Geom 23:381–388 (2000) Discrete & Computational DOI: 10.1007/s004540010006 Geometry © 2000 Springer-Verlag New York Inc. The Missing Boundaries of the Santal´o Diagrams for the Cases (d,ω,R) and (ω, R, r) M. A. Hern´andez Cifre and S. Segura Gomis Departamento de Matem´aticas, Universidad de Murcia, 30100 Murcia, Spain {mhcifre,salsegom}@fcu.um.es Abstract. In this paper we solve two open problems posed by Santal´o: to obtain complete systems of inequalities for some triples of measures of a planar convex set. 1. Introduction There is abundant literature on geometric inequalities for planar figures. These inequali- ties connect several geometric quantities and in many cases determine the extremal sets which satisfy the equality conditions. Each new inequality obtained is interesting on its own, but it is also possible to ask if a collection of inequalities concerning several geometric magnitudes is large enough to determine the existence of the figure. Such a collection is called a complete system of inequalities: a system of inequalities relating all the geometric characteristics such that, for any set of numbers satisfying those conditions, a planar figure with these values of the characteristics exists in the given class. Santal´o [7] considered the area, the perimeter, the diameter, the minimal width, the circumradius, and the inradius of a planar convex K (A, p, d, ω, R and r, respectively). He tried to find complete systems of inequalities concerning either two or three of these measures. He found that, for pairs of measures, the known classic inequalities between them already form a complete system. -
Systems of Lines Amd Planes on the Quadric
Systems of lines and planes on the quadric hypersurface in spaces of four and five dimensions Item Type text; Thesis-Reproduction (electronic) Authors Fish, Anne, 1924- Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 01/10/2021 14:55:47 Link to Item http://hdl.handle.net/10150/319188 SYSTEMS OF LINES■AMD PLANES ON THE QUADRIC HYPERSURFACE SPACES OF FOUR AND- FIVE DIMENSIONS - by Anne Fish A TLes is submitted to the faculty of the Department of Mathematics in partial fullfilment of the requirements for the degree of Master of Science in the Graduate College University of Arizona 1948 Approved: Direeto^ of Thesis t Date & R A& TABLE OF CONTENTS I? O (TU. O "fc X OH a a © © a © o © © © © o © o © © ©ooo oool Chapter I Geometry of M-Dirnensiona 1 Projective Space © » 2 II Ruled Surfaces in Three-Dimensional Projective Space © © © © © © © © © © © © 10 III Systems of Lines on a Quadric Surface in Three-Dimensional Projective Space © © « © 13 IV Systems of Lines on a Quadric Hypersurface in Four-Dimensional Projective Space « © © © 23 V Systems of Planes on a Quadric Hypersurface in a Projective Space of Five Dimensions © © 34 Bibliography © © © © © © © © © © © © © © © © © © © s © © © 4V 194189 INTRODUCTION It has long been common knowledge -
Systolic Invariants of Groups and 2-Complexes Via Grushko Decomposition
SYSTOLIC INVARIANTS OF GROUPS AND 2-COMPLEXES VIA GRUSHKO DECOMPOSITION YULI B. RUDYAK∗ AND STEPHANE´ SABOURAU Abstract. We prove a finiteness result for the systolic area of groups. Namely, we show that there are only finitely many pos- sible unfree factors of fundamental groups of 2-complexes whose systolic area is uniformly bounded. We also show that the number of freely indecomposable such groups grows at least exponentially with the bound on the systolic area. Furthermore, we prove a uniform systolic inequality for all 2-complexes with unfree funda- mental group that improves the previously known bounds in this dimension. Resum´ e.´ Nous prouvons un r´esultat de finitude pour l’aire sys- tolique des groupes. Pr´ecis´ement, nous montrons qu’il n’existe qu’un nombre fini de facteurs non-libres dans les groupes fonda- mentaux des 2-complexes d’aire systolique uniform´ement born´ee. Nous montrons aussi que le nombre de tels groupes librement ind´ecomposables croˆıt au moins exponentiellement avec la borne sur l’aire systolique. De plus, nous prouvons une in´egalit´esys- tolique uniforme pour tous les 2-complexes de groupe fondamental non-libre qui am´elioreles bornes pr´ec´edemment connues dans cette dimension. Contents 1. Introduction 2 2. Topological preliminaries 5 3. Complexes of zero Grushko free index 7 4. Existence of ε-regular metrics 9 5. Counting fundamental groups 12 6. Two systolic finiteness results 14 Date: July 17, 2007. 2000 Mathematics Subject Classification. Primary 53C23; Secondary 20E06 . Key words and phrases. systole, systolic area, systolic ratio, 2-complex, Grushko decomposition.