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Closed geodesics on connected sums and $3$-manifolds 连通和和$3$-流形上的闭测地线
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-12 DOI: 10.4310/jdg/1649953350
H. Rademacher, I. Taimanov
{"title":"Closed geodesics on connected sums and $3$-manifolds","authors":"H. Rademacher, I. Taimanov","doi":"10.4310/jdg/1649953350","DOIUrl":"https://doi.org/10.4310/jdg/1649953350","url":null,"abstract":"We study the asymptotics of the number N(t) of geometrically distinct closed geodesics of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups and apply this result to the prime decomposition of a three-manifold. In particular we show that the function N(t) grows at least like the prime numbers on a compact 3-manifold with infinite fundamental group. It follows that a generic Riemannian metric on a compact 3-manifold has infinitely many geometrically distinct closed geodesics. We also consider the case of a connected sum of a compact manifold with positive first Betti number and a simply-connected manifold which is not homeomorphic to a sphere.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47842181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
An optimal $L^2$ extension theorem on weakly pseudoconvex Kähler manifolds 弱伪凸Kähler流形上的最优L^2扩展定理
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-01 DOI: 10.4310/JDG/1536285628
Xiangyu Zhou, Langfeng Zhu
{"title":"An optimal $L^2$ extension theorem on weakly pseudoconvex Kähler manifolds","authors":"Xiangyu Zhou, Langfeng Zhu","doi":"10.4310/JDG/1536285628","DOIUrl":"https://doi.org/10.4310/JDG/1536285628","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/JDG/1536285628","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70470792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 43
The $L_p$-Aleksandrov problem for $L_p$-integral curvature $L_p$积分曲率的$L_p$-Alekandrov问题
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-01 DOI: 10.4310/JDG/1536285625
Yong Huang, E. Lutwak, Deane Yang, Gaoyong Zhang
{"title":"The $L_p$-Aleksandrov problem for $L_p$-integral curvature","authors":"Yong Huang, E. Lutwak, Deane Yang, Gaoyong Zhang","doi":"10.4310/JDG/1536285625","DOIUrl":"https://doi.org/10.4310/JDG/1536285625","url":null,"abstract":"It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp extension, for all real p. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the Lp-integral curvature of a convex body. This problem is solved for positive p and is answered for negative p provided the given measure is even.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/JDG/1536285625","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49578384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 47
A note on Selberg’s lemma and negatively curved Hadamard manifolds 关于Selberg引理和负弯曲Hadamard流形的注解
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-08-05 DOI: 10.4310/jdg/1649953550
M. Kapovich
{"title":"A note on Selberg’s lemma and negatively curved Hadamard manifolds","authors":"M. Kapovich","doi":"10.4310/jdg/1649953550","DOIUrl":"https://doi.org/10.4310/jdg/1649953550","url":null,"abstract":"Author(s): Kapovich, Michael | Abstract: Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43789812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
On type-preserving representations of thrice punctured projective plane group 关于三次删截投影平面群的保型表示
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-07-22 DOI: 10.4310/jdg/1635368618
Sara Maloni, Frédéric Palesi, Tian Yang
{"title":"On type-preserving representations of thrice punctured projective plane group","authors":"Sara Maloni, Frédéric Palesi, Tian Yang","doi":"10.4310/jdg/1635368618","DOIUrl":"https://doi.org/10.4310/jdg/1635368618","url":null,"abstract":"In this paper we consider type-preserving representations of the fundamental group of the three--holed projective plane into $mathrm{PGL}(2, R) =mathrm{Isom}(HH^2)$ and study the connected components with non-maximal euler class. We show that in euler class zero for all such representations there is a one simple closed curve which is non-hyperbolic, while in euler class $pm 1$ we show that there are $6$ components where all the simple closed curves are sent to hyperbolic elements and $2$ components where there are simple closed curves sent to non-hyperbolic elements. This answer a question asked by Brian Bowditch. In addition, we show also that in most of these components the action of the mapping class group on these non-maximal component is ergodic. In this work, we use an extension of Kashaev's theory of decorated character varieties to the context of non-orientable surfaces.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47850304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets 存在有界对称区域的局部全纯曲线的渐近全测地线及其在代数子集均匀化问题中的应用
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-07-19 DOI: 10.4310/jdg/1641413830
S. Chan, N. Mok
{"title":"Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets","authors":"S. Chan, N. Mok","doi":"10.4310/jdg/1641413830","DOIUrl":"https://doi.org/10.4310/jdg/1641413830","url":null,"abstract":"The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincare disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain $Omega$ must necessarily be asymptotically totally geodesic. Assuming otherwise we derive by the method of rescaling a hypothetical holomorphic isometric embedding of the Poincare disk with ${rm Aut}(Omega')$-equivalent tangent spaces into a tube domain $Omega' subset Omega$ and derive a contradiction by means of the Poincare-Lelong equation. We deduce that equivariant holomorphic embeddings between bounded symmetric domains must be totally geodesic. Furthermore, we solve a uniformization problem on algebraic subsets $Z subset Omega$. More precisely, if $check Gammasubset {rm Aut}(Omega)$ is a torsion-free discrete subgroup leaving $Z$ invariant such that $Z/check Gamma$ is compact, we prove that $Z subset Omega$ is totally geodesic. In particular, letting $Gamma subset{rm Aut}(Omega)$ be a torsion-free lattice, and $pi: Omega to Omega/Gamma =: X_Gamma$ be the uniformization map, a subvariety $Y subset X_Gamma$ must be totally geodesic whenever some (and hence any) irreducible component $Z$ of $pi^{-1}(Y)$ is an algebraic subset of $Omega$. For cocompact lattices this yields a characterization of totally geodesic subsets of $X_Gamma$ by means of bi-algebraicity without recourse to the celebrated monodromy result of Andre-Deligne on subvarieties of Shimura varieties, and as such our proof applies to not necessarily arithmetic cocompact lattices.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45266175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On non-diffractive cones 关于非衍射锥
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-07-13 DOI: 10.4310/jdg/1649953486
J. Galkowski, J. Wunsch
{"title":"On non-diffractive cones","authors":"J. Galkowski, J. Wunsch","doi":"10.4310/jdg/1649953486","DOIUrl":"https://doi.org/10.4310/jdg/1649953486","url":null,"abstract":"A subject of recent interest in inverse problems is whether a corner must diffract fixed frequency waves. We generalize this question somewhat and study cones $[0,infty)times Y$ which do not diffract high frequency waves. We prove that if $Y$ is analytic and does not diffract waves at high frequency then every geodesic on $Y$ is closed with period $2pi$. Moreover, we show that if $dim Y=2$, then $Y$ is isometric to either the sphere of radius 1 or its $mathbb{Z}^2$ quotient, $mathbb{R}mathbb{P}^2$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46672329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Beloshapka’s rigidity conjecture for real submanifolds in complex space 复空间中实子流形的Beloshapka刚性猜想
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-07-10 DOI: 10.4310/jdg/1664378617
Jan Gregorovič
{"title":"On Beloshapka’s rigidity conjecture for real submanifolds in complex space","authors":"Jan Gregorovič","doi":"10.4310/jdg/1664378617","DOIUrl":"https://doi.org/10.4310/jdg/1664378617","url":null,"abstract":"A well known Conjecture due to Beloshapka asserts that all totally nondegenerate polynomial models with the length $lgeq 3$ of their Levi-Tanaka algebra are {em rigid}, that is, any point preserving automorphism of them is completely determined by the restriction of its differential at the fixed point onto the complex tangent space. For the length $l=3$, Beloshapka's Conjecture was proved by Gammel and Kossovskiy in 2006. In this paper, we prove the Conjecture for arbitrary length $lgeq 3$. \u0000As another application of our method, we construct polynomial models of length $lgeq 3$, which are not totally nondegenerate and admit large groups of point preserving nonlinear automorphisms.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49519321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Growth of quantum $6j$-symbols and applications to the volume conjecture 量子$6j$的增长-符号及其在体积猜想中的应用
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-07-09 DOI: 10.4310/jdg/1645207506
G. Belletti, Renaud Detcherry, Efstratia Kalfagianni, Tian Yang
{"title":"Growth of quantum $6j$-symbols and applications to the volume conjecture","authors":"G. Belletti, Renaud Detcherry, Efstratia Kalfagianni, Tian Yang","doi":"10.4310/jdg/1645207506","DOIUrl":"https://doi.org/10.4310/jdg/1645207506","url":null,"abstract":"We prove the Turaev-Viro invariants volume conjecture for complements of fundamental shadow links: an infinite family of hyperbolic link complements in connected sums of copies of $S^1times S^2$. The main step of the proof is to find a sharp upper bound on the growth rate of the quantum $6j-$symbol evaluated at $e^{frac{2pi i}{r}}.$ As an application of the main result, we show that the volume of any hyperbolic 3-manifold with empty or toroidal boundary can be estimated in terms of the Turaev-Viro invariants of an appropriate link contained in it. We also build additional evidence for a conjecture of Andersen, Masbaum and Ueno (AMU conjecture) about the geometric properties of surface mapping class groups detected by the quantum representations.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43976923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 20
Subspace concentration of dual curvature measures of symmetric convex bodies 对称凸体对偶曲率测度的子空间集中
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-07-01 DOI: 10.4310/JDG/1531188189
K. Böröczky, M. Henk, H. Pollehn
{"title":"Subspace concentration of dual curvature measures of symmetric convex bodies","authors":"K. Böröczky, M. Henk, H. Pollehn","doi":"10.4310/JDG/1531188189","DOIUrl":"https://doi.org/10.4310/JDG/1531188189","url":null,"abstract":"We prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body. © 2018 International Press of Boston, Inc. All Rights Reserved.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/JDG/1531188189","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45338206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 89
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