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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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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
Blowups and blowdowns of geodesics in Carnot groups 卡诺群中测地线的爆破和爆破
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-06-25 DOI: 10.4310/jdg/1680883578
Eero Hakavuori, E. Donne
{"title":"Blowups and blowdowns of geodesics in Carnot groups","authors":"Eero Hakavuori, E. Donne","doi":"10.4310/jdg/1680883578","DOIUrl":"https://doi.org/10.4310/jdg/1680883578","url":null,"abstract":"This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents of Carnot geodesics are geodesics in some groups of lower nilpotency step. Namely, every blowup curve of every geodesic in every Carnot group is still a geodesic in the group modulo its last layer. Then as a consequence we get that in every sub-Riemannian manifold any $s$ times iterated tangent of any geodesic is a line, where $s$ is the step of the sub-Riemannian manifold in question. With a similar approach, we also show that blowdown curves of geodesics in sub-Riemannian Carnot groups are contained in subgroups of lower rank. This latter result is also extended to rough geodesics.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43756235","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: 16
Weakly symmetric pseudo–Riemannian nilmanifolds 弱对称伪黎曼零流形
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-06-21 DOI: 10.4310/jdg/1664378619
J. Wolf, Zhiqi Chen
{"title":"Weakly symmetric pseudo–Riemannian nilmanifolds","authors":"J. Wolf, Zhiqi Chen","doi":"10.4310/jdg/1664378619","DOIUrl":"https://doi.org/10.4310/jdg/1664378619","url":null,"abstract":"In an earlier paper we developed the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derived the classification from the cases where $G$ is compact. As a consequence we obtained the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans--lorentzian signature $(n-2,2)$. Here we work out the classification of weakly symmetric pseudo--riemannian nilmanifolds $G/H$ from the classification for the case $G = Nrtimes H$ with $H$ compact and $N$ nilpotent. It turns out that there is a plethora of new examples that merit further study. Starting with that riemannian case, we see just when a given involutive automorphism of $H$ extends to an involutive automorphism of $G$, and we show that any two such extensions result in isometric pseudo--riemannian nilmanifolds. The results are tabulated in the last two sections of the paper.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43135438","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: 6
Poles of cubic differentials and ends of convex $mathbb{RP}^2$-surfaces 三次微分的极点和凸$mathbb｛RP｝^2$曲面的端点
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-06-17 DOI: 10.4310/jdg/1679503805
Xin Nie
{"title":"Poles of cubic differentials and ends of convex $mathbb{RP}^2$-surfaces","authors":"Xin Nie","doi":"10.4310/jdg/1679503805","DOIUrl":"https://doi.org/10.4310/jdg/1679503805","url":null,"abstract":"On any oriented surface, the affine sphere construction gives a one-to-one correspondence between convex $mathbb{RP}^2$-structures and holomorphic cubic differentials. Generalizing results of Benoist-Hulin, Loftin and Dumas-Wolf, we show that poles of order less than $3$ of cubic differentials correspond to finite volume ends of convex $mathbb{RP}^2$-structures, while poles of order at least $3$ correspond to geodesic or piecewise geodesic boundary components. More specifically, in the latter situation, we prove asymptotic results for the convex $mathbb{RP}^2$-structure around the pole in terms of the cubic differential.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41379947","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
Refined disk potentials for immersed Lagrangian surfaces 浸没拉格朗日曲面的精细圆盘势
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-06-10 DOI: 10.4310/jdg/1664378618
Georgios Dimitroglou Rizell, T. Ekholm, D. Tonkonog
{"title":"Refined disk potentials for immersed Lagrangian surfaces","authors":"Georgios Dimitroglou Rizell, T. Ekholm, D. Tonkonog","doi":"10.4310/jdg/1664378618","DOIUrl":"https://doi.org/10.4310/jdg/1664378618","url":null,"abstract":"We define a refined Gromov-Witten disk potential of self-transverse monotone immersed Lagrangian surfaces in a symplectic 4-manifold as an element in a capped version of the Chekanov--Eliashberg dg-algebra of the singularity links of the double points (a collection of Legendrian Hopf links). We give a surgery formula that expresses the potential after smoothing a double point. \u0000We study refined potentials of monotone immersed Lagrangian spheres in the complex projective plane and find monotone spheres that cannot be displaced from complex lines and conics by symplectomorphisms. We also derive general restrictions on sphere potentials using Legendrian lifts to the contact 5-sphere.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44733601","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: 13
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