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Adiabatic limits of anti-self-dual connections on collapsed $K3$ surfaces 坍塌$K3$表面上反自对偶连接的绝热极限
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-23 DOI: 10.4310/JDG/1622743140
V. Datar, Adam Jacob, Yuguang Zhang
{"title":"Adiabatic limits of anti-self-dual connections on collapsed $K3$ surfaces","authors":"V. Datar, Adam Jacob, Yuguang Zhang","doi":"10.4310/JDG/1622743140","DOIUrl":"https://doi.org/10.4310/JDG/1622743140","url":null,"abstract":"We prove a convergence result for a family of Yang-Mills connections over an elliptic $K3$ surface $M$ as the fibers collapse. In particular, assume $M$ is projective, admits a section, and has singular fibers of Kodaira type $I_1$ and type $II$. Let $Xi_{t_k}$ be a sequence of $SU(n)$ connections on a principal $SU(n)$ bundle over $M$, that are anti-self-dual with respect to a sequence of Ricci flat metrics collapsing the fibers of $M$. Given certain non-degeneracy assumptions on the spectral covers induced by $barpartial_{Xi_{t_k}}$, we show that away from a finite number of fibers, the curvature $F_{Xi_{t_k}}$ is locally bounded in $C^0$, the connections converge along a subsequence (and modulo unitary gauge change) in $L^p_1$ to a limiting $L^p_1$ connection $Xi_0$, and the restriction of $Xi_0$ to any fiber is $C^{1,alpha}$ gauge equivalent to a flat connection with holomorphic structure determined by the sequence of spectral covers. Additionally, we relate the connections $Xi_{t_k}$ to a converging family of special Lagrangian multi-sections in the mirror HyperK\"ahler structure, addressing a conjecture of Fukaya in this setting.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42240777","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
Improvements for eigenfunction averages: An application of geodesic beams 本征函数平均值的改进：测地光束的应用
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-17 DOI: 10.4310/jdg/1689262062
Y. Canzani, J. Galkowski
{"title":"Improvements for eigenfunction averages: An application of geodesic beams","authors":"Y. Canzani, J. Galkowski","doi":"10.4310/jdg/1689262062","DOIUrl":"https://doi.org/10.4310/jdg/1689262062","url":null,"abstract":"Let $(M,g)$ be a smooth, compact Riemannian manifold and ${phi_lambda }$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-Delta_gphi_lambda =lambda^2 phi_lambda$. Given a smooth submanifold $H subset M$ of codimension $kgeq 1$, we find conditions on the pair $(M,H)$, even when $H={x}$, for which $$ Big|int_Hphi_lambda dsigma_HBig|=OBig(frac{lambda^{frac{k-1}{2}}}{sqrt{log lambda}}Big)qquad text{or}qquad |phi_lambda(x)|=OBig(frac{lambda ^{frac{n-1}{2}}}{sqrt{log lambda}}Big), $$ as $lambdato infty$. These conditions require no global assumption on the manifold $M$ and instead relate to the structure of the set of recurrent directions in the unit normal bundle to $H$. Our results extend all previously known conditions guaranteeing improvements on averages, including those on sup-norms. For example, we show that if $(M,g)$ is a surface with Anosov geodesic flow, then there are logarithmically improved averages for any $Hsubset M$. We also find weaker conditions than having no conjugate points which guarantee $sqrt{log lambda}$ improvements for the $L^infty$ norm of eigenfunctions. Our results are obtained using geodesic beam techniques, which yield a mechanism for obtaining general quantitative improvements for averages and sup-norms.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42279484","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: 7
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":" ","pages":""},"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":"1 1","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","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
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