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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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, $mathbb{R}mathbb{P}^2$.
{"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}
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 $lgeq 3$. As 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.
{"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}
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