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.
{"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":null,"pages":null},"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}
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.
在以前的一篇论文中，我们发展了弱对称伪黎曼流形$G/H$的分类，其中$G$是半单李群，$H$是约化子群。我们从$G$紧的情况推导出分类。由此我们得到了洛伦兹签名$(n-1,1)$和反洛伦兹签名$(n-2,2)$的半简单弱对称流形的分类。本文从$H$紧且$N$幂零的情况$G = Nr * H$的分类出发，给出了弱对称伪黎曼零流形$G/H$的分类。事实证明，有大量的新例子值得进一步研究。从黎曼情形开始，我们看到当一个给定的H$对合自同构扩展到G$对合自同构时，我们证明了任意两个这样的扩展都会产生等距伪黎曼零流形。结果列在论文的最后两部分。
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
Georgios Dimitroglou Rizell, T. Ekholm, D. Tonkonog
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. We 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.
{"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":null,"pages":null},"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}
We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with cylindrical ends.
{"title":"Index theory and deformations of open nonnegatively curved manifolds","authors":"I. Belegradek","doi":"10.4310/jdg/1669998181","DOIUrl":"https://doi.org/10.4310/jdg/1669998181","url":null,"abstract":"We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with cylindrical ends.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47745625","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 improve a result by Chodosh and Ketover. We prove that, in an asymptotically flat $3$-manifold $M$ that contains no closed minimal surfaces, fixing $qin M$ and a $2$-plane $V$ in $T_qM$ there is a properly embedded minimal plane $Sigma$ in $M$ such that $qinSigma$ and $T_qSigma=V$. We also prove that fixing three points in $M$ there is a properly embedded minimal plane passing through these three points.
{"title":"Minimal planes in asymptotically flat three-manifolds","authors":"L. Mazet, H. Rosenberg","doi":"10.4310/jdg/1649953568","DOIUrl":"https://doi.org/10.4310/jdg/1649953568","url":null,"abstract":"In this paper, we improve a result by Chodosh and Ketover. We prove that, in an asymptotically flat $3$-manifold $M$ that contains no closed minimal surfaces, fixing $qin M$ and a $2$-plane $V$ in $T_qM$ there is a properly embedded minimal plane $Sigma$ in $M$ such that $qinSigma$ and $T_qSigma=V$. We also prove that fixing three points in $M$ there is a properly embedded minimal plane passing through these three points.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48580191","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 solve the problem of determining the fundamental degrees of freedom underlying a generalized Kahler structure of symplectic type. For a usual Kahler structure, it is well-known that the geometry ...
{"title":"Morita equivalence and the generalized Kähler potential","authors":"Francis Bischoff, M. Gualtieri, M. Zabzine","doi":"10.4310/jdg/1659987891","DOIUrl":"https://doi.org/10.4310/jdg/1659987891","url":null,"abstract":"We solve the problem of determining the fundamental degrees of freedom underlying a generalized Kahler structure of symplectic type. For a usual Kahler structure, it is well-known that the geometry ...","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48637629","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 prove that a geodesic net with three boundary (= unbalanced) vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degrees of unbalanced vertices. The result seems to be new even in the Euclidean case. We demonstrate by examples that the result is not true for metrics of positive curvature on the plane, and that there are no immediate generalizations of this result for geodesic nets with four unbalanced vertices.
{"title":"Geodesic nets with three boundary vertices","authors":"Fabian Parsch","doi":"10.4310/jdg/1631124286","DOIUrl":"https://doi.org/10.4310/jdg/1631124286","url":null,"abstract":"We prove that a geodesic net with three boundary (= unbalanced) vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degrees of unbalanced vertices. The result seems to be new even in the Euclidean case. We demonstrate by examples that the result is not true for metrics of positive curvature on the plane, and that there are no immediate generalizations of this result for geodesic nets with four unbalanced vertices.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48277955","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}
Given an involution on a rational homology 3-sphere $Y$ with quotient the $3$-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg--Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Fr{o}yshov invariants associated to spin structures on $Y$. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.
{"title":"On the Frøyshov invariant and monopole Lefschetz number","authors":"Jianfeng Lin, Daniel Ruberman, N. Saveliev","doi":"10.4310/jdg/1683307008","DOIUrl":"https://doi.org/10.4310/jdg/1683307008","url":null,"abstract":"Given an involution on a rational homology 3-sphere $Y$ with quotient the $3$-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg--Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Fr{o}yshov invariants associated to spin structures on $Y$. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46174205","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}
Jaroslaw Buczy'nski, Jaroslaw A. Wi'sniewski, Andrzej Weber
We prove LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most 9 with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Bialynicki-Birula decomposition and equivariant Riemann-Roch theorems. From the point of view of T-varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most (dim X+5)/2 with dim X arbitrarily high, but we require this special (contact) structure of X. Previous methods for studying T-varieties in general usually only apply for complexity at most 2 or 3.
{"title":"Algebraic torus actions on contact manifolds","authors":"Jaroslaw Buczy'nski, Jaroslaw A. Wi'sniewski, Andrzej Weber","doi":"10.4310/jdg/1659987892","DOIUrl":"https://doi.org/10.4310/jdg/1659987892","url":null,"abstract":"We prove LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most 9 with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Bialynicki-Birula decomposition and equivariant Riemann-Roch theorems. From the point of view of T-varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most (dim X+5)/2 with dim X arbitrarily high, but we require this special (contact) structure of X. Previous methods for studying T-varieties in general usually only apply for complexity at most 2 or 3.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2018-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46260595","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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