Pub Date : 2018-11-09DOI: 10.4310/ajm.2021.v25.n4.a4
C. Plaut
We show how to extend the Covering Spectrum (CS) of Sormani-Wei to two spectra, called the Extended Covering Spectrum (ECS) and Entourage Spectrum (ES) that are new for Riemannian manifolds but defined with useful properties on any metric on a Peano continuum. We do so by measuring in two different ways the "size" of a topological generalization of the $delta$-covers of Sormani-Wei called "entourage covers". For Riemannian manifolds $M$ of dimension at least 3, we characterize entourage covers as those covers corresponding to the normal closures of finite subsets of $pi_{1}(M)$. We show that CS$subset$ES$subset$MLS and that for Riemannian manifolds these inclusions may be strict, where MLS is the set of lengths of curves that are shortest in their free homotopy classes. We give equivalent definitions for all of these spectra that do not actually involve lengths of curves. Of particular interest are resistance metrics on fractals for which there are no non-constant rectifiable curves, but where there is a reasonable notion of Laplace Spectrum (LaS). The paper opens new fronts for questions about the relationship between LaS and subsets of the length spectrum for a range of spaces from Riemannian manifolds to resistance metric spaces.
{"title":"Spectra related to the length spectrum","authors":"C. Plaut","doi":"10.4310/ajm.2021.v25.n4.a4","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n4.a4","url":null,"abstract":"We show how to extend the Covering Spectrum (CS) of Sormani-Wei to two spectra, called the Extended Covering Spectrum (ECS) and Entourage Spectrum (ES) that are new for Riemannian manifolds but defined with useful properties on any metric on a Peano continuum. We do so by measuring in two different ways the \"size\" of a topological generalization of the $delta$-covers of Sormani-Wei called \"entourage covers\". For Riemannian manifolds $M$ of dimension at least 3, we characterize entourage covers as those covers corresponding to the normal closures of finite subsets of $pi_{1}(M)$. We show that CS$subset$ES$subset$MLS and that for Riemannian manifolds these inclusions may be strict, where MLS is the set of lengths of curves that are shortest in their free homotopy classes. We give equivalent definitions for all of these spectra that do not actually involve lengths of curves. Of particular interest are resistance metrics on fractals for which there are no non-constant rectifiable curves, but where there is a reasonable notion of Laplace Spectrum (LaS). The paper opens new fronts for questions about the relationship between LaS and subsets of the length spectrum for a range of spaces from Riemannian manifolds to resistance metric spaces.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48673807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-11-07DOI: 10.4310/AJM.2020.V24.N5.A3
Yunxia Chen, Yongdong Huang, N. Leung
Every compact symmetric space $M$ admits a dual noncompact symmetric space $check{M}$. When $M$ is a generalized Grassmannian, we can view $check{M}$ as a open submanifold of it consisting of space-like subspaces cite{HL}. Motivated from this, we study the embeddings from noncompact symmetric spaces to their compact duals, including space-like embedding for generalized Grassmannians, Borel embedding for Hermitian symmetric spaces and the generalized embedding for symmetric R-spaces. We will compare these embeddings and describe their images using cut loci.
{"title":"Embeddings from noncompact symmetric spaces to their compact duals","authors":"Yunxia Chen, Yongdong Huang, N. Leung","doi":"10.4310/AJM.2020.V24.N5.A3","DOIUrl":"https://doi.org/10.4310/AJM.2020.V24.N5.A3","url":null,"abstract":"Every compact symmetric space $M$ admits a dual noncompact symmetric space $check{M}$. When $M$ is a generalized Grassmannian, we can view $check{M}$ as a open submanifold of it consisting of space-like subspaces cite{HL}. Motivated from this, we study the embeddings from noncompact symmetric spaces to their compact duals, including space-like embedding for generalized Grassmannians, Borel embedding for Hermitian symmetric spaces and the generalized embedding for symmetric R-spaces. We will compare these embeddings and describe their images using cut loci.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46156424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-11-05DOI: 10.4310/AJM.2020.V24.N4.A7
C. Scarpa, J. Stoppa
We discuss a natural extension of the K"ahler reduction of Fujiki and Donaldson, which realises the scalar curvature of K"ahler metrics as a moment map, to a hyperk"ahler reduction. Our approach is based on an explicit construction of hyperk"ahler metrics due to Biquard and Gauduchon. This extension is reminiscent of how one derives Hitchin's equations for harmonic bundles, and yields real and complex moment map equations which deform the constant scalar curvature K"ahler (cscK) condition. In the special case of complex curves we recover previous results of Donaldson. We focus on the case of complex surfaces. In particular we show the existence of solutions to the moment map equations on a class of ruled surfaces which do not admit cscK metrics.
{"title":"Scalar curvature and an infinite-dimensional hyperkähler reduction","authors":"C. Scarpa, J. Stoppa","doi":"10.4310/AJM.2020.V24.N4.A7","DOIUrl":"https://doi.org/10.4310/AJM.2020.V24.N4.A7","url":null,"abstract":"We discuss a natural extension of the K\"ahler reduction of Fujiki and Donaldson, which realises the scalar curvature of K\"ahler metrics as a moment map, to a hyperk\"ahler reduction. Our approach is based on an explicit construction of hyperk\"ahler metrics due to Biquard and Gauduchon. This extension is reminiscent of how one derives Hitchin's equations for harmonic bundles, and yields real and complex moment map equations which deform the constant scalar curvature K\"ahler (cscK) condition. In the special case of complex curves we recover previous results of Donaldson. We focus on the case of complex surfaces. In particular we show the existence of solutions to the moment map equations on a class of ruled surfaces which do not admit cscK metrics.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43118127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-11-01DOI: 10.4310/ajm.2021.v25.n3.a6
V. Mazorchuk, E. Vishnyakova
We construct a new class of algebras resembling enveloping algebras and generalizing orthogonal Gelfand-Zeitlin algebras and rational Galois algebras studied by [EMV,FuZ,RZ,Har]. The algebras are defined via a geometric realization in terms of sheaves of functions invariant under an action of a finite group. A natural class of modules over these algebra can be constructed via a similar geometric realization. In the special case of a local reflection group, these modules are shown to have an explicit basis, generalizing similar results for orthogonal Gelfand-Zeitlin algebras from [EMV] and for rational Galois algebras from [FuZ]. We also construct a family of canonical simple Harish-Chandra modules and give sufficient conditions for simplicity of some modules.
{"title":"Harish–Chandra modules over invariant subalgebras in a skew-group ring","authors":"V. Mazorchuk, E. Vishnyakova","doi":"10.4310/ajm.2021.v25.n3.a6","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n3.a6","url":null,"abstract":"We construct a new class of algebras resembling enveloping algebras and generalizing orthogonal Gelfand-Zeitlin algebras and rational Galois algebras studied by [EMV,FuZ,RZ,Har]. The algebras are defined via a geometric realization in terms of sheaves of functions invariant under an action of a finite group. A natural class of modules over these algebra can be constructed via a similar geometric realization. In the special case of a local reflection group, these modules are shown to have an explicit basis, generalizing similar results for orthogonal Gelfand-Zeitlin algebras from [EMV] and for rational Galois algebras from [FuZ]. We also construct a family of canonical simple Harish-Chandra modules and give sufficient conditions for simplicity of some modules.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-31DOI: 10.4310/AJM.2018.V22.N4.A8
Fei Ye, Tongde Zhang, Zhixian Zhu
In this paper, we present a characterization of a big Q-divisor D on a smooth projective surface S with D2 > 0 and H1(OS(−D)) = 0, which generalizes a result of Sakai [Sak90] for D integral. As applications of this result for Q-divisors, we prove results on base-pointfreeness and very-ampleness of the adjoint linear system |KS + D|. These results can be viewed as refinements of previous results on smooth surfaces of Ein-Lazarsfeld [EL93] and Ma¸sek [Ma¸s99].
{"title":"Sakai’s theorem for $mathbb{Q}$-divisors on surfaces and applications","authors":"Fei Ye, Tongde Zhang, Zhixian Zhu","doi":"10.4310/AJM.2018.V22.N4.A8","DOIUrl":"https://doi.org/10.4310/AJM.2018.V22.N4.A8","url":null,"abstract":"In this paper, we present a characterization of a big Q-divisor D on a smooth \u0000projective surface S with D2 > 0 and H1(OS(−D)) = 0, which generalizes a result of Sakai \u0000[Sak90] for D integral. As applications of this result for Q-divisors, we prove results on base-pointfreeness and very-ampleness of the adjoint linear system |KS + D|. These results can be viewed as \u0000refinements of previous results on smooth surfaces of Ein-Lazarsfeld [EL93] and Ma¸sek [Ma¸s99].","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42388224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-14DOI: 10.4310/ajm.2020.v24.n6.a5
B. Street
Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are real analytic. We give necessary and sufficient, coordinate-free conditions for the existence of such a coordinate system. Moreover, we present a quantitative study of these coordinate charts. This is the third part in a three-part series of papers. The first part, joint with Stovall, lay the groundwork for the coordinate system we use in this paper and showed how such coordinate charts can be viewed as scaling maps for sub-Riemannian geometry. The second part dealt with the analogous questions with real analytic replaced by $C^infty$ and Zygmund spaces.
{"title":"Coordinates adapted to vector fields III: Real analyticity","authors":"B. Street","doi":"10.4310/ajm.2020.v24.n6.a5","DOIUrl":"https://doi.org/10.4310/ajm.2020.v24.n6.a5","url":null,"abstract":"Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are real analytic. We give necessary and sufficient, coordinate-free conditions for the existence of such a coordinate system. Moreover, we present a quantitative study of these coordinate charts. This is the third part in a three-part series of papers. The first part, joint with Stovall, lay the groundwork for the coordinate system we use in this paper and showed how such coordinate charts can be viewed as scaling maps for sub-Riemannian geometry. The second part dealt with the analogous questions with real analytic replaced by $C^infty$ and Zygmund spaces.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47584348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-06DOI: 10.4310/ajm.2020.v24.n6.a4
A. Fern'andez-P'erez, Rogério Mol, R. Rosas
A singular real analytic foliation $mathcal{F}$ of real codimension one on an $n$-dimensional complex manifold $M$ is Levi-flat if each of its leaves is foliated by immersed complex manifolds of dimension $n-1$. These complex manifolds are leaves of a singular real analytic foliation $mathcal{L}$ which is tangent to $mathcal{F}$. In this article, we classify germs of Levi-flat foliations at $(mathbb{C}^{n},0)$ under the hypothesis that $mathcal{L}$ is a germ holomorphic foliation. Essentially, we prove that there are two possibilities for $mathcal{L}$, from which the classification of $mathcal{F}$ derives: either it has a meromorphic first integral or is defined by a closed rational $1-$form. Our local results also allow us to classify real algebraic Levi-flat foliations on the complex projective space $mathbb{P}^{n} = mathbb{P}^{n}_{mathbb{C}}$.
{"title":"On singular real analytic Levi-flat foliations","authors":"A. Fern'andez-P'erez, Rogério Mol, R. Rosas","doi":"10.4310/ajm.2020.v24.n6.a4","DOIUrl":"https://doi.org/10.4310/ajm.2020.v24.n6.a4","url":null,"abstract":"A singular real analytic foliation $mathcal{F}$ of real codimension one on an $n$-dimensional complex manifold $M$ is Levi-flat if each of its leaves is foliated by immersed complex manifolds of dimension $n-1$. These complex manifolds are leaves of a singular real analytic foliation $mathcal{L}$ which is tangent to $mathcal{F}$. In this article, we classify germs of Levi-flat foliations at $(mathbb{C}^{n},0)$ under the hypothesis that $mathcal{L}$ is a germ holomorphic foliation. Essentially, we prove that there are two possibilities for $mathcal{L}$, from which the classification of $mathcal{F}$ derives: either it has a meromorphic first integral or is defined by a closed rational $1-$form. Our local results also allow us to classify real algebraic Levi-flat foliations on the complex projective space $mathbb{P}^{n} = mathbb{P}^{n}_{mathbb{C}}$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45232786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-06-16DOI: 10.4310/ajm.2022.v26.n1.a2
Gunhee Cho
We show that on a certain class of bounded, complete Reinhardt domains in C that enjoy a lot of symmetries, the Carathéodory pseudo-distance and the geodesic distance of the complete Kähler-Einstein metric with Ricci curvature −1 are different.
{"title":"Comparing the Carathéodory pseudo-distance and the Kähler–Einstein distance on complete reinhardt domains","authors":"Gunhee Cho","doi":"10.4310/ajm.2022.v26.n1.a2","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n1.a2","url":null,"abstract":"We show that on a certain class of bounded, complete Reinhardt domains in C that enjoy a lot of symmetries, the Carathéodory pseudo-distance and the geodesic distance of the complete Kähler-Einstein metric with Ricci curvature −1 are different.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41354546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-06-13DOI: 10.4310/AJM.2020.V24.N4.A6
A. Fern'andez-P'erez, J. Tamara
Let $mathcal{F}$ be a singular codimension one holomorphic foliation on a compact complex manifold $X$ of dimension at least three such that its singular set has codimension at least two. In this paper, we determine Lehmann-Suwa residues of $mathcal{F}$ as multiples of complex numbers by integration currents along irreducible complex subvarieties of $X$. We then prove a formula that determines the Baum-Bott residue of simple almost Liouvillian foliations of codimension one, in terms of Lehmann-Suwa residues, generalizing a result of Marco Brunella. As an application, we give sufficient conditions for the existence of dicritical singularities of a singular real-analytic Levi-flat hypersurface $Msubset X$ tangent to $mathcal{F}$.
{"title":"Lehmann–Suwa residues of codimension one holomorphic foliations and applications","authors":"A. Fern'andez-P'erez, J. Tamara","doi":"10.4310/AJM.2020.V24.N4.A6","DOIUrl":"https://doi.org/10.4310/AJM.2020.V24.N4.A6","url":null,"abstract":"Let $mathcal{F}$ be a singular codimension one holomorphic foliation on a compact complex manifold $X$ of dimension at least three such that its singular set has codimension at least two. In this paper, we determine Lehmann-Suwa residues of $mathcal{F}$ as multiples of complex numbers by integration currents along irreducible complex subvarieties of $X$. We then prove a formula that determines the Baum-Bott residue of simple almost Liouvillian foliations of codimension one, in terms of Lehmann-Suwa residues, generalizing a result of Marco Brunella. As an application, we give sufficient conditions for the existence of dicritical singularities of a singular real-analytic Levi-flat hypersurface $Msubset X$ tangent to $mathcal{F}$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41783546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-05-01DOI: 10.4310/AJM.2020.V24.N5.A5
O. García-Prada, D. Salamon, Samuel Trautwein
The Ricci form is a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. This observation yields a new approach to the Weil-Petersson symplectic form on the Teichmuller space of isotopy classes of complex structures with real first Chern class zero and nonempty Kahler cone.
{"title":"Complex structures, moment maps, and the Ricci form","authors":"O. García-Prada, D. Salamon, Samuel Trautwein","doi":"10.4310/AJM.2020.V24.N5.A5","DOIUrl":"https://doi.org/10.4310/AJM.2020.V24.N5.A5","url":null,"abstract":"The Ricci form is a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. This observation yields a new approach to the Weil-Petersson symplectic form on the Teichmuller space of isotopy classes of complex structures with real first Chern class zero and nonempty Kahler cone.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48714764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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