Minoru Tanaka, T. Akamatsu, R. Sinclair, M. Yamaguchi
It is known that if the Gaussian curvature function along each meridian on a surface of revolution ( R 2 , dr 2 + m ( r ) 2 dθ 2 ) is decreasing, then the cut locus of each point of θ − 1 (0) is empty or a subarc of the opposite meridian θ − 1 ( π ) . Such a surface is called a von Mangoldt’s surface of revolution . A surface of revolution ( R 2 , dr 2 + m ( r ) 2 dθ 2 ) is called a generalized von Mangoldt surface of revolution if the cut locus of each point of θ − 1 (0) is empty or a subarc of the opposite meridian θ − 1 ( π ) . For example, the surface of revolution ( R 2 , dr 2 + m 0 ( r ) 2 dθ 2 ) , where m 0 ( x ) = x/ (1 + x 2 ) , has the same cut locus structure as above and the cut locus of each point in r − 1 ((0 , ∞ )) is nonempty. Note that the Gaussian curvature function is not decreasing along a meridian for this surface. In this article, we give sufficient conditions for a surface of revolution ( R 2 , dr 2 + m ( r ) 2 dθ 2 ) to be a generalized von Mangoldt surface of revolution. Moreover, we prove that for any surface of revolution with finite total curvature c, there exists a generalized von Mangoldt surface of revolution with the same total curvature c such that the Gaussian curvature function along a meridian is not monotone on [ a, ∞ ) for any a > 0 .
{"title":"Generalized von Mangoldt surfaces of revolution and asymmetric two-spheres of revolution with simple cut locus structure","authors":"Minoru Tanaka, T. Akamatsu, R. Sinclair, M. Yamaguchi","doi":"10.2969/jmsj/88838883","DOIUrl":"https://doi.org/10.2969/jmsj/88838883","url":null,"abstract":"It is known that if the Gaussian curvature function along each meridian on a surface of revolution ( R 2 , dr 2 + m ( r ) 2 dθ 2 ) is decreasing, then the cut locus of each point of θ − 1 (0) is empty or a subarc of the opposite meridian θ − 1 ( π ) . Such a surface is called a von Mangoldt’s surface of revolution . A surface of revolution ( R 2 , dr 2 + m ( r ) 2 dθ 2 ) is called a generalized von Mangoldt surface of revolution if the cut locus of each point of θ − 1 (0) is empty or a subarc of the opposite meridian θ − 1 ( π ) . For example, the surface of revolution ( R 2 , dr 2 + m 0 ( r ) 2 dθ 2 ) , where m 0 ( x ) = x/ (1 + x 2 ) , has the same cut locus structure as above and the cut locus of each point in r − 1 ((0 , ∞ )) is nonempty. Note that the Gaussian curvature function is not decreasing along a meridian for this surface. In this article, we give sufficient conditions for a surface of revolution ( R 2 , dr 2 + m ( r ) 2 dθ 2 ) to be a generalized von Mangoldt surface of revolution. Moreover, we prove that for any surface of revolution with finite total curvature c, there exists a generalized von Mangoldt surface of revolution with the same total curvature c such that the Gaussian curvature function along a meridian is not monotone on [ a, ∞ ) for any a > 0 .","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47244357","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}
In this article we show a Cartan decomposition for reductive Riemannian Gelfand pairs and an induction of spherical functions for Riemannian Gelfand pairs. With the induction we find that the property of the symmetry of spherical functions, which is known for Riemannian symmetric pairs, can also be induced from the corresponding property of smaller dimension. A Fourier transform of a positive function for a Riemannian Gelfand pair with abelian unipotent radical is also given under some condition on its support by using the symmetry of spherical function. 0. Introduction In this article we prove a Cartan decomposition for reductive Riemannian Gelfand pairs and show an application to spherical functions for Riemannian Gelfand pairs. A pair (G,H) of a real Lie group G and its compact subgroup H with G/H connected is a Riemannian Gelfand pair if the algebra (under convolution) of H-biinvariant finite complex Radon measures on G is commutative. A reductive Riemannian symmetric pair is a typical example of Riemannian Gelfand pairs. The reader is referred to [Wo07] for the general theory (G is not necessarily a Lie group) of Gelfand pair and [Ya05] for the classification. Our first result is a Cartan decomposition (Theorem 2.5) of the form G = HAH with A an abelian Lie subgroup of G for a reductive Riemannian Gelfand pair (G,H), which is proved in Section 2. The proof uses the induction on the dimension of G. We find all the reductive Riemannian Gelfand pairs for which we cannot reduce a Cartan decomposition to more smaller dimensional cases with the Cartan decomposition for reductive Riemannian symmetric pairs [He78] in Section 1 by inspecting Krämer’s classification of reductive spherical subalgebras [Kr79]. In Section 3 we show an induction of spherical functions (Theorem 3.1) for a Riemannian Gelfand pair (G,H). The induction is given as the integration on H, whose integral kernel is provided from the Iwasawa projection on the reductive part. In Section 4 we show that the property of the symmetry of spherical functions, which is known for reductive Riemannian symmetric pairs, can also be induced from the corresponding property of smaller dimension by using the induction of spherical functions (Lemma 4.8), and that the property holds in the case when the unipotent radical of G is abelian (Theorem 4.19). As an application of this property we find that the convolution product of a compactly supported function and a spherical function takes a simple 2020 Mathematics Subject Classification. primary 22E46; secondary 43A90; 53C30. Date: June 29, 2021.
{"title":"A Cartan decomposition for Gelfand pairs and induction of spherical functions","authors":"Yu-ichi Tanaka","doi":"10.2969/jmsj/85588558","DOIUrl":"https://doi.org/10.2969/jmsj/85588558","url":null,"abstract":"In this article we show a Cartan decomposition for reductive Riemannian Gelfand pairs and an induction of spherical functions for Riemannian Gelfand pairs. With the induction we find that the property of the symmetry of spherical functions, which is known for Riemannian symmetric pairs, can also be induced from the corresponding property of smaller dimension. A Fourier transform of a positive function for a Riemannian Gelfand pair with abelian unipotent radical is also given under some condition on its support by using the symmetry of spherical function. 0. Introduction In this article we prove a Cartan decomposition for reductive Riemannian Gelfand pairs and show an application to spherical functions for Riemannian Gelfand pairs. A pair (G,H) of a real Lie group G and its compact subgroup H with G/H connected is a Riemannian Gelfand pair if the algebra (under convolution) of H-biinvariant finite complex Radon measures on G is commutative. A reductive Riemannian symmetric pair is a typical example of Riemannian Gelfand pairs. The reader is referred to [Wo07] for the general theory (G is not necessarily a Lie group) of Gelfand pair and [Ya05] for the classification. Our first result is a Cartan decomposition (Theorem 2.5) of the form G = HAH with A an abelian Lie subgroup of G for a reductive Riemannian Gelfand pair (G,H), which is proved in Section 2. The proof uses the induction on the dimension of G. We find all the reductive Riemannian Gelfand pairs for which we cannot reduce a Cartan decomposition to more smaller dimensional cases with the Cartan decomposition for reductive Riemannian symmetric pairs [He78] in Section 1 by inspecting Krämer’s classification of reductive spherical subalgebras [Kr79]. In Section 3 we show an induction of spherical functions (Theorem 3.1) for a Riemannian Gelfand pair (G,H). The induction is given as the integration on H, whose integral kernel is provided from the Iwasawa projection on the reductive part. In Section 4 we show that the property of the symmetry of spherical functions, which is known for reductive Riemannian symmetric pairs, can also be induced from the corresponding property of smaller dimension by using the induction of spherical functions (Lemma 4.8), and that the property holds in the case when the unipotent radical of G is abelian (Theorem 4.19). As an application of this property we find that the convolution product of a compactly supported function and a spherical function takes a simple 2020 Mathematics Subject Classification. primary 22E46; secondary 43A90; 53C30. Date: June 29, 2021.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49317267","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}
Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the second dynamical degrees distinct. We give a boundedness result about heights of its periodic points. This is motivated by a conjecture of Silverman for polynomial automorphisms of affine spaces. We also study the Kawaguchi--Silverman conjecture concerning dynamical and arithmetic degrees for certain rational self-maps in dimension two. In particular, we reduce the problem to the dynamical Mordell--Lang conjecture and verify the Kawaguchi--Silverman conjecture for some new cases. As a byproduct of the argument, we show the existence of Zariski dense orbits in these cases.
{"title":"Periodic points and arithmetic degrees of certain rational self-maps","authors":"Long Wang","doi":"10.2969/jmsj/89568956","DOIUrl":"https://doi.org/10.2969/jmsj/89568956","url":null,"abstract":"Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the second dynamical degrees distinct. We give a boundedness result about heights of its periodic points. This is motivated by a conjecture of Silverman for polynomial automorphisms of affine spaces. We also study the Kawaguchi--Silverman conjecture concerning dynamical and arithmetic degrees for certain rational self-maps in dimension two. In particular, we reduce the problem to the dynamical Mordell--Lang conjecture and verify the Kawaguchi--Silverman conjecture for some new cases. As a byproduct of the argument, we show the existence of Zariski dense orbits in these cases.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43065553","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}
We consider the Moore-Nehari equation, u′′+h(x, λ)|u|p−1u = 0 in (−1, 1) with u(−1) = u(1) = 0, where p > 1, h(x, λ) = 0 for |x| < λ, h(x, λ) = 1 for λ ≤ |x| ≤ 1 and λ ∈ (0, 1) is a parameter. We prove the existence of a solution which has exactly m zeros in the interval (−1, 0) and exactly n zeros in (0, 1) for given nonnegative integers m and n.
{"title":"Symmetric and asymmetric nodal solutions for the Moore–Nehari differential equation","authors":"R. Kajikiya","doi":"10.2969/jmsj/86168616","DOIUrl":"https://doi.org/10.2969/jmsj/86168616","url":null,"abstract":"We consider the Moore-Nehari equation, u′′+h(x, λ)|u|p−1u = 0 in (−1, 1) with u(−1) = u(1) = 0, where p > 1, h(x, λ) = 0 for |x| < λ, h(x, λ) = 1 for λ ≤ |x| ≤ 1 and λ ∈ (0, 1) is a parameter. We prove the existence of a solution which has exactly m zeros in the interval (−1, 0) and exactly n zeros in (0, 1) for given nonnegative integers m and n.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42083547","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}
We show the strong cohomological rigidity of Hirzebruch surface bundles over Bott manifolds. As a corollary, we have that the strong cohomological rigidity conjecture is true for Bott manifolds of dimension $8$.
{"title":"Strong cohomological rigidity of Hirzebruch surface bundles in Bott towers","authors":"Hiroaki Ishida","doi":"10.2969/jmsj/88718871","DOIUrl":"https://doi.org/10.2969/jmsj/88718871","url":null,"abstract":"We show the strong cohomological rigidity of Hirzebruch surface bundles over Bott manifolds. As a corollary, we have that the strong cohomological rigidity conjecture is true for Bott manifolds of dimension $8$.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41606144","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}
We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne-Mostow theory and periods of K3 surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of complex hyperbolic balls. We show in Theorem 7.4 that the two ball-quotient constructions can be unified in a geometric way.
{"title":"The complex ball-quotient structure of the moduli space of certain sextic curves","authors":"Zhiwei Zheng, Yiming Zhong","doi":"10.2969/jmsj/88318831","DOIUrl":"https://doi.org/10.2969/jmsj/88318831","url":null,"abstract":"We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne-Mostow theory and periods of K3 surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of complex hyperbolic balls. We show in Theorem 7.4 that the two ball-quotient constructions can be unified in a geometric way.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44959504","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}
(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show the existence of a singular level set of a convex Hamiltonian, which is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.
{"title":"Rigid fibers of integrable systems on cotangent bundles","authors":"Morimichi Kawasaki, Ryuma Orita","doi":"10.2969/jmsj/84278427","DOIUrl":"https://doi.org/10.2969/jmsj/84278427","url":null,"abstract":"(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show the existence of a singular level set of a convex Hamiltonian, which is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47935909","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}
A handlebody-link is a disjoint union of handlebodies embedded in the 3-sphere $S^3$. A multiple conjugation quandle is an algebraic system whose axioms are derived from the Reidemeister moves for handlebody-link diagrams. In this paper, we introduce the notion of a presentation of a multiple conjugation quandle and define the fundamental multiple conjugation quandle of a handlebody-link. We also see that the fundamental multiple conjugation quandle is an invariant of handlebody-links.
{"title":"The fundamental multiple conjugation quandle of a handlebody-link","authors":"Atsushi Ishii","doi":"10.2969/jmsj/84308430","DOIUrl":"https://doi.org/10.2969/jmsj/84308430","url":null,"abstract":"A handlebody-link is a disjoint union of handlebodies embedded in the 3-sphere $S^3$. A multiple conjugation quandle is an algebraic system whose axioms are derived from the Reidemeister moves for handlebody-link diagrams. In this paper, we introduce the notion of a presentation of a multiple conjugation quandle and define the fundamental multiple conjugation quandle of a handlebody-link. We also see that the fundamental multiple conjugation quandle is an invariant of handlebody-links.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47459818","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}
The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched m-primary ideals in the case where the reduction number attains almost minimal value in a Cohen-Macaulay local ring (A,m). As an application, we present complete descriptions of the associated graded ring of stretched m-primary ideals with small reduction number.
{"title":"The reduction number of stretched ideals","authors":"K. Ozeki","doi":"10.2969/jmsj/86498649","DOIUrl":"https://doi.org/10.2969/jmsj/86498649","url":null,"abstract":"The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched m-primary ideals in the case where the reduction number attains almost minimal value in a Cohen-Macaulay local ring (A,m). As an application, we present complete descriptions of the associated graded ring of stretched m-primary ideals with small reduction number.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48792082","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}
Hiraku Atobe, Masataka Chida, T. Ibukiyama, H. Katsurada, Takuya Yamauchi
Let $f$ be a primitive form with respect to $SL_2(Z)$. Then we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Duke-Imamoglu-Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $Sp_2(Z)$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.
{"title":"Harder's conjecture I","authors":"Hiraku Atobe, Masataka Chida, T. Ibukiyama, H. Katsurada, Takuya Yamauchi","doi":"10.2969/jmsj/87988798","DOIUrl":"https://doi.org/10.2969/jmsj/87988798","url":null,"abstract":"Let $f$ be a primitive form with respect to $SL_2(Z)$. Then we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Duke-Imamoglu-Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $Sp_2(Z)$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44438657","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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