In this paper we show that every three-dimensional closed hyperbolic manifold admits no locally geometric $1$-parameter family of closed minimal surfaces.
本文证明了每一个三维闭双曲流形不存在局部几何$1$参数的闭极小曲面族。
{"title":"Non-existence of geometric minimal foliations in hyperbolic three-manifolds","authors":"Michael Wolf, Yunhui Wu","doi":"10.4171/CMH/484","DOIUrl":"https://doi.org/10.4171/CMH/484","url":null,"abstract":"In this paper we show that every three-dimensional closed hyperbolic manifold admits no locally geometric $1$-parameter family of closed minimal surfaces.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42874725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We find the minimal size of 4 dimensional balls and polydisks into which product Lagrangian tori can be mapped by a Hamiltonian diffeomorphism.
我们找到了四维球和多盘的最小尺寸，其中积拉格朗日环面可以用哈密顿微分同构映射。
{"title":"Squeezing Lagrangian tori in dimension 4","authors":"R. Hind, E. Opshtein","doi":"10.4171/cmh/496","DOIUrl":"https://doi.org/10.4171/cmh/496","url":null,"abstract":"We find the minimal size of 4 dimensional balls and polydisks into which product Lagrangian tori can be mapped by a Hamiltonian diffeomorphism.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/496","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47530057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We recover results by Ullmo-Yafaev and Peterzil-Starchenko on the closure of the image of an algebraic variety in a compact complex torus. Our approach uses directed closed currents and allows us to extend the result for dimension 1 flows to the setting of commutative complex Lie groups which are not necessarily compact. A version of the classical Ax-Lindemann-Weierstrass theorem for commutative complex Lie groups is also given.
{"title":"Algebraic flows on commutative complex Lie groups","authors":"T. Dinh, Duc-Viet Vu","doi":"10.4171/cmh/492","DOIUrl":"https://doi.org/10.4171/cmh/492","url":null,"abstract":"We recover results by Ullmo-Yafaev and Peterzil-Starchenko on the closure of the image of an algebraic variety in a compact complex torus. Our approach uses directed closed currents and allows us to extend the result for dimension 1 flows to the setting of commutative complex Lie groups which are not necessarily compact. A version of the classical Ax-Lindemann-Weierstrass theorem for commutative complex Lie groups is also given.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/492","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49006217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Frei, D. Loughran, Rachel Newton, with an appendix by Yonatan Harpaz, Olivier Wittenberg
We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.
{"title":"Number fields with prescribed norms (with an appendix by Yonatan Harpaz and Olivier Wittenberg)","authors":"C. Frei, D. Loughran, Rachel Newton, with an appendix by Yonatan Harpaz, Olivier Wittenberg","doi":"10.4171/CMH/528","DOIUrl":"https://doi.org/10.4171/CMH/528","url":null,"abstract":"We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44493462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small deformation of the coefficients of the original equations. This method is based on the properties of Zariski equisingular families of varieties. Moreover we construct an algorithm, that, given a system of equations defining a variety $V$, produces a system of equations with algebraic coefficients of a variety homeomorphic to $V$
{"title":"Algebraic varieties are homeomorphic to varieties defined over number fields","authors":"A. Parusiński, G. Rond","doi":"10.4171/cmh/490","DOIUrl":"https://doi.org/10.4171/cmh/490","url":null,"abstract":"We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small deformation of the coefficients of the original equations. This method is based on the properties of Zariski equisingular families of varieties. \u0000Moreover we construct an algorithm, that, given a system of equations defining a variety $V$, produces a system of equations with algebraic coefficients of a variety homeomorphic to $V$","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/490","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42873545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We relate the Gromov norm on homology classes to the harmonic norm on the dual cohomology and obtain double sided bounds in terms of the volume and other geometric quantities of the underlying manifold. Along the way, we provide comparisons to other related norms and quantities as well.
{"title":"Homological norms on nonpositively curved manifolds","authors":"C. Connell, Shi Wang","doi":"10.4171/cmh/550","DOIUrl":"https://doi.org/10.4171/cmh/550","url":null,"abstract":"We relate the Gromov norm on homology classes to the harmonic norm on the dual cohomology and obtain double sided bounds in terms of the volume and other geometric quantities of the underlying manifold. Along the way, we provide comparisons to other related norms and quantities as well.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47368235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L in X to the skyscraper sheaf of a point y of Y. We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the `Beauville--Voisin subring' in the Chow groups of Y, and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.
{"title":"Rational equivalence and Lagrangian tori on K3 surfaces","authors":"Nick Sheridan, I. Smith","doi":"10.4171/CMH/489","DOIUrl":"https://doi.org/10.4171/CMH/489","url":null,"abstract":"Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L in X to the skyscraper sheaf of a point y of Y. We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the `Beauville--Voisin subring' in the Chow groups of Y, and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43546149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles, without any restrictions on the magnetic form, using the dissipative method for compactness introduced in cite{Groman2015}. As an application, we deduce that if $N$ is a closed manifold and $ sigma$ is a magnetic form that is not weakly exact, then the $ pi_1$-sensitive Hofer-Zehnder capacity of any compact set in the magnetic cotangent bundle determined by $ sigma$ is finite.
{"title":"The symplectic cohomology of magnetic cotangent bundles","authors":"Yoel Groman, W. Merry","doi":"10.4171/cmh/555","DOIUrl":"https://doi.org/10.4171/cmh/555","url":null,"abstract":"We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles, without any restrictions on the magnetic form, using the dissipative method for compactness introduced in cite{Groman2015}. As an application, we deduce that if $N$ is a closed manifold and $ sigma$ is a magnetic form that is not weakly exact, then the $ pi_1$-sensitive Hofer-Zehnder capacity of any compact set in the magnetic cotangent bundle determined by $ sigma$ is finite.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45904381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that if $K$ is a monogenic, primitive, totally real number field, that contains units of every signature, then there exists a lower bound for the rank of integer universal quadratic forms defined over $K$. In particular, we extend the work of Blomer and Kala, to show that there exist infinitely many totally real cubic number fields that do not have a universal quadratic form of a given rank defined over them. For the real quadratic number fields with a unit of negative norm, we show that the minimal rank of a universal quadratic form goes to infinity as the discriminant of the number field grows. These results follow from the study of interlacing polynomials. Specifically, we show that there are only finitely many irreducible monic polynomials related to primitive number fields of a given degree, that have a bounded number of interlacing polynomials.
{"title":"A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real number field","authors":"Pavlo Yatsyna","doi":"10.4171/CMH/459","DOIUrl":"https://doi.org/10.4171/CMH/459","url":null,"abstract":"We show that if $K$ is a monogenic, primitive, totally real number field, that contains units of every signature, then there exists a lower bound for the rank of integer universal quadratic forms defined over $K$. In particular, we extend the work of Blomer and Kala, to show that there exist infinitely many totally real cubic number fields that do not have a universal quadratic form of a given rank defined over them. For the real quadratic number fields with a unit of negative norm, we show that the minimal rank of a universal quadratic form goes to infinity as the discriminant of the number field grows. These results follow from the study of interlacing polynomials. Specifically, we show that there are only finitely many irreducible monic polynomials related to primitive number fields of a given degree, that have a bounded number of interlacing polynomials.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/459","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45733702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pointwise convergence of spherical averages is proved for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen-Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by the first author for the case of free groups may be applied.
{"title":"Convergence of spherical averages for actions of Fuchsian groups","authors":"A. Bufetov, A. Klimenko, C. Series","doi":"10.4171/cmh/548","DOIUrl":"https://doi.org/10.4171/cmh/548","url":null,"abstract":"Pointwise convergence of spherical averages is proved for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen-Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by the first author for the case of free groups may be applied.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47134249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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