We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to the round metric. A key tool is to employ the smooth Cheeger-Gromov compactness theorem to obtain general a priori estimates for Gauss curvatures $K$ which are in stable orbits of the conformal group $mathrm{Conf}(S^{2})$. We prove that in such a stable region, the map $u rightarrow K_{g}$, $g = e^{2u}g_{+1}$ is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. We also present a new proof and generalization of the Moser theorem on Gauss curvatures of even conformal metrics on $S^{2}$. In contrast to previous work, the work here does not use any of the Sobolev-type inequalities of Trudinger-Moser-Aubin-Onofri.
{"title":"The Nirenberg problem of prescribed Gauss curvature on $S^2$","authors":"Michael T. Anderson","doi":"10.4171/cmh/512","DOIUrl":"https://doi.org/10.4171/cmh/512","url":null,"abstract":"We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to the round metric. A key tool is to employ the smooth Cheeger-Gromov compactness theorem to obtain general a priori estimates for Gauss curvatures $K$ which are in stable orbits of the conformal group $mathrm{Conf}(S^{2})$. We prove that in such a stable region, the map $u rightarrow K_{g}$, $g = e^{2u}g_{+1}$ is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. \u0000We also present a new proof and generalization of the Moser theorem on Gauss curvatures of even conformal metrics on $S^{2}$. In contrast to previous work, the work here does not use any of the Sobolev-type inequalities of Trudinger-Moser-Aubin-Onofri.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91308426","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 exhibit the first examples of contact structures on $S^{2n-1}$ with $ngeq 4$ and on $S^3times S^2$, all equipped with their standard smooth structures, for which every Reeb flow has positive topological entropy. As a new technical tool for the study of the volume growth of Reeb flows we introduce the notion of algebraic growth of wrapped Floer homology. Its power stems from its stability under several geometric operations on Liouville domains.
{"title":"Dynamically exotic contact spheres in dimensions $geq 7$","authors":"Marcelo R. R. Alves, Matthias Meiwes","doi":"10.4171/cmh/468","DOIUrl":"https://doi.org/10.4171/cmh/468","url":null,"abstract":"We exhibit the first examples of contact structures on $S^{2n-1}$ with $ngeq 4$ and on $S^3times S^2$, all equipped with their standard smooth structures, for which every Reeb flow has positive topological entropy. As a new technical tool for the study of the volume growth of Reeb flows we introduce the notion of algebraic growth of wrapped Floer homology. Its power stems from its stability under several geometric operations on Liouville domains.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/468","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43803909","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}
Let $Lambda$ be a quasi-projective variety and assume that, either $Lambda$ is a subvariety of the moduli space $mathcal{M}_d$ of degree $d$ rational maps, or $Lambda$ parametrizes an algebraic family $(f_lambda)_{lambdainLambda}$ of degree $d$ rational maps on $mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$-th bifurcation current letting the periods of the cycles go to $infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the $log^+$ of the modulus of the multipliers of periodic points.
{"title":"Hyperbolic components of rational maps: Quantitative equidistribution and counting","authors":"T. Gauthier, Y. Okuyama, Gabriel Vigny","doi":"10.4171/CMH/462","DOIUrl":"https://doi.org/10.4171/CMH/462","url":null,"abstract":"Let $Lambda$ be a quasi-projective variety and assume that, either $Lambda$ is a subvariety of the moduli space $mathcal{M}_d$ of degree $d$ rational maps, or $Lambda$ parametrizes an algebraic family $(f_lambda)_{lambdainLambda}$ of degree $d$ rational maps on $mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$-th bifurcation current letting the periods of the cycles go to $infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the $log^+$ of the modulus of the multipliers of periodic points.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/462","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48372205","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 prove that isogenous K3 surfaces have isomorphic Chow motives. This provides a motivic interpretation of a long standing conjecture of Safarevich which has been settled only recently by Buskin. The main step consists of a new proof of Safarevich's conjecture that circumvents the analytic parts in Buskin's approach, avoiding twistor spaces and non-algebraic K3 surfaces.
{"title":"Motives of isogenous K3 surfaces","authors":"D. Huybrechts","doi":"10.4171/cmh/465","DOIUrl":"https://doi.org/10.4171/cmh/465","url":null,"abstract":"We prove that isogenous K3 surfaces have isomorphic Chow motives. This provides a motivic interpretation of a long standing conjecture of Safarevich which has been settled only recently by Buskin. The main step consists of a new proof of Safarevich's conjecture that circumvents the analytic parts in Buskin's approach, avoiding twistor spaces and non-algebraic K3 surfaces.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/465","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49584930","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}
M. Dajczer, Theodoros Kasioumis, A. Savas-Halilaj, T. Vlachos
In this paper we investigate m-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least m−2 at any point. These are austere submanifolds in the sense of Harvey and Lawson [19] and were initially studied by Bryant [3]. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit [7] in terms of a class of surfaces, called elliptic, for which the ellipse of curvature of a certain order is a circle at any point. Under the assumption of completeness, it turns out that any submanifold is either totally geodesic or has dimension three. In the latter case there are plenty of examples, even compact ones. Under the mild assumption that the Omori-Yau maximum principle holds on the manifold, a trivial condition in the compact case, we provide a complete local parametric description of the submanifolds in terms of 1-isotropic surfaces in Euclidean space. These are the minimal surfaces for which the standard ellipse of curvature is a circle at any point. For these surfaces, there exists a Weierstrass type representation that generates all simply connected ones. Let M be a complete m-dimensional Riemannian manifold. In [10] we considered the case of minimal isometric immersions into Euclidean space f : M → R, m ≥ 3, satisfying that the index of relative nullity is at least m − 2 at any point. Under the mild assumption that the Omori-Yau maximum principle holds on M, we concluded that any f must be “trivial”, namely, just a cylinder over a complete minimal surface. This result is global in nature since for any dimension there are plenty of non-complete examples other than open subsets of cylinders. It is natural to expect rather different type of conclusions when considering a similar global problem for minimal isometric immersions into nonflat space forms. For instance, for submanifolds in the hyperbolic space one would guess that under the same condition on the relative nullity index there exist many non-trivial examples, and that a kind of triviality conclusion will only hold under a strong additional assumption. The third author would like to acknowledge financial support from the grant DFG SM 78/6-1.
在本文中，我们研究了欧氏球中的m维完全极小子流形，其相对零度指数在任意点至少为m−2。这些是Harvey和Lawson[19]意义上的严格子流形，最初由Bryant[3]研究。对于任何维度和余维度，都有大量由Dajczer和Florit[7]根据一类称为椭圆的曲面充分描述的非完全例子，其中某阶曲率的椭圆在任何点上都是圆。在完备性假设下，证明了任何子流形要么是全测地的，要么是三维的。在后一种情况下，有很多例子，甚至是紧凑的例子。在一个温和的假设下，即Omori-Yau极大值原理在流形上成立，这是紧致情况下的一个平凡条件，我们在欧氏空间中用1-各向同性曲面提供了子流形的完整局部参数描述。这些是标准曲率椭圆在任何点都是圆的最小曲面。对于这些曲面，存在一个Weierstrass类型表示，它生成所有简单连接的曲面。设M是一个完全的M维黎曼流形。在[10]中，我们考虑了欧几里得空间f:M中最小等距浸入的情况→ R、 m≥3，满足相对零度指数在任意点至少为m−2。在大森-尤极大值原理对M成立的温和假设下，我们得出结论，任何f都必须是“平凡的”，即，只是一个完全极小表面上的圆柱体。这个结果本质上是全局的，因为对于任何维度，除了圆柱体的开子集之外，还有很多不完全的例子。当考虑一个类似的全局问题，将最小等距浸入非平面空间形式时，很自然地会得出截然不同的结论。例如，对于双曲空间中的子流形，人们可以猜测，在相对零度指数上的相同条件下，存在许多非平凡的例子，并且一种平凡的结论只有在一个强的附加假设下才成立。第三作者感谢DFG SM 78/6-1的资助。
{"title":"Complete minimal submanifolds with nullity in Euclidean spheres","authors":"M. Dajczer, Theodoros Kasioumis, A. Savas-Halilaj, T. Vlachos","doi":"10.4171/CMH/446","DOIUrl":"https://doi.org/10.4171/CMH/446","url":null,"abstract":"In this paper we investigate m-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least m−2 at any point. These are austere submanifolds in the sense of Harvey and Lawson [19] and were initially studied by Bryant [3]. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit [7] in terms of a class of surfaces, called elliptic, for which the ellipse of curvature of a certain order is a circle at any point. Under the assumption of completeness, it turns out that any submanifold is either totally geodesic or has dimension three. In the latter case there are plenty of examples, even compact ones. Under the mild assumption that the Omori-Yau maximum principle holds on the manifold, a trivial condition in the compact case, we provide a complete local parametric description of the submanifolds in terms of 1-isotropic surfaces in Euclidean space. These are the minimal surfaces for which the standard ellipse of curvature is a circle at any point. For these surfaces, there exists a Weierstrass type representation that generates all simply connected ones. Let M be a complete m-dimensional Riemannian manifold. In [10] we considered the case of minimal isometric immersions into Euclidean space f : M → R, m ≥ 3, satisfying that the index of relative nullity is at least m − 2 at any point. Under the mild assumption that the Omori-Yau maximum principle holds on M, we concluded that any f must be “trivial”, namely, just a cylinder over a complete minimal surface. This result is global in nature since for any dimension there are plenty of non-complete examples other than open subsets of cylinders. It is natural to expect rather different type of conclusions when considering a similar global problem for minimal isometric immersions into nonflat space forms. For instance, for submanifolds in the hyperbolic space one would guess that under the same condition on the relative nullity index there exist many non-trivial examples, and that a kind of triviality conclusion will only hold under a strong additional assumption. The third author would like to acknowledge financial support from the grant DFG SM 78/6-1.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/446","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49013349","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 prove that the Stern diatomic sequence is asymptotically distributed according to a normal law, on a logarithmic scale. This is obtained by studying complex moments, and the analytic properties of a transfer operator.
{"title":"Statistical distribution of the Stern sequence","authors":"S. Bettin, S. Drappeau, Lukas Spiegelhofer","doi":"10.4171/CMH/460","DOIUrl":"https://doi.org/10.4171/CMH/460","url":null,"abstract":"We prove that the Stern diatomic sequence is asymptotically distributed according to a normal law, on a logarithmic scale. This is obtained by studying complex moments, and the analytic properties of a transfer operator.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/460","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46311898","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}
Federico Franceschini, R. Frigerio, M. B. Pozzetti, A. Sisto
We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In the appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichm"uller translation distance.
{"title":"The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups","authors":"Federico Franceschini, R. Frigerio, M. B. Pozzetti, A. Sisto","doi":"10.4171/CMH/456","DOIUrl":"https://doi.org/10.4171/CMH/456","url":null,"abstract":"We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In the appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichm\"uller translation distance.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/456","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48725562","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}
In this article, we provide bounds on systoles associated to a holomorphic $1$-form $omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.
{"title":"The maximum number of systoles for genus two Riemann surfaces with abelian differentials","authors":"C. Judge, H. Parlier","doi":"10.4171/CMH/463","DOIUrl":"https://doi.org/10.4171/CMH/463","url":null,"abstract":"In this article, we provide bounds on systoles associated to a holomorphic $1$-form $omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/463","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44315036","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}
Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.
{"title":"Generalized triangle groups, expanders, and a problem of Agol and Wise","authors":"A. Lubotzky, J. Manning, H. Wilton","doi":"10.17863/CAM.27372","DOIUrl":"https://doi.org/10.17863/CAM.27372","url":null,"abstract":"Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49427426","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}
Let $F_n$ be the free group on $n$ generators and $Gamma_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $Gamma_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on $F_2$ that are $Aut(F_2)$-invariant. This answers an open problem posed by Miklos Abert.
{"title":"Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups","authors":"Michael Brandenbursky, Michał Marcinkowski","doi":"10.4171/cmh/470","DOIUrl":"https://doi.org/10.4171/cmh/470","url":null,"abstract":"Let $F_n$ be the free group on $n$ generators and $Gamma_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $Gamma_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on $F_2$ that are $Aut(F_2)$-invariant. This answers an open problem posed by Miklos Abert.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/470","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49095817","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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