We prove that all spaces of finite Assouad-Nagata dimension admit a good order for Travelling Salesman Problem, and provide sufficient conditions under which the converse is true. We formulate a conjectural characterisation of spaces of finite $AN$-dimension, which would yield a gap statement for the efficiency of orders on metric spaces. Under assumption of doubling, we prove a stronger gap phenomenon about all orders on a given metric space.
{"title":"Assouad–Nagata dimension and gap for ordered metric spaces","authors":"A. Erschler, I. Mitrofanov","doi":"10.4171/cmh/549","DOIUrl":"https://doi.org/10.4171/cmh/549","url":null,"abstract":"We prove that all spaces of finite Assouad-Nagata dimension admit a good order for Travelling Salesman Problem, and provide sufficient conditions under which the converse is true. We formulate a conjectural characterisation of spaces of finite $AN$-dimension, which would yield a gap statement for the efficiency of orders on metric spaces. Under assumption of doubling, we prove a stronger gap phenomenon about all orders on a given metric space.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43013292","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}
. This article gives a characterization of quotients of complex tori by finite groups acting freely in codimension two in terms of a numerical vanishing condition on the first and second Chern class. This generalizes results previously obtained by Greb–Kebekus–Peternell in the projective setting, and by Kirschner and the second author in dimension three. As a key ingredient to the proof, we obtain a version of the Bogomolov–Gieseker inequality for stable sheaves on singular spaces, including a discussion of the case of equality.
{"title":"Numerical characterization of complex torus quotients","authors":"B. Claudon, Patrick Graf, Henri Guenancia","doi":"10.4171/cmh/543","DOIUrl":"https://doi.org/10.4171/cmh/543","url":null,"abstract":". This article gives a characterization of quotients of complex tori by finite groups acting freely in codimension two in terms of a numerical vanishing condition on the first and second Chern class. This generalizes results previously obtained by Greb–Kebekus–Peternell in the projective setting, and by Kirschner and the second author in dimension three. As a key ingredient to the proof, we obtain a version of the Bogomolov–Gieseker inequality for stable sheaves on singular spaces, including a discussion of the case of equality.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47778336","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 K be a totally real number field of degree n ≥ 2. The inverse different of K gives rise to a lattice in R n . We prove that the space of Schwartz Fourier eigenfunctions on R n which vanish on the “component-wise square root” of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres √ mS n − 1 for integers m ≥ 0 and, as m → ∞ , there are ∼ c K m n − 1 many points on the m -th sphere for some explicit constant c K , proportional to the square root of the discriminant of K . This contrasts a recent Fourier uniqueness result by Stoller [17, Cor. 1.1]. Using a different construction involving the codifferent of K , we prove an analogue for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each. We also study a related question about existence of Fourier interpolation formulas with nodes “ √ Λ” for general lattices Λ ⊂ R n . Using results about lattices in Lie groups of higher rank we prove that if n ≥ 2 and a certain group Γ Λ ≤ PSL 2 ( R ) n is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all n ≥ 5 and all real λ > 2, Fourier interpolation results for sequences of spheres (cid:112) 2 m/λS n − 1 , where m ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincar´e type for Hecke groups of infinite covolume and is similar to the one in [17, § 4].
设K是n≥2次的全实数域。K的倒数在Rn中产生了一个晶格。我们证明了在该晶格的“分量平方根”上消失的Rn上的Schwartz-Fourier本征函数的空间是有限维的。由此获得的傅立叶非唯一性集是所有球面并集的离散子集√mS n−1，对于整数m≥0和，作为m→ ∞ , 对于某个显式常数c K，在第m个球面上有~c K m n−1个多点，与K的判别式的平方根成比例。这与Stoller[17，Cor.1-1]最近的傅立叶唯一性结果形成了对比。使用涉及K的协差的不同构造，我们证明了椭球离散子集的相似性。在特殊情况下，这些集合也位于半径更密集的球体上，但每个球体上的点更少。我们还研究了一般格∧⊂Rn的节点为“√∧”的傅立叶插值公式的存在性的一个相关问题。利用关于高阶李群中格的结果，我们证明了如果n≥2，并且某个群Γ∧≤PSL2（R）n是离散的，那么这样的插值公式不可能存在。出于这些更一般的考虑，我们重新审视了一个径向变量的情况，并证明了对于所有n≥5和所有实数λ>2，球面序列（cid:112）2 m/λS n−1的傅立叶插值结果，其中m的范围在任何固定的非负整数集上。该证明依赖于有限体积Hecke群的一系列Poincar´e型，与[17，§4]中的证明类似。
{"title":"Fourier non-uniqueness sets from totally real number fields","authors":"D. Radchenko, Martin Stoller","doi":"10.4171/cmh/538","DOIUrl":"https://doi.org/10.4171/cmh/538","url":null,"abstract":". Let K be a totally real number field of degree n ≥ 2. The inverse different of K gives rise to a lattice in R n . We prove that the space of Schwartz Fourier eigenfunctions on R n which vanish on the “component-wise square root” of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres √ mS n − 1 for integers m ≥ 0 and, as m → ∞ , there are ∼ c K m n − 1 many points on the m -th sphere for some explicit constant c K , proportional to the square root of the discriminant of K . This contrasts a recent Fourier uniqueness result by Stoller [17, Cor. 1.1]. Using a different construction involving the codifferent of K , we prove an analogue for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each. We also study a related question about existence of Fourier interpolation formulas with nodes “ √ Λ” for general lattices Λ ⊂ R n . Using results about lattices in Lie groups of higher rank we prove that if n ≥ 2 and a certain group Γ Λ ≤ PSL 2 ( R ) n is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all n ≥ 5 and all real λ > 2, Fourier interpolation results for sequences of spheres (cid:112) 2 m/λS n − 1 , where m ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincar´e type for Hecke groups of infinite covolume and is similar to the one in [17, § 4].","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"61 25","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41246446","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}
For a given pseudo-Anosov homeomorphism $varphi$ of a closed surface $S$, the action of $varphi$ on the Teichm"uller space $mathcal T(S)$ preserves the Weil-Petersson symplectic form. We give explicit formulae for two invariant functions $mathcal T(S)to mathbb R$ whose symplectic gradients generate autonomous Hamiltonian flows that coincide with the action of $varphi$ at time one. We compute the Poisson bracket between these two functions. This amounts to computing the variation of length of a H"older cocyle on one lamination along a shear vector field defined by another. For a measurably generic set of laminations, we prove that the variation of length is expressed as the cosine of the angle between the two laminations integrated against the product H"older distribution, generalizing a result of Kerckhoff. We also obtain rates of convergence for the supports of germs of differentiable paths of measured laminations in the Hausdorff metric on a hyperbolic surface, which may be of independent interest.
{"title":"Hamiltonian flows for pseudo-Anosov mapping classes","authors":"James Farre","doi":"10.4171/cmh/551","DOIUrl":"https://doi.org/10.4171/cmh/551","url":null,"abstract":"For a given pseudo-Anosov homeomorphism $varphi$ of a closed surface $S$, the action of $varphi$ on the Teichm\"uller space $mathcal T(S)$ preserves the Weil-Petersson symplectic form. We give explicit formulae for two invariant functions $mathcal T(S)to mathbb R$ whose symplectic gradients generate autonomous Hamiltonian flows that coincide with the action of $varphi$ at time one. We compute the Poisson bracket between these two functions. This amounts to computing the variation of length of a H\"older cocyle on one lamination along a shear vector field defined by another. For a measurably generic set of laminations, we prove that the variation of length is expressed as the cosine of the angle between the two laminations integrated against the product H\"older distribution, generalizing a result of Kerckhoff. We also obtain rates of convergence for the supports of germs of differentiable paths of measured laminations in the Hausdorff metric on a hyperbolic surface, which may be of independent interest.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47226955","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 give asymptotic upper and lower bounds for the real and imaginary parts of cycle integrals of the classical modular j-function along geodesics that correspond to Markov irrationalities.
给出了经典模j函数沿测地线的实部和虚部积分的渐近上界和下界。
{"title":"Cycle integrals of the $j$-function on Markov geodesics","authors":"P. Bengoechea","doi":"10.4171/cmh/535","DOIUrl":"https://doi.org/10.4171/cmh/535","url":null,"abstract":"We give asymptotic upper and lower bounds for the real and imaginary parts of cycle integrals of the classical modular j-function along geodesics that correspond to Markov irrationalities.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41499068","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}
. For hypersurfaces moving by standard mean curvature flow with boundary, we show that if the tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in R 3 that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces. Let κ be the largest number with the following property: if M is a minimal surface in R 3 bounded by a smooth simple closed curve of total curvature < κ , then M is a disk. Examples show that κ < 4 π . In this paper, we use mean curvature flow to show that κ > 3 π . We get a slightly larger lower bound for orientable surfaces.
{"title":"Boundary singularities in mean curvature flow and total curvature of minimal surface boundaries","authors":"B. White","doi":"10.4171/cmh/542","DOIUrl":"https://doi.org/10.4171/cmh/542","url":null,"abstract":". For hypersurfaces moving by standard mean curvature flow with boundary, we show that if the tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in R 3 that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces. Let κ be the largest number with the following property: if M is a minimal surface in R 3 bounded by a smooth simple closed curve of total curvature < κ , then M is a disk. Examples show that κ < 4 π . In this paper, we use mean curvature flow to show that κ > 3 π . We get a slightly larger lower bound for orientable surfaces.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49412718","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 give effective bilipschitz bounds on the change in metric between thick parts of a cusped hyperbolic 3-manifold and its long Dehn fillings. In the thin parts of the manifold, we give effective bounds on the change in complex length of a short closed geodesic. These results quantify the filling theorem of Brock and Bromberg, and extend previous results of the authors from finite volume hyperbolic 3-manifolds to any tame hyperbolic 3-manifold. To prove the main results, we assemble tools from Kleinian group theory into a template for transferring theorems about finite-volume manifolds into theorems about infinite-volume manifolds. We also prove and apply an infinite-volume version of the 6-Theorem.
{"title":"Effective drilling and filling of tame hyperbolic 3-manifolds","authors":"D. Futer, J. Purcell, S. Schleimer","doi":"10.4171/CMH/536","DOIUrl":"https://doi.org/10.4171/CMH/536","url":null,"abstract":"We give effective bilipschitz bounds on the change in metric between thick parts of a cusped hyperbolic 3-manifold and its long Dehn fillings. In the thin parts of the manifold, we give effective bounds on the change in complex length of a short closed geodesic. These results quantify the filling theorem of Brock and Bromberg, and extend previous results of the authors from finite volume hyperbolic 3-manifolds to any tame hyperbolic 3-manifold. To prove the main results, we assemble tools from Kleinian group theory into a template for transferring theorems about finite-volume manifolds into theorems about infinite-volume manifolds. We also prove and apply an infinite-volume version of the 6-Theorem.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42048255","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 an upper bound on the rank of the abelianised revised fundamental group (called"revised first Betti number") of a compact $RCD^{*}(K,N)$ space, in the same spirit of the celebrated Gromov-Gallot upper bound on the first Betti number for a smooth compact Riemannian manifold with Ricci curvature bounded below. When the synthetic lower Ricci bound is close enough to (negative) zero and the aforementioned upper bound on the revised first Betti number is saturated (i.e. equal to the integer part of $N$, denoted by $lfloor N rfloor$), then we establish a torus stability result stating that the space is $lfloor N rfloor$-rectifiable as a metric measure space, and a finite cover must be mGH-close to an $lfloor N rfloor$-dimensional flat torus; moreover, in case $N$ is an integer, we prove that the space itself is bi-H"older homeomorphic to a flat torus. This second result extends to the class of non-smooth $RCD^{*}(-delta, N)$ spaces a celebrated torus stability theorem by Colding (later refined by Cheeger-Colding).
我们证明了紧$RCD^{*}(K,N)$空间的阿贝列化修正基本群(称为“修正第一Betti数”)的秩上界，其精神与著名的具有Ricci曲率的光滑紧黎曼流形的第一Betti数的Gromov-Gallot上界相同。当合成下界足够接近于(负)零，且修正后的第一Betti数上的上界饱和(即等于$N$的整数部分，记为$lfloor N rfloor$)，则我们建立了环面稳定性结果，表明该空间作为度量度量空间是$lfloor N rfloor$-可整流的，并且有限覆盖必须mgh -接近$lfloor N rfloor$-维平面环面;此外，当$N$是整数时，我们证明了空间本身是平面环面的bi-H old同胚。第二个结果推广到一类非光滑的$RCD^{*}(-delta, N)$空间，这是由Colding提出的著名环面稳定性定理(后来由Cheeger-Colding改进)。
{"title":"An upper bound on the revised first Betti number and a torus stability result for RCD spaces","authors":"Ilaria Mondello, A. Mondino, Raquel Perales","doi":"10.4171/CMH/540","DOIUrl":"https://doi.org/10.4171/CMH/540","url":null,"abstract":"We prove an upper bound on the rank of the abelianised revised fundamental group (called\"revised first Betti number\") of a compact $RCD^{*}(K,N)$ space, in the same spirit of the celebrated Gromov-Gallot upper bound on the first Betti number for a smooth compact Riemannian manifold with Ricci curvature bounded below. When the synthetic lower Ricci bound is close enough to (negative) zero and the aforementioned upper bound on the revised first Betti number is saturated (i.e. equal to the integer part of $N$, denoted by $lfloor N rfloor$), then we establish a torus stability result stating that the space is $lfloor N rfloor$-rectifiable as a metric measure space, and a finite cover must be mGH-close to an $lfloor N rfloor$-dimensional flat torus; moreover, in case $N$ is an integer, we prove that the space itself is bi-H\"older homeomorphic to a flat torus. This second result extends to the class of non-smooth $RCD^{*}(-delta, N)$ spaces a celebrated torus stability theorem by Colding (later refined by Cheeger-Colding).","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47919805","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 study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four.
研究了非闭域上两个二次曲面光滑完全交的合理性构造。在实数上，我们建立了四维合理性标准。
{"title":"Rationality of even-dimensional intersections of two real quadrics","authors":"B. Hassett, J. Koll'ar, Y. Tschinkel","doi":"10.4171/cmh/529","DOIUrl":"https://doi.org/10.4171/cmh/529","url":null,"abstract":"We study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44611617","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 study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a Folner-type characterization, and discuss closure properties of the class of skew-amenable topological groups. Moreover, we isolate a dynamical sufficient condition for skew-amenability and provide several concrete variations of this criterion in the context of transformation groups. These results are then used to decide skew-amenability for a number of examples of topological groups built from or related to Thompson's group $F$ and Monod's group of piecewise projective homeomorphisms of the real line.
{"title":"Skew-amenability of topological groups","authors":"K. Juschenko, Friedrich Martin Schneider","doi":"10.4171/CMH/525","DOIUrl":"https://doi.org/10.4171/CMH/525","url":null,"abstract":"We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a Folner-type characterization, and discuss closure properties of the class of skew-amenable topological groups. Moreover, we isolate a dynamical sufficient condition for skew-amenability and provide several concrete variations of this criterion in the context of transformation groups. These results are then used to decide skew-amenability for a number of examples of topological groups built from or related to Thompson's group $F$ and Monod's group of piecewise projective homeomorphisms of the real line.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45447970","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：