Lagrangian surgery and Lagrangian cobordism give geometric interpretations to exact triangles in Floer cohomology. Lagrangian $k$-surgery modifies an immersed Lagrangian submanifold by topological $k$-surgery while removing a self-intersection point of the immersion. Associated to a $k$-surgery is a Lagrangian surgery trace cobordism. We prove that every Lagrangian cobordism is exactly homotopic to a concatenation of suspension cobordisms and Lagrangian surgery traces. Furthermore, we show that each Lagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain associated to the handle attachment with the Floer cochain generated by the self-intersection. We give a sample computation for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds, recover the Lagrangian surgery exact sequence, and provide conditions for when non-monotone Lagrangian cobordisms yield continuation maps in the Fukaya category.
{"title":"Lagrangian cobordisms and Lagrangian surgery","authors":"Jeff Hicks","doi":"10.4171/cmh/554","DOIUrl":"https://doi.org/10.4171/cmh/554","url":null,"abstract":"Lagrangian surgery and Lagrangian cobordism give geometric interpretations to exact triangles in Floer cohomology. Lagrangian $k$-surgery modifies an immersed Lagrangian submanifold by topological $k$-surgery while removing a self-intersection point of the immersion. Associated to a $k$-surgery is a Lagrangian surgery trace cobordism. We prove that every Lagrangian cobordism is exactly homotopic to a concatenation of suspension cobordisms and Lagrangian surgery traces. Furthermore, we show that each Lagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain associated to the handle attachment with the Floer cochain generated by the self-intersection. We give a sample computation for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds, recover the Lagrangian surgery exact sequence, and provide conditions for when non-monotone Lagrangian cobordisms yield continuation maps in the Fukaya category.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135774997","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}
Sébastien Gouëzel, Camille Noûs, Barbara Schapira, Samuel Tapie, Felipe Riquelme
In the context of geodesic flows of noncompact negatively curved manifolds, we propose three different definitions of entropy and pressure at infinity, through growth of periodic orbits, critical exponents of Poincaré series, and entropy (pressure) of invariant measures. We show that these notions coincide. Thanks to these entropy and pressure at infinity, we investigate thoroughly the notion of strong positive recurrence in this geometric context. A potential is said to be strongly positively recurrent when its pressure at infinity is strictly smaller than the full topological pressure. We show, in particular, that if a potential is strongly positively recurrent, then it admits a finite Gibbs measure. We also provide easy criteria allowing to build such strong positively recurrent potentials and many examples.
{"title":"Pressure at infinity and strong positive recurrence in negative curvature","authors":"Sébastien Gouëzel, Camille Noûs, Barbara Schapira, Samuel Tapie, Felipe Riquelme","doi":"10.4171/cmh/552","DOIUrl":"https://doi.org/10.4171/cmh/552","url":null,"abstract":"In the context of geodesic flows of noncompact negatively curved manifolds, we propose three different definitions of entropy and pressure at infinity, through growth of periodic orbits, critical exponents of Poincaré series, and entropy (pressure) of invariant measures. We show that these notions coincide. Thanks to these entropy and pressure at infinity, we investigate thoroughly the notion of strong positive recurrence in this geometric context. A potential is said to be strongly positively recurrent when its pressure at infinity is strictly smaller than the full topological pressure. We show, in particular, that if a potential is strongly positively recurrent, then it admits a finite Gibbs measure. We also provide easy criteria allowing to build such strong positively recurrent potentials and many examples.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135220348","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}
{"title":"Ricci flow of $W^{2,2}$-metrics in four dimensions","authors":"Tobias Lamm, M. Simon","doi":"10.4171/cmh/553","DOIUrl":"https://doi.org/10.4171/cmh/553","url":null,"abstract":"","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41888105","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}
Luke Jeffreys, Carlos Matheus, Carlos Gustavo Moreira, Clément Rieutord
{"title":"Corrigendum and addendum to Appendix A of “Fractal geometry of the complement of Lagrange spectrum in Markov spectrum”","authors":"Luke Jeffreys, Carlos Matheus, Carlos Gustavo Moreira, Clément Rieutord","doi":"10.4171/cmh/558","DOIUrl":"https://doi.org/10.4171/cmh/558","url":null,"abstract":"","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48801169","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 the transverse geometric behavior of 2-dimensional foliations in 3-manifolds. We show that an $mathbb{R}$-covered, transversely orientable foliation with Gromov hyperbolic leaves in a closed 3-manifold admits a regulating, transverse pseudo-Anosov flow (in the appropriate sense) in each atoroidal piece of the manifold. The flow is a blow up of a one prong pseudo-Anosov flow. In addition we show that there is a regulating flow for the whole foliation. We also determine how deck transformations act on the universal circle of the foliation.
{"title":"$mathbb{R}$-covered foliations and transverse pseudo-Anosov flows in atoroidal pieces","authors":"Sergio R. Fenley","doi":"10.4171/cmh/547","DOIUrl":"https://doi.org/10.4171/cmh/547","url":null,"abstract":"We study the transverse geometric behavior of 2-dimensional foliations in 3-manifolds. We show that an $mathbb{R}$-covered, transversely orientable foliation with Gromov hyperbolic leaves in a closed 3-manifold admits a regulating, transverse pseudo-Anosov flow (in the appropriate sense) in each atoroidal piece of the manifold. The flow is a blow up of a one prong pseudo-Anosov flow. In addition we show that there is a regulating flow for the whole foliation. We also determine how deck transformations act on the universal circle of the foliation.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135183733","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 establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $operatorname{Ric}_{infty} ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry–Ledoux’s Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag’s needle decomposition (also called localization), and is inspired by a recent work of Cavalletti, Maggi and Mondino on compact spaces. Besides the quantitative isoperimetry, a reverse Poincaré inequality for the guiding function that we have as a key step, as well as the way we use it, are of independent interest.
利用$operatorname{Ric}_{infty} ge 1$建立了加权黎曼流形的定量等周不等式。准确地说，我们根据Bakry-Ledoux高斯等周不等式的亏缺，给出了Borel集与相关引导函数(由针分解产生)的子层(或超层)集之间对称差的体积上界。这是除欧几里德空间和高斯空间外，在非紧空间上的第一个定量等周不等式。我们的论证利用了Klartag的针状分解(也称为局部化)，并受到Cavalletti, Maggi和Mondino最近关于紧空间的工作的启发。除了定量等径法，我们作为关键步骤的指导函数的逆庞加莱不等式，以及我们使用它的方式，都是独立的兴趣。
{"title":"Quantitative estimates for the Bakry–Ledoux isoperimetric inequality","authors":"Cong Hung Mai, Shin-ichi Ohta","doi":"10.4171/cmh/523","DOIUrl":"https://doi.org/10.4171/cmh/523","url":null,"abstract":"We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $operatorname{Ric}_{infty} ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry–Ledoux’s Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag’s needle decomposition (also called localization), and is inspired by a recent work of Cavalletti, Maggi and Mondino on compact spaces. Besides the quantitative isoperimetry, a reverse Poincaré inequality for the guiding function that we have as a key step, as well as the way we use it, are of independent interest.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138540715","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 paper, we investigate manifolds for which the curvature of the second kind (following the terminology of Nishikawa) satisfies certain positivity conditions. Our main result settles Nishikawa's conjecture that manifolds for which the curvature (operator) of the second kind are positive are diffeomorphic to a sphere, by showing that such manifolds satisfy Brendle's PIC1 condition. In dimension four we show that curvature of the second kind has a canonical normal form, and use this to classify Einstein four-manifolds for which the curvature (operator) of the second kind is five-non-negative. We also calculate the normal form for some explicit examples in order to show that this assumption is sharp.
{"title":"Curvature of the second kind and a conjecture of Nishikawa","authors":"M. Gursky, Xiaodong Cao, Hung Tran","doi":"10.4171/cmh/545","DOIUrl":"https://doi.org/10.4171/cmh/545","url":null,"abstract":"In this paper, we investigate manifolds for which the curvature of the second kind (following the terminology of Nishikawa) satisfies certain positivity conditions. Our main result settles Nishikawa's conjecture that manifolds for which the curvature (operator) of the second kind are positive are diffeomorphic to a sphere, by showing that such manifolds satisfy Brendle's PIC1 condition. In dimension four we show that curvature of the second kind has a canonical normal form, and use this to classify Einstein four-manifolds for which the curvature (operator) of the second kind is five-non-negative. We also calculate the normal form for some explicit examples in order to show that this assumption is sharp.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42823620","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 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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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