{"title":"Erratum to “The cyclic homology of the group rings”","authors":"D. Burghelea","doi":"10.4171/cmh/561","DOIUrl":"https://doi.org/10.4171/cmh/561","url":null,"abstract":"","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"59 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139262026","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}
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":"221 2","pages":"0"},"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":"40 1","pages":"0"},"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":"1 1","pages":""},"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":" ","pages":""},"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":"24 1","pages":"0"},"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":"47 1","pages":""},"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":" ","pages":""},"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}
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：