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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.
{"title":"Ulrich bundles on cubic fourfolds","authors":"Daniele Faenzi, Yeongrak Kim","doi":"10.4171/cmh/546","DOIUrl":"https://doi.org/10.4171/cmh/546","url":null,"abstract":"We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43334076","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 define a new invariant of finitely generated representations of a finite group, with coefficients in a commutative noetherian ring. This invariant uses group cohomology and takes values in the singularity category of the coefficient ring. It detects which representations are controlled by permutation modules.
{"title":"Permutation modules and cohomological singularity","authors":"Paul Balmer, Martin Gallauer","doi":"10.4171/cmh/534","DOIUrl":"https://doi.org/10.4171/cmh/534","url":null,"abstract":". We define a new invariant of finitely generated representations of a finite group, with coefficients in a commutative noetherian ring. This invariant uses group cohomology and takes values in the singularity category of the coefficient ring. It detects which representations are controlled by permutation modules.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44949368","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：