{"title":"A Riemann–Hilbert correspondence in positive characteristic","authors":"B. Bhatt, J. Lurie","doi":"10.4310/CJM.2019.V7.N1.A3","DOIUrl":"https://doi.org/10.4310/CJM.2019.V7.N1.A3","url":null,"abstract":"We explain a version of the Riemann-Hilbert correspondence for $p$-torsion 'etale sheaves on an arbitrary $mathbf{F}_p$-scheme.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2017-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49105284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-10-18DOI: 10.4310/cjm.2021.v9.n1.a4
Miaofen Chen, Laurent Fargues, Xu Shen
We prove the Fargues-Rapoport conjecture for p-adic period domains: for a reductive group G over a p-adic field and a minuscule cocharacter {mu} of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G,{mu}) is fully Hodge-Newton decomposable.
{"title":"On the structure of some $p$-adic period domains","authors":"Miaofen Chen, Laurent Fargues, Xu Shen","doi":"10.4310/cjm.2021.v9.n1.a4","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n1.a4","url":null,"abstract":"We prove the Fargues-Rapoport conjecture for p-adic period domains: for a reductive group G over a p-adic field and a minuscule cocharacter {mu} of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G,{mu}) is fully Hodge-Newton decomposable.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2017-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45364649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-14DOI: 10.4310/cjm.2019.v7.n4.a2
Shuang Miao, S. Shahshahani
In this work, which is based on an essential linear analysis carried out by Christodoulou, we study the evolution of tidal energy for the motion of two gravitating incompressible fluid balls with free boundaries obeying the Euler-Poisson equations. The orbital energy is defined as the mechanical energy of the two bodies' center of mass. According to the classical analysis of Kepler and Newton, when the fluids are replaced by point masses, the conic curve describing the trajectories of the masses is a hyperbola when the orbital energy is positive and an ellipse when the orbital energy is negative. The orbital energy is conserved in the case of point masses. If the point masses are initially very far, then the orbital energy is positive, corresponding to hyperbolic motion. However, in the motion of fluid bodies the orbital energy is no longer conserved because part of the conserved energy is used in deforming the boundaries of the bodies. In this case the total energy $tilde{mathcal{E}}$ can be decomposed into a sum $tilde{mathcal{E}}:=widetilde{mathcal{E}_{{mathrm{orbital}}}}+widetilde{mathcal{E}_{{mathrm{tidal}}}}$, with $widetilde{mathcal{E}_{{mathrm{tidal}}}}$ measuring the energy used in deforming the boundaries, such that if $widetilde{mathcal{E}_{{mathrm{orbital}}}} 0$, then the orbit of the bodies must be bounded. In this work we prove that under appropriate conditions on the initial configuration of the system, the fluid boundaries and velocity remain regular up to the point of the first closest approach in the orbit, and that the tidal energy $widetilde{mathcal{E}_{{mathrm{tidal}}}}$ can be made arbitrarily large relative to the total energy $tilde{mathcal{E}}$. In particular under these conditions $widetilde{mathcal{E}_{{mathrm{orbital}}}}$, which is initially positive, becomes negative before the point of the first closest approach.
{"title":"On tidal energy in Newtonian two-body motion","authors":"Shuang Miao, S. Shahshahani","doi":"10.4310/cjm.2019.v7.n4.a2","DOIUrl":"https://doi.org/10.4310/cjm.2019.v7.n4.a2","url":null,"abstract":"In this work, which is based on an essential linear analysis carried out by Christodoulou, we study the evolution of tidal energy for the motion of two gravitating incompressible fluid balls with free boundaries obeying the Euler-Poisson equations. The orbital energy is defined as the mechanical energy of the two bodies' center of mass. According to the classical analysis of Kepler and Newton, when the fluids are replaced by point masses, the conic curve describing the trajectories of the masses is a hyperbola when the orbital energy is positive and an ellipse when the orbital energy is negative. The orbital energy is conserved in the case of point masses. If the point masses are initially very far, then the orbital energy is positive, corresponding to hyperbolic motion. However, in the motion of fluid bodies the orbital energy is no longer conserved because part of the conserved energy is used in deforming the boundaries of the bodies. In this case the total energy $tilde{mathcal{E}}$ can be decomposed into a sum $tilde{mathcal{E}}:=widetilde{mathcal{E}_{{mathrm{orbital}}}}+widetilde{mathcal{E}_{{mathrm{tidal}}}}$, with $widetilde{mathcal{E}_{{mathrm{tidal}}}}$ measuring the energy used in deforming the boundaries, such that if $widetilde{mathcal{E}_{{mathrm{orbital}}}} 0$, then the orbit of the bodies must be bounded. In this work we prove that under appropriate conditions on the initial configuration of the system, the fluid boundaries and velocity remain regular up to the point of the first closest approach in the orbit, and that the tidal energy $widetilde{mathcal{E}_{{mathrm{tidal}}}}$ can be made arbitrarily large relative to the total energy $tilde{mathcal{E}}$. In particular under these conditions $widetilde{mathcal{E}_{{mathrm{orbital}}}}$, which is initially positive, becomes negative before the point of the first closest approach.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2017-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44140504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-06-30DOI: 10.4310/CJM.2019.V7.N1.A1
Morten Lüders
We prove a restriction isomorphism for zero-cycles with coefficients in Milnor K-theory for smooth projective schemes over excellent henselian discrete valuation rings. Furthermore we relate zero-cycles with coefficients in Milnor K-theory to 'etale cohomology and certain Kato complexes and deduce finiteness results for zero-cycles with coefficients in Milnor K-theory over local fields.
{"title":"A restriction isomorphism for zero-cycles with coefficients in Milnor K-theory","authors":"Morten Lüders","doi":"10.4310/CJM.2019.V7.N1.A1","DOIUrl":"https://doi.org/10.4310/CJM.2019.V7.N1.A1","url":null,"abstract":"We prove a restriction isomorphism for zero-cycles with coefficients in Milnor K-theory for smooth projective schemes over excellent henselian discrete valuation rings. Furthermore we relate zero-cycles with coefficients in Milnor K-theory to 'etale cohomology and certain Kato complexes and deduce finiteness results for zero-cycles with coefficients in Milnor K-theory over local fields.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2017-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43325839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-06-29DOI: 10.4310/CJM.2019.V7.N1.A2
Gang Liu
Let $M^n$ be a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, we prove that $M$ is biholomorphic to $mathbb{C}^n$. This confirms Yau's uniformization conjecture when M has maximal volume growth.
{"title":"On Yau’s uniformization conjecture","authors":"Gang Liu","doi":"10.4310/CJM.2019.V7.N1.A2","DOIUrl":"https://doi.org/10.4310/CJM.2019.V7.N1.A2","url":null,"abstract":"Let $M^n$ be a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, we prove that $M$ is biholomorphic to $mathbb{C}^n$. This confirms Yau's uniformization conjecture when M has maximal volume growth.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2016-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-10-09DOI: 10.4310/cjm.2021.v9.n3.a1
Alexandre Afgoustidis
George Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact reductive Lie group $G$ and those of its Cartan motion group $G_0$ $-$ the semidirect product of a maximal compact subgroup of $G$ and a vector space. He conjectured the existence of a natural one-to-one correspondence between "most" irreducible (tempered) representations of $G$ and "most" irreducible (unitary) representations of $G_0$. We here describe a simple and natural bijection between the tempered duals of both groups, and an extension to a one-to-one correspondence between the admissible duals.
George Mackey在1975年提出了非紧约李群$G$的不可约酉表示与它的Cartan运动群$G_0$ $-$ G$的极大紧子群与向量空间的半直积的不可约酉表示之间存在类比。他推测在$G$的“最”不可约(调质)表示和$G_0$的“最”不可约(酉)表示之间存在一种自然的一对一对应关系。我们在这里描述了两个群的调和对偶之间的简单和自然的双射，以及可容许对偶之间一对一对应的扩展。
{"title":"On the analogy between real reductive groups and Cartan motion groups: the Mackey–Higson bijection","authors":"Alexandre Afgoustidis","doi":"10.4310/cjm.2021.v9.n3.a1","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n3.a1","url":null,"abstract":"George Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact reductive Lie group $G$ and those of its Cartan motion group $G_0$ $-$ the semidirect product of a maximal compact subgroup of $G$ and a vector space. He conjectured the existence of a natural one-to-one correspondence between \"most\" irreducible (tempered) representations of $G$ and \"most\" irreducible (unitary) representations of $G_0$. We here describe a simple and natural bijection between the tempered duals of both groups, and an extension to a one-to-one correspondence between the admissible duals.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2015-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-08-08DOI: 10.4310/cjm.2020.v8.n2.a4
Tristan C. Collins, Adam Jacob, S. Yau
Let $(X,alpha)$ be a K"ahler manifold of dimension n, and let $[omega] in H^{1,1}(X,mathbb{R})$. We study the problem of specifying the Lagrangian phase of $omega$ with respect to $alpha$, which is described by the nonlinear elliptic equation [ sum_{i=1}^{n} arctan(lambda_i)= h(x) ] where $lambda_i$ are the eigenvalues of $omega$ with respect to $alpha$. When $h(x)$ is a topological constant, this equation corresponds to the deformed Hermitian-Yang-Mills (dHYM) equation, and is related by Mirror Symmetry to the existence of special Lagrangian submanifolds of the mirror. We introduce a notion of subsolution for this equation, and prove a priori $C^{2,beta}$ estimates when $|h|>(n-2)frac{pi}{2}$ and a subsolution exists. Using the method of continuity we show that the dHYM equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists. Finally, we discover some stability-type cohomological obstructions to the existence of solutions to the dHYM equation and we conjecture that when these obstructions vanish the dHYM equation admits a solution. We confirm this conjecture for complex surfaces.
设$(X,alpha)$是一个n维的Kähler流形，设$[omega] in H^{1,1}(X,mathbb{R})$。研究了求解$omega$相对于$alpha$的拉格朗日相的问题，该问题用非线性椭圆方程[ sum_{i=1}^{n} arctan(lambda_i)= h(x) ]来描述，其中$lambda_i$为$omega$相对于$alpha$的特征值。当$h(x)$为拓扑常数时，该方程对应于变形的Hermitian-Yang-Mills (dHYM)方程，并通过镜像对称性与该镜像的特殊拉格朗日子流形的存在性联系起来。我们引入了该方程的子解的概念，并证明了$|h|>(n-2)frac{pi}{2}$和子解存在时的先验$C^{2,beta}$估计。用连续性方法证明了在超临界相情况下，只要存在子解，dHYM方程就有光滑解。最后，我们发现了dHYM方程解存在的一些稳定型上同调障碍，并推测当这些障碍消失时，dHYM方程存在解。我们在复杂曲面上证实了这个猜想。
{"title":"$(1,1)$ forms with specified Lagrangian phase: a priori estimates and algebraic obstructions","authors":"Tristan C. Collins, Adam Jacob, S. Yau","doi":"10.4310/cjm.2020.v8.n2.a4","DOIUrl":"https://doi.org/10.4310/cjm.2020.v8.n2.a4","url":null,"abstract":"Let $(X,alpha)$ be a K\"ahler manifold of dimension n, and let $[omega] in H^{1,1}(X,mathbb{R})$. We study the problem of specifying the Lagrangian phase of $omega$ with respect to $alpha$, which is described by the nonlinear elliptic equation [ sum_{i=1}^{n} arctan(lambda_i)= h(x) ] where $lambda_i$ are the eigenvalues of $omega$ with respect to $alpha$. When $h(x)$ is a topological constant, this equation corresponds to the deformed Hermitian-Yang-Mills (dHYM) equation, and is related by Mirror Symmetry to the existence of special Lagrangian submanifolds of the mirror. We introduce a notion of subsolution for this equation, and prove a priori $C^{2,beta}$ estimates when $|h|>(n-2)frac{pi}{2}$ and a subsolution exists. Using the method of continuity we show that the dHYM equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists. Finally, we discover some stability-type cohomological obstructions to the existence of solutions to the dHYM equation and we conjecture that when these obstructions vanish the dHYM equation admits a solution. We confirm this conjecture for complex surfaces.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2015-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-04-03DOI: 10.4310/cjm.2020.v8.n1.a1
G. Garkusha, I. Panin
The category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [12]. Based on the theory, framed motives are introduced and studied in [7]. The main aim of this paper is to prove that for any $mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $mathcal F$, the associated Nisnevich sheaf $mathcal F_{nis}$ is $mathbb A^1$-invariant whenever the base field $k$ is infinite of characteristic different from 2. Moreover, if the base field $k$ is infinite perfect of characteristic different from 2, then every $mathbb A^1$-invariant quasi-stable Nisnevich framed sheaf of Abelian groups is strictly $mathbb A^1$-invariant and quasi-stable. Furthermore, the same statements are true in characteristic 2 if we also assume that the $mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $mathcal F$ is a presheaf of $mathbb Z[1/2]$-modules. This result and the paper are inspired by Voevodsky's paper [13].
{"title":"Homotopy invariant presheaves with framed transfers","authors":"G. Garkusha, I. Panin","doi":"10.4310/cjm.2020.v8.n1.a1","DOIUrl":"https://doi.org/10.4310/cjm.2020.v8.n1.a1","url":null,"abstract":"The category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [12]. Based on the theory, framed motives are introduced and studied in [7]. The main aim of this paper is to prove that for any $mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $mathcal F$, the associated Nisnevich sheaf $mathcal F_{nis}$ is $mathbb A^1$-invariant whenever the base field $k$ is infinite of characteristic different from 2. Moreover, if the base field $k$ is infinite perfect of characteristic different from 2, then every $mathbb A^1$-invariant quasi-stable Nisnevich framed sheaf of Abelian groups is strictly $mathbb A^1$-invariant and quasi-stable. Furthermore, the same statements are true in characteristic 2 if we also assume that the $mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $mathcal F$ is a presheaf of $mathbb Z[1/2]$-modules. This result and the paper are inspired by Voevodsky's paper [13].","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2015-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-08-22DOI: 10.4310/cjm.2020.v8.n3.a2
W. Meeks, G. Tinaglia
In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $mathbb{R}^3$ with constant mean curvature which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [24] to prove to prove a weak chord arc type result for these disks. In Section 4 we apply this weak chord arc result to obtain an intrinsic version of the one-sided curvature estimate for disks embedded in $mathbb{R}^3$ with constant mean curvature. In a natural sense, these one-sided curvature estimates generalize respectively, the extrinsic and intrinsic one-sided curvature estimates for minimal disks embedded in $mathbb{R}^3$ given by Colding and Minicozzi in Theorem 0.2 of [8] and in Corollary 0.8 of [9].
{"title":"One-sided curvature estimates for H-disks","authors":"W. Meeks, G. Tinaglia","doi":"10.4310/cjm.2020.v8.n3.a2","DOIUrl":"https://doi.org/10.4310/cjm.2020.v8.n3.a2","url":null,"abstract":"In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $mathbb{R}^3$ with constant mean curvature which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [24] to prove to prove a weak chord arc type result for these disks. In Section 4 we apply this weak chord arc result to obtain an intrinsic version of the one-sided curvature estimate for disks embedded in $mathbb{R}^3$ with constant mean curvature. In a natural sense, these one-sided curvature estimates generalize respectively, the extrinsic and intrinsic one-sided curvature estimates for minimal disks embedded in $mathbb{R}^3$ given by Colding and Minicozzi in Theorem 0.2 of [8] and in Corollary 0.8 of [9].","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2014-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-10-31DOI: 10.4310/cjm.2020.v8.n1.a2
A. Dranishnikov, S. Ferry, S. Weinberger
We show that there are homotopy equivalences $h:Nto M$ between closed manifolds which are induced by cell-like maps $p:Nto X$ and $q:Mto X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $mathbb L$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $>6$ we classify all such homotopy equivalences. As an application, we show that such homotopy equivalences are realized by deformations of Riemannian manifolds in Gromov-Hausdorff space preserving a contractibility function.
{"title":"An infinite-dimensional phenomenon in finite-dimensional metric topology","authors":"A. Dranishnikov, S. Ferry, S. Weinberger","doi":"10.4310/cjm.2020.v8.n1.a2","DOIUrl":"https://doi.org/10.4310/cjm.2020.v8.n1.a2","url":null,"abstract":"We show that there are homotopy equivalences $h:Nto M$ between closed manifolds which are induced by cell-like maps $p:Nto X$ and $q:Mto X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $mathbb L$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $>6$ we classify all such homotopy equivalences. As an application, we show that such homotopy equivalences are realized by deformations of Riemannian manifolds in Gromov-Hausdorff space preserving a contractibility function.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2006-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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