Pub Date : 2017-11-06DOI: 10.4310/acta.2019.v223.n1.a3
M. Salle
We prove that every lattice in a product of higher rank simple Lie groups or higher rank simple algebraic groups over local fields has Vincent Lafforgue's strong property (T). Over non-archimedean local fields, we also prove that they have strong Banach proerty (T) with respect to all Banach spaces with nontrivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-cocompact lattices, such as $mathrm{SL}_n(Z)$ for $n geq 3$. To do so, we introduce a stronger form of strong property (T) which allows us to deal with more general objects than group representations on Banach spaces that we call two-step representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher rank groups have this property and that this property passes to undistorted lattices.
{"title":"Strong property (T) for higher rank lattices","authors":"M. Salle","doi":"10.4310/acta.2019.v223.n1.a3","DOIUrl":"https://doi.org/10.4310/acta.2019.v223.n1.a3","url":null,"abstract":"We prove that every lattice in a product of higher rank simple Lie groups or higher rank simple algebraic groups over local fields has Vincent Lafforgue's strong property (T). Over non-archimedean local fields, we also prove that they have strong Banach proerty (T) with respect to all Banach spaces with nontrivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-cocompact lattices, such as $mathrm{SL}_n(Z)$ for $n geq 3$. To do so, we introduce a stronger form of strong property (T) which allows us to deal with more general objects than group representations on Banach spaces that we call two-step representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher rank groups have this property and that this property passes to undistorted lattices.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2017-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46328008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-10-27DOI: 10.4310/acta.2019.v223.n2.a2
L. Guth, J. Hickman, Marina Iliopoulou
The sharp range of $L^p$-estimates for the class of H"ormander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments.
{"title":"Sharp estimates for oscillatory integral operators via polynomial partitioning","authors":"L. Guth, J. Hickman, Marina Iliopoulou","doi":"10.4310/acta.2019.v223.n2.a2","DOIUrl":"https://doi.org/10.4310/acta.2019.v223.n2.a2","url":null,"abstract":"The sharp range of $L^p$-estimates for the class of H\"ormander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2017-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42141769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-10-17DOI: 10.4310/acta.2021.v227.n2.a3
H. Hedenmalm, Aron Wennman
We prove that the planar normalized orthogonal polynomials $P_{n,m}(z)$ of degree $n$ with respect to an exponentially varying planar measure $e^{-2mQ(z)}mathrm{dA}(z)$ enjoy an asymptotic expansion [ P_{n,m}(z)sim m^{frac{1}{4}}sqrt{phi_tau'(z)}[phi_tau(z)]^n e^{mmathcal{Q}_tau(z)}left(mathcal{B}_{0,tau}(z) +frac{1}{m}mathcal{B}_{1,tau}(z)+frac{1}{m^2} mathcal{B}_{2,tau}(z)+ldotsright), ] as $n=mtautoinfty$. Here $mathcal{S}_tau$ denotes the droplet, the boundary of which is assumed to be a smooth, simple, closed curve, and $phi_tau$ is a conformal mapping $mathcal{S}_tau^ctomathbb{D}_e$. The functions $mathcal{Q}_tau$ and $mathcal{B}_{j,tau}(z)$ are bounded holomorphic functions which may be computed in terms of $Q$ and $mathcal{S}_tau$. We apply these results to prove universality at the boundary for regular droplets in the random normal matrix model, i.e., that the limiting rescaled process is the random process with correlation kernel $$ mathrm{k}(xi,eta)= e^{xibareta,-frac12(lvertxirvert^2+lvert etarvert^2)} operatorname{erf}(xi+bar{eta}). $$ A key ingredient in the proof of the asymptotic expansion is the construction of an {orthogonal foliation} -- a smooth flow of closed curves near $partialmathcal{S}_tau$, on each of which $P_{n,m}$ is orthogonal to lower order polynomials, with respect to an induced measure. To compute the coefficients, we develop an algorithm which determines $mathcal{B}_{j,tau}$ up to any desired order in terms of inhomogeneous Toeplitz kernel conditions.
{"title":"Planar orthogonal polynomials and boundary universality in the random normal matrix model","authors":"H. Hedenmalm, Aron Wennman","doi":"10.4310/acta.2021.v227.n2.a3","DOIUrl":"https://doi.org/10.4310/acta.2021.v227.n2.a3","url":null,"abstract":"We prove that the planar normalized orthogonal polynomials $P_{n,m}(z)$ of degree $n$ with respect to an exponentially varying planar measure $e^{-2mQ(z)}mathrm{dA}(z)$ enjoy an asymptotic expansion [ P_{n,m}(z)sim m^{frac{1}{4}}sqrt{phi_tau'(z)}[phi_tau(z)]^n e^{mmathcal{Q}_tau(z)}left(mathcal{B}_{0,tau}(z) +frac{1}{m}mathcal{B}_{1,tau}(z)+frac{1}{m^2} mathcal{B}_{2,tau}(z)+ldotsright), ] as $n=mtautoinfty$. Here $mathcal{S}_tau$ denotes the droplet, the boundary of which is assumed to be a smooth, simple, closed curve, and $phi_tau$ is a conformal mapping $mathcal{S}_tau^ctomathbb{D}_e$. The functions $mathcal{Q}_tau$ and $mathcal{B}_{j,tau}(z)$ are bounded holomorphic functions which may be computed in terms of $Q$ and $mathcal{S}_tau$. We apply these results to prove universality at the boundary for regular droplets in the random normal matrix model, i.e., that the limiting rescaled process is the random process with correlation kernel $$ mathrm{k}(xi,eta)= e^{xibareta,-frac12(lvertxirvert^2+lvert etarvert^2)} operatorname{erf}(xi+bar{eta}). $$ A key ingredient in the proof of the asymptotic expansion is the construction of an {orthogonal foliation} -- a smooth flow of closed curves near $partialmathcal{S}_tau$, on each of which $P_{n,m}$ is orthogonal to lower order polynomials, with respect to an induced measure. To compute the coefficients, we develop an algorithm which determines $mathcal{B}_{j,tau}$ up to any desired order in terms of inhomogeneous Toeplitz kernel conditions.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2017-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42156274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-13DOI: 10.4310/acta.2022.v228.n1.a1
R. Bamler, B. Kleiner
We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman's conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three manifolds --- in particular to the Generalized Smale Conjecture --- which will appear in a subsequent paper.
{"title":"Uniqueness and stability of Ricci flow through singularities","authors":"R. Bamler, B. Kleiner","doi":"10.4310/acta.2022.v228.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2022.v228.n1.a1","url":null,"abstract":"We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman's conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. \u0000Our results have applications to the study of diffeomorphism groups of three manifolds --- in particular to the Generalized Smale Conjecture --- which will appear in a subsequent paper.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2017-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44402472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-07-10DOI: 10.4310/ACTA.2022.v229.n2.a2
Benjamin Gammage, V. Shende
We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective DM toric stack is equivalent to the wrapped Fukaya category of a hypersurface in a complex torus. Hypersurfaces with every Newton polytope can be obtained. Our proof has the following ingredients. Using Mikhalkin-Viro patchworking, we compute the skeleton of the hypersurface. The result matches the [FLTZ] skeleton and is naturally realized as a Legendrian in the cosphere bundle of a torus. By [GPS1, GPS2, GPS3], we trade wrapped Fukaya categories for microlocal sheaf theory. By proving a new functoriality result for Bondal's coherent-constructible correspondence, we reduce the sheaf calculation to Kuwagaki's recent theorem on mirror symmetry for toric varieties.
{"title":"Mirror symmetry for very affine hypersurfaces","authors":"Benjamin Gammage, V. Shende","doi":"10.4310/ACTA.2022.v229.n2.a2","DOIUrl":"https://doi.org/10.4310/ACTA.2022.v229.n2.a2","url":null,"abstract":"We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective DM toric stack is equivalent to the wrapped Fukaya category of a hypersurface in a complex torus. Hypersurfaces with every Newton polytope can be obtained. \u0000Our proof has the following ingredients. Using Mikhalkin-Viro patchworking, we compute the skeleton of the hypersurface. The result matches the [FLTZ] skeleton and is naturally realized as a Legendrian in the cosphere bundle of a torus. By [GPS1, GPS2, GPS3], we trade wrapped Fukaya categories for microlocal sheaf theory. By proving a new functoriality result for Bondal's coherent-constructible correspondence, we reduce the sheaf calculation to Kuwagaki's recent theorem on mirror symmetry for toric varieties.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2017-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41520437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-07-01Epub Date: 2017-02-20DOI: 10.1007/s00018-017-2472-6
Edward J Andress, Michael Nicolaou, Farrell McGeoghan, Kenneth J Linton
Bile salts are natural detergents required to solubilise dietary fat and lipid soluble vitamins. They are synthesised in hepatocytes and secreted into the luminal space of the biliary tree by the bile salt export pump (BSEP), an ATP-binding cassette (ABC) transporter in the canalicular membrane. BSEP deficiency causes cytotoxic accumulation of bile salts in the hepatocyte that results in mild-to-severe forms of cholestasis. The resulting inflammation can also progress to hepatocellular cancer via a novel mechanism involving upregulation of proliferative signalling pathways. A second ABC transporter of the canalicular membrane is also critical for bile formation. ABCB4 flops phosphatidylcholine into the outer leaflet of the membrane to be extracted by bile salts in the canalicular space. These mixed micelles reduce the detergent action of the bile salts and protect the biliary tree from their cytotoxic activity. ABCB4 deficiency also causes cholestasis, and might be expected to cause cholangitis and predispose to liver cancer. Non-synonymous SNPs in ABCB4 have now been described in patients with liver cancer or with inflammatory liver diseases that are known to predispose to cancer, but data showing that the SNPs are sufficiently deleterious to be an etiological factor are lacking. Here, we report the first characterisation at the protein level of six ABCB4 variants (D243A, K435T, G535D, I490T, R545C, and S978P) previously found in patients with inflammatory liver diseases or liver cancer. All significantly impair the transporter with a range of phenotypes exhibited, including low abundance, intracellular retention, and reduced floppase activity, suggesting that ABCB4 deficiency is the root cause of the pathology in these cases.
{"title":"ABCB4 missense mutations D243A, K435T, G535D, I490T, R545C, and S978P significantly impair the lipid floppase and likely predispose to secondary pathologies in the human population.","authors":"Edward J Andress, Michael Nicolaou, Farrell McGeoghan, Kenneth J Linton","doi":"10.1007/s00018-017-2472-6","DOIUrl":"10.1007/s00018-017-2472-6","url":null,"abstract":"<p><p>Bile salts are natural detergents required to solubilise dietary fat and lipid soluble vitamins. They are synthesised in hepatocytes and secreted into the luminal space of the biliary tree by the bile salt export pump (BSEP), an ATP-binding cassette (ABC) transporter in the canalicular membrane. BSEP deficiency causes cytotoxic accumulation of bile salts in the hepatocyte that results in mild-to-severe forms of cholestasis. The resulting inflammation can also progress to hepatocellular cancer via a novel mechanism involving upregulation of proliferative signalling pathways. A second ABC transporter of the canalicular membrane is also critical for bile formation. ABCB4 flops phosphatidylcholine into the outer leaflet of the membrane to be extracted by bile salts in the canalicular space. These mixed micelles reduce the detergent action of the bile salts and protect the biliary tree from their cytotoxic activity. ABCB4 deficiency also causes cholestasis, and might be expected to cause cholangitis and predispose to liver cancer. Non-synonymous SNPs in ABCB4 have now been described in patients with liver cancer or with inflammatory liver diseases that are known to predispose to cancer, but data showing that the SNPs are sufficiently deleterious to be an etiological factor are lacking. Here, we report the first characterisation at the protein level of six ABCB4 variants (D243A, K435T, G535D, I490T, R545C, and S978P) previously found in patients with inflammatory liver diseases or liver cancer. All significantly impair the transporter with a range of phenotypes exhibited, including low abundance, intracellular retention, and reduced floppase activity, suggesting that ABCB4 deficiency is the root cause of the pathology in these cases.</p>","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"113 1","pages":"2513-2524"},"PeriodicalIF":8.0,"publicationDate":"2017-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00018-017-2472-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73653219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-09DOI: 10.4310/acta.2021.v227.n2.a1
P. Berger
For any $2le rle infty$, $nge 2$, we prove the existence of an open set $U$ of $C^r$-self-mappings of any $n$-manifold so that a generic map $f$ in $U$ displays a fast growth of the number of periodic points: the number of its $N$-periodic points grows as fast as asked. This complements the works of Martens-de Melo-van Strien, Gochenko-Shil'nikov-Turaev, Kaloshin, Bonatti-Diaz-Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness. Furthermore for any $2le r
对于任何$2leinfty$，$nge2$，我们证明了任何$n$-流形的$C^r$-自映射的开集$U$的存在，使得$U$中的泛型映射$f$显示周期点数量的快速增长：其$n$-周期点的数量增长得与要求的一样快。这补充了Martens de Melo van Strien、Gochenko-Shil'nikov-Turaev、Kaloshin、Bonatti Diaz Fisher和Turaev的作品，为Smale在1967年、Bowen在1978年和Arnold在1989年提出的任何维度的流形和任何光滑度的问题提供了完整的答案。此外，对于任何$2le r
{"title":"Generic family displaying robustly a fast growth of the number of periodic points","authors":"P. Berger","doi":"10.4310/acta.2021.v227.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2021.v227.n2.a1","url":null,"abstract":"For any $2le rle infty$, $nge 2$, we prove the existence of an open set $U$ of $C^r$-self-mappings of any $n$-manifold so that a generic map $f$ in $U$ displays a fast growth of the number of periodic points: the number of its $N$-periodic points grows as fast as asked. This complements the works of Martens-de Melo-van Strien, Gochenko-Shil'nikov-Turaev, Kaloshin, Bonatti-Diaz-Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness. \u0000Furthermore for any $2le r<infty$ and any $kge 0$, we prove the existence of an open set $hat U$ of $k$-parameter families in $U$ so that for a generic $(f_a)_ain hat U$, for every $|a|le 1$, the map $f_a$ displays a fast growth of periodic points. This gives a negative answer to a problem asked by Arnold in 1992 in the finitely smooth case.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2017-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45288215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-12-30DOI: 10.4310/ACTA.2021.v227.n1.a3
K. Matetski, J. Quastel, Daniel Remenik
An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process with arbitrary initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [Sas05; BFPS07]. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data. In the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula, in terms of analogous kernels based on Brownian motion, for the transition probabilities of the scaling invariant Markov process at the centre of the KPZ universality class. The formula readily reproduces known special self-similar solutions such as the Airy$_1$ and Airy$_2$ processes. The invariant Markov process takes values in real valued functions which look locally like Brownian motion, and is H"older $1/3-$ in time.
{"title":"The KPZ fixed point","authors":"K. Matetski, J. Quastel, Daniel Remenik","doi":"10.4310/ACTA.2021.v227.n1.a3","DOIUrl":"https://doi.org/10.4310/ACTA.2021.v227.n1.a3","url":null,"abstract":"An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process with arbitrary initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [Sas05; BFPS07]. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data. In the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula, in terms of analogous kernels based on Brownian motion, for the transition probabilities of the scaling invariant Markov process at the centre of the KPZ universality class. The formula readily reproduces known special self-similar solutions such as the Airy$_1$ and Airy$_2$ processes. The invariant Markov process takes values in real valued functions which look locally like Brownian motion, and is H\"older $1/3-$ in time.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2016-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71153243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-11-16DOI: 10.1007/S11511-016-0140-6
Yong Huang, E. Lutwak, Deane Yang, Gaoyong Zhang
{"title":"Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems","authors":"Yong Huang, E. Lutwak, Deane Yang, Gaoyong Zhang","doi":"10.1007/S11511-016-0140-6","DOIUrl":"https://doi.org/10.1007/S11511-016-0140-6","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"216 1","pages":"325-388"},"PeriodicalIF":3.7,"publicationDate":"2016-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S11511-016-0140-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53064695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-11-16DOI: 10.4310/ACTA.2018.V221.N1.A5
A. Lytchak, S. Wenger
We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.
{"title":"Isoperimetric characterization of upper curvature bounds","authors":"A. Lytchak, S. Wenger","doi":"10.4310/ACTA.2018.V221.N1.A5","DOIUrl":"https://doi.org/10.4310/ACTA.2018.V221.N1.A5","url":null,"abstract":"We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"221 1","pages":"159-202"},"PeriodicalIF":3.7,"publicationDate":"2016-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71152818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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