{"title":"Sur la répresentation analytique d'une branche uniforme d'une fonction monogène","authors":"G. Mittag-Leffler","doi":"10.1007/BF02404411","DOIUrl":"https://doi.org/10.1007/BF02404411","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2016-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/BF02404411","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52296821","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 : 2016-05-09DOI: 10.4310/acta.2019.v223.n1.a2
W. Meeks, Joaquín Pérez, A. Ros
We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has at least two ends implies that $M$ has finite stability index which is bounded by a constant that only depends on its genus.
{"title":"Bounds on the topology and index of minimal surfaces","authors":"W. Meeks, Joaquín Pérez, A. Ros","doi":"10.4310/acta.2019.v223.n1.a2","DOIUrl":"https://doi.org/10.4310/acta.2019.v223.n1.a2","url":null,"abstract":"We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has at least two ends implies that $M$ has finite stability index which is bounded by a constant that only depends on its genus.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2016-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71152957","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 : 2016-03-11DOI: 10.4310/ACTA.2020.V225.N2.A3
David Gabai, Mehdi Yazdi
In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any Euler class with norm equal to one is Euler class of a taut foliation. We construct counterexamples to this conjecture and suggest an alternative conjecture.
{"title":"On Thurston’s Euler class-one conjecture","authors":"David Gabai, Mehdi Yazdi","doi":"10.4310/ACTA.2020.V225.N2.A3","DOIUrl":"https://doi.org/10.4310/ACTA.2020.V225.N2.A3","url":null,"abstract":"In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any Euler class with norm equal to one is Euler class of a taut foliation. We construct counterexamples to this conjecture and suggest an alternative conjecture.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2016-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71153048","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 : 2016-01-25DOI: 10.4310/ACTA.2019.v222.n1.a1
Mihalis Dafermos, G. Holzegel, I. Rodnianski
We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components). These latter statements are in fact deduced in the course of the proof by exploiting associated quantities shown to satisfy the Regge--Wheeler equation, for which appropriate decay can be obtained easily by adapting previous work on the linear scalar wave equation. The bounds on the rate of decay to linearised Kerr are inverse polynomial, suggesting that dispersion is sufficient to control the non-linearities of the Einstein equations in a potential future proof of nonlinear stability. This paper is self-contained and includes a physical-space derivation of the equations of linearised gravity around Schwarzschild from the full non-linear Einstein vacuum equations expressed in a double null gauge.
{"title":"The linear stability of the Schwarzschild solution to gravitational perturbations","authors":"Mihalis Dafermos, G. Holzegel, I. Rodnianski","doi":"10.4310/ACTA.2019.v222.n1.a1","DOIUrl":"https://doi.org/10.4310/ACTA.2019.v222.n1.a1","url":null,"abstract":"We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components). These latter statements are in fact deduced in the course of the proof by exploiting associated quantities shown to satisfy the Regge--Wheeler equation, for which appropriate decay can be obtained easily by adapting previous work on the linear scalar wave equation. The bounds on the rate of decay to linearised Kerr are inverse polynomial, suggesting that dispersion is sufficient to control the non-linearities of the Einstein equations in a potential future proof of nonlinear stability. This paper is self-contained and includes a physical-space derivation of the equations of linearised gravity around Schwarzschild from the full non-linear Einstein vacuum equations expressed in a double null gauge.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2016-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71153002","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 : 2015-10-20DOI: 10.1007/S11511-015-0128-7
A. Avila
{"title":"Global theory of one-frequency Schrödinger operators","authors":"A. Avila","doi":"10.1007/S11511-015-0128-7","DOIUrl":"https://doi.org/10.1007/S11511-015-0128-7","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2015-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S11511-015-0128-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53064153","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 : 2014-10-08DOI: 10.1007/s11511-017-0146-8
G. Williamson
{"title":"Local Hodge theory of Soergel bimodules","authors":"G. Williamson","doi":"10.1007/s11511-017-0146-8","DOIUrl":"https://doi.org/10.1007/s11511-017-0146-8","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2014-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11511-017-0146-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53064769","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 : 2014-10-02DOI: 10.4310/ACTA.2017.v218.n2.a1
F. Bonahon, G. Dreyer
For a closed surface S, the Hitchin component Hitn(S) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group �1(S) to the Lie group PSLn(R). We construct a parametrization of the Hitchin component that is well-adapted to a geodesic laminationon the surface. This is a natural extension of Thurston's parametrization of the Teichmuller space T(S) by shear coordinates associated to �, corresponding to the case n = 2. However, significantly new ideas are needed in this higher dimensional case. The article concludes with a few applications.
{"title":"HITCHIN CHARACTERS AND GEODESIC LAMINATIONS","authors":"F. Bonahon, G. Dreyer","doi":"10.4310/ACTA.2017.v218.n2.a1","DOIUrl":"https://doi.org/10.4310/ACTA.2017.v218.n2.a1","url":null,"abstract":"For a closed surface S, the Hitchin component Hitn(S) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group �1(S) to the Lie group PSLn(R). We construct a parametrization of the Hitchin component that is well-adapted to a geodesic laminationon the surface. This is a natural extension of Thurston's parametrization of the Teichmuller space T(S) by shear coordinates associated to �, corresponding to the case n = 2. However, significantly new ideas are needed in this higher dimensional case. The article concludes with a few applications.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2014-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71152805","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 : 2014-07-29DOI: 10.1007/S11511-016-0135-3
A. Gogolev, P. Ontaneda, F. R. Hertz
{"title":"New partially hyperbolic dynamical systems I","authors":"A. Gogolev, P. Ontaneda, F. R. Hertz","doi":"10.1007/S11511-016-0135-3","DOIUrl":"https://doi.org/10.1007/S11511-016-0135-3","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2014-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S11511-016-0135-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53064416","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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