Totally geodesic subvarieties. Let Mg denote the moduli space of Riemann surfaces X of genus g. If we also record n unordered marked points on X, we obtain the moduli space Mg,n. A subvariety V of moduli space is totally geodesic if it contains every Teichmüller geodesic that is tangent to it. It is primitive if it does not arise from a lower– dimensional moduli space via a covering construction. The first family of primitive, totally geodesic varieties of dimension one in Mg was discovered by Veech in the 1980s [V2]. These rare and remarkable Teichmüller curves are related to Jacobians with real multiplication and polygonal billiard tables with optimal dynamical properties. A second family was discovered shortly thereafter [Wa]. To date only a handful of families of Teichmüller curves are known. The first known primitive, totally geodesic variety of dimension larger than one is the recently discovered flex surface F ⊂ M1,3 [MMW]. The surface F is closely related to a new type of SL2(R)–invariant subvariety ΩG in the moduli space of
{"title":"Billiards, quadrilaterals and moduli spaces","authors":"A. Eskin, C. McMullen, R. E. Mukamel, A. Wright","doi":"10.1090/jams/950","DOIUrl":"https://doi.org/10.1090/jams/950","url":null,"abstract":"Totally geodesic subvarieties. Let Mg denote the moduli space of Riemann surfaces X of genus g. If we also record n unordered marked points on X, we obtain the moduli space Mg,n. A subvariety V of moduli space is totally geodesic if it contains every Teichmüller geodesic that is tangent to it. It is primitive if it does not arise from a lower– dimensional moduli space via a covering construction. The first family of primitive, totally geodesic varieties of dimension one in Mg was discovered by Veech in the 1980s [V2]. These rare and remarkable Teichmüller curves are related to Jacobians with real multiplication and polygonal billiard tables with optimal dynamical properties. A second family was discovered shortly thereafter [Wa]. To date only a handful of families of Teichmüller curves are known. The first known primitive, totally geodesic variety of dimension larger than one is the recently discovered flex surface F ⊂ M1,3 [MMW]. The surface F is closely related to a new type of SL2(R)–invariant subvariety ΩG in the moduli space of","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/950","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43211750","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}
In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with respect to a group whose connected component is a torus. Character computation is done in two steps. First, we treat the case of distinguished p p -characters: those that are not contained in a proper Levi. Here we essentially show that the category of equivariant modules we consider is a cell quotient of an affine parabolic category O mathcal {O} . For this, we prove an equivalence between two categorifications of a parabolically induced module over the affine Hecke algebra conjectured by the first named author. For the general nilpotent p p -character, we get character formulas by explicitly computing the duality operator on a suitable equivariant K-group.
{"title":"Dimensions of modular irreducible representations of semisimple Lie algebras","authors":"R. Bezrukavnikov, I. Losev","doi":"10.1090/jams/1017","DOIUrl":"https://doi.org/10.1090/jams/1017","url":null,"abstract":"In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with respect to a group whose connected component is a torus. Character computation is done in two steps. First, we treat the case of distinguished \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-characters: those that are not contained in a proper Levi. Here we essentially show that the category of equivariant modules we consider is a cell quotient of an affine parabolic category \u0000\u0000 \u0000 \u0000 O\u0000 \u0000 mathcal {O}\u0000 \u0000\u0000. For this, we prove an equivalence between two categorifications of a parabolically induced module over the affine Hecke algebra conjectured by the first named author. For the general nilpotent \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-character, we get character formulas by explicitly computing the duality operator on a suitable equivariant K-group.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2020-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48883015","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}
We consider the energy supercritical d + 1 d+1 -dimensional semi-linear heat equation ∂ t u = Δ u + u p , x ∈ R d + 1 , p ≥ 3 , d ≥ 14. begin{equation*} partial _tu=Delta u+u^{p}, xin Bbb R^{d+1}, pgeq 3, dgeq 14. end{equation*} A fundamental open problem on this canonical nonlinear model is to understand the possible blow-up profiles appearing after renormalisation of a singularity. We exhibit in this paper a new scenario corresponding to the first example of a strongly anisotropic blow-up bubble: the solution displays a completely different behaviour depending on the considered direction in space. A fundamental step of the analysis is to solve the reconnection problem in order to produce finite energy solutions which is the heart of the matter. The corresponding anistropic mechanism is expected to be of fundamental importance in other settings in particular in fluid mechanics. The proof relies on a new functional framework for the construction and stabilisation of type II bubbles in the parabolic setting using energy estimates only, and allows us to exhibit new unexpected blow-up speeds.
{"title":"Strongly anisotropic type II blow up at an isolated point","authors":"Charles Collot, F. Merle, Pierre Raphael","doi":"10.1090/jams/941","DOIUrl":"https://doi.org/10.1090/jams/941","url":null,"abstract":"We consider the energy supercritical \u0000\u0000 \u0000 \u0000 d\u0000 +\u0000 1\u0000 \u0000 d+1\u0000 \u0000\u0000-dimensional semi-linear heat equation \u0000\u0000 \u0000 \u0000 \u0000 ∂\u0000 t\u0000 \u0000 u\u0000 =\u0000 Δ\u0000 u\u0000 +\u0000 \u0000 u\u0000 \u0000 p\u0000 \u0000 \u0000 ,\u0000 \u0000 \u0000 x\u0000 ∈\u0000 \u0000 \u0000 R\u0000 \u0000 \u0000 d\u0000 +\u0000 1\u0000 \u0000 \u0000 ,\u0000 \u0000 \u0000 p\u0000 ≥\u0000 3\u0000 ,\u0000 \u0000 d\u0000 ≥\u0000 14.\u0000 \u0000 begin{equation*} partial _tu=Delta u+u^{p}, xin Bbb R^{d+1}, pgeq 3, dgeq 14. end{equation*}\u0000 \u0000\u0000\u0000 A fundamental open problem on this canonical nonlinear model is to understand the possible blow-up profiles appearing after renormalisation of a singularity. We exhibit in this paper a new scenario corresponding to the first example of a strongly anisotropic blow-up bubble: the solution displays a completely different behaviour depending on the considered direction in space. A fundamental step of the analysis is to solve the reconnection problem in order to produce finite energy solutions which is the heart of the matter. The corresponding anistropic mechanism is expected to be of fundamental importance in other settings in particular in fluid mechanics. The proof relies on a new functional framework for the construction and stabilisation of type II bubbles in the parabolic setting using energy estimates only, and allows us to exhibit new unexpected blow-up speeds.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"33 1","pages":"527-607"},"PeriodicalIF":3.9,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/941","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46658049","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}
We prove the existence of a locally dense set of real polynomial automorphisms of C 2 mathbb C^2 displaying a wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These Fatou components have non-empty real trace and their statistical behavior is historic with high emergence. The proof is based on a geometric model for parameter families of surface real mappings. At a dense set of parameters, we show that the dynamics of the model displays a historic, high emergent, stable domain. We show that this model can be embedded into families of Hénon maps of explicit degree and also in an open and dense set of 5 5 -parameter C r C^r -families of surface diffeomorphisms in the Newhouse domain, for every 2 ≤ r ≤ ∞ 2le rle infty and r = ω r=omega . This implies a complement of the work of Kiriki and Soma [Adv. Math. 306 (2017), pp. 524–588], a proof of the last Taken’s problem in the C ∞ C^{infty } and C ω C^omega -case. The main difficulty is that here perturbations are done only along finite-dimensional parameter families. The proof is based on the multi-renormalization introduced by Berger [Zoology in the Hénon family: twin babies and Milnor’s swallows, 2018].
我们证明了具有游荡Fatou分量的C2mathbb C^2的实多项式自同构的局部稠密集的存在性；特别是这解决了Bedford和Smillie在1991年报道的它们的存在问题。这些法图分量具有非空实迹，其统计行为具有历史性，出现率高。证明是基于曲面实映射参数族的几何模型。在一组密集的参数下，我们表明模型的动力学显示出一个历史的、高度涌现的、稳定的域。我们证明了这个模型可以嵌入到显式度的Hénon映射族中，也可以嵌入到Newhouse域中的5个5参数C r C^r-族的表面微分同胚的开稠密集合中，对于每2≤r≤∞2个r和r=ωr=ω。这意味着对Kiriki和Soma的工作的补充[Adv.Math.306（2017），pp.524–588]，在C∞C^和CωC^ω情况下最后一个Taken问题的证明。主要的困难是，这里的扰动只沿着有限维参数族进行。该证明基于Berger提出的多重重整[赫农家族动物学：双胞胎婴儿和米尔诺燕子，2018]。
{"title":"Emergence of wandering stable components","authors":"P. Berger, S'ebastien Biebler","doi":"10.1090/jams/1005","DOIUrl":"https://doi.org/10.1090/jams/1005","url":null,"abstract":"We prove the existence of a locally dense set of real polynomial automorphisms of \u0000\u0000 \u0000 \u0000 \u0000 C\u0000 \u0000 2\u0000 \u0000 mathbb C^2\u0000 \u0000\u0000 displaying a wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These Fatou components have non-empty real trace and their statistical behavior is historic with high emergence. The proof is based on a geometric model for parameter families of surface real mappings. At a dense set of parameters, we show that the dynamics of the model displays a historic, high emergent, stable domain. We show that this model can be embedded into families of Hénon maps of explicit degree and also in an open and dense set of \u0000\u0000 \u0000 5\u0000 5\u0000 \u0000\u0000-parameter \u0000\u0000 \u0000 \u0000 C\u0000 r\u0000 \u0000 C^r\u0000 \u0000\u0000-families of surface diffeomorphisms in the Newhouse domain, for every \u0000\u0000 \u0000 \u0000 2\u0000 ≤\u0000 r\u0000 ≤\u0000 ∞\u0000 \u0000 2le rle infty\u0000 \u0000\u0000 and \u0000\u0000 \u0000 \u0000 r\u0000 =\u0000 ω\u0000 \u0000 r=omega\u0000 \u0000\u0000. This implies a complement of the work of Kiriki and Soma [Adv. Math. 306 (2017), pp. 524–588], a proof of the last Taken’s problem in the \u0000\u0000 \u0000 \u0000 C\u0000 \u0000 ∞\u0000 \u0000 \u0000 C^{infty }\u0000 \u0000\u0000 and \u0000\u0000 \u0000 \u0000 C\u0000 ω\u0000 \u0000 C^omega\u0000 \u0000\u0000-case. The main difficulty is that here perturbations are done only along finite-dimensional parameter families. The proof is based on the multi-renormalization introduced by Berger [Zoology in the Hénon family: twin babies and Milnor’s swallows, 2018].","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47573275","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}
We prove that the conformal group of a closed, simply connected, real analytic Lorentzian manifold is compact. D'Ambra proved in 1988 that the isometry group of such a manifold is compact. Our result implies the Lorentzian Lichnerowicz Conjecture for real analytic Lorentzian manifolds with finite fundamental group. Second version adds some clarifications and corrections.
{"title":"The conformal group of a compact simply connected Lorentzian manifold","authors":"K. Melnick, V. Pecastaing","doi":"10.1090/JAMS/976","DOIUrl":"https://doi.org/10.1090/JAMS/976","url":null,"abstract":"We prove that the conformal group of a closed, simply connected, real analytic Lorentzian manifold is compact. D'Ambra proved in 1988 that the isometry group of such a manifold is compact. Our result implies the Lorentzian Lichnerowicz Conjecture for real analytic Lorentzian manifolds with finite fundamental group. \u0000Second version adds some clarifications and corrections.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2019-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48150865","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}
<p>After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental (<italic>i.e.</italic>, they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow from bar x plus h"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">↦<!-- ↦ --></mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>h</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">xmapsto x+h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h element-of double-struck upper C Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">hin mathbb {C}^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>), the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-difference operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow from bar q x"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">↦<!-- ↦ --></mml:mo> <mml:mi>q</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">xmapsto qx</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q element-of double-struck upper C Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">qin mathbb {C}^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> not a root of unity), and the Mahler operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x rig
在Hölder证明了他关于伽玛函数的经典定理之后，已经有一大堆结果表明，线性差分方程的解往往是超遍历的（即，它们不可能是代数微分方程的解）。在本文中，我们得到了与移位算子x相关的一般线性差分方程解的第一个完整结果↦ x+hxmapstox+h（h∈C*hinmathbb｛C｝^*），q q-差分算子x↦ qxmapsto qx（q∈C*qinmathbb｛C｝^*不是单位根），以及Mahler算子x↦ x p x mapsto x ^p（p≥2 p geq 2整数）。唯一的限制是，我们约束我们的解在具有复系数的变量x x中（或者在与移位算子相关的一些特殊情况下在变量1/x1/x中）表示为（可能是分支的）Laurent级数。我们的证明是基于Hardouin和Singer提出的参数化差分伽罗瓦理论。我们还从我们的主要结果中推导了关于Mahler函数及其导数在代数点上的值的代数独立性的一般性陈述。
{"title":"Hypertranscendence and linear difference equations","authors":"B. Adamczewski, T. Dreyfus, C. Hardouin","doi":"10.1090/jams/960","DOIUrl":"https://doi.org/10.1090/jams/960","url":null,"abstract":"<p>After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental (<italic>i.e.</italic>, they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift operator <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x right-arrow from bar x plus h\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">↦<!-- ↦ --></mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>h</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">xmapsto x+h</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h element-of double-struck upper C Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>h</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">hin mathbb {C}^*</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>), the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q\">\u0000 <mml:semantics>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">q</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-difference operator <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x right-arrow from bar q x\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">↦<!-- ↦ --></mml:mo>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mi>x</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">xmapsto qx</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q element-of double-struck upper C Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">qin mathbb {C}^*</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> not a root of unity), and the Mahler operator <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x rig","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2019-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44873591","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}
We derive $C^infty$ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to point-wise bounds on their mass, energy and entropy densities. We also establish decay estimates for large velocities, for all derivatives of the solution.
{"title":"Global regularity estimates for the Boltzmann equation without cut-off","authors":"C. Imbert, L. Silvestre","doi":"10.1090/JAMS/986","DOIUrl":"https://doi.org/10.1090/JAMS/986","url":null,"abstract":"We derive $C^infty$ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to point-wise bounds on their mass, energy and entropy densities. We also establish decay estimates for large velocities, for all derivatives of the solution.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2019-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46176378","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}
We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago-Ivanov-Polterovich. To do this, we introduce an analogue of the curve graph from the theory of mapping class groups. We show that it is hyperbolic and that the natural group action by isometries satisfies the criterion of Bestvina-Fujiwara.
{"title":"Quasi-morphisms on surface diffeomorphism groups","authors":"Jonathan Bowden, S. Hensel, Richard C. H. Webb","doi":"10.1090/JAMS/981","DOIUrl":"https://doi.org/10.1090/JAMS/981","url":null,"abstract":"We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago-Ivanov-Polterovich. \u0000To do this, we introduce an analogue of the curve graph from the theory of mapping class groups. We show that it is hyperbolic and that the natural group action by isometries satisfies the criterion of Bestvina-Fujiwara.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2019-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43553009","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}
We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works (see Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339] and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794]), which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339 and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794] and rely on the fractal uncertainty principle of Jean Bourgain and Semyon Dyatlov [Ann. of Math. (2) 187 (2018), pp. 825–867]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used by Semyon Dyatlov and Joshua Zahl [Geom. Funct. Anal. 26 (2016), pp. 1011–1094]. Instead, our argument uses Egorov’s theorem up to local Ehrenfest time and the hyperbolic parametrix of Stéphane Nonnenmacher and Maciej Zworski [Acta Math. 203 (2009), pp. 149–233], together with the C 1 + C^{1+} regularity of the stable/unstable foliations.
{"title":"Control of eigenfunctions on surfaces of variable curvature","authors":"S. Dyatlov, Long Jin, S. Nonnenmacher","doi":"10.1090/jams/979","DOIUrl":"https://doi.org/10.1090/jams/979","url":null,"abstract":"We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works (see Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339] and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794]), which considered the setting of surfaces of constant negative curvature.\u0000\u0000The proofs use the strategy of Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339 and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794] and rely on the fractal uncertainty principle of Jean Bourgain and Semyon Dyatlov [Ann. of Math. (2) 187 (2018), pp. 825–867]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used by Semyon Dyatlov and Joshua Zahl [Geom. Funct. Anal. 26 (2016), pp. 1011–1094]. Instead, our argument uses Egorov’s theorem up to local Ehrenfest time and the hyperbolic parametrix of Stéphane Nonnenmacher and Maciej Zworski [Acta Math. 203 (2009), pp. 149–233], together with the \u0000\u0000 \u0000 \u0000 C\u0000 \u0000 1\u0000 +\u0000 \u0000 \u0000 C^{1+}\u0000 \u0000\u0000 regularity of the stable/unstable foliations.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2019-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43735354","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}
{"title":"Duality between the pseudoeffective and the movable cone on a projective manifold","authors":"David Witt Nyström","doi":"10.1090/JAMS/922","DOIUrl":"https://doi.org/10.1090/JAMS/922","url":null,"abstract":"","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"32 1","pages":"675-689"},"PeriodicalIF":3.9,"publicationDate":"2019-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/922","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42647432","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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