Pub Date : 2019-12-18DOI: 10.4007/annals.2023.198.1.3
I. Rodnianski, Yakov Shlapentokh-Rothman
In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in $3+1$ dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our introduction of a new type of self-similarity for the Einstein vacuum equations. Connected to this is a new geometric twisting phenomenon which plays the leading role in singularity formation. Prior to this work, the only known examples of naked singularities were the solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar-field system, as well as other solutions explored numerically for either the spherically symmetric Einstein equations coupled to suitable matter models or for the Einstein equations in higher dimensions.
{"title":"Naked singularities for the Einstein vacuum equations: The exterior solution","authors":"I. Rodnianski, Yakov Shlapentokh-Rothman","doi":"10.4007/annals.2023.198.1.3","DOIUrl":"https://doi.org/10.4007/annals.2023.198.1.3","url":null,"abstract":"In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in $3+1$ dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our introduction of a new type of self-similarity for the Einstein vacuum equations. Connected to this is a new geometric twisting phenomenon which plays the leading role in singularity formation. \u0000Prior to this work, the only known examples of naked singularities were the solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar-field system, as well as other solutions explored numerically for either the spherically symmetric Einstein equations coupled to suitable matter models or for the Einstein equations in higher dimensions.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2019-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41711310","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 introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet--Rallis relative trace formula, we complete the proof of the global Gan--Gross--Prasad conjecture and its refinement Ichino--Ikeda conjecture for $mathrm{U}(n)timesmathrm{U}(n+1)$ in the stable case.
{"title":"Isolation of cuspidal spectrum, with application to the\u0000 Gan–Gross–Prasad conjecture","authors":"Raphael Beuzart-Plessis, Yifeng Liu, Wei Zhang, Xinwen Zhu","doi":"10.4007/annals.2021.194.2.5","DOIUrl":"https://doi.org/10.4007/annals.2021.194.2.5","url":null,"abstract":"We introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet--Rallis relative trace formula, we complete the proof of the global Gan--Gross--Prasad conjecture and its refinement Ichino--Ikeda conjecture for $mathrm{U}(n)timesmathrm{U}(n+1)$ in the stable case.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43374548","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 : 2019-12-16DOI: 10.4007/annals.2022.195.3.2
M. Hausmann
We prove that the homotopical $A$-equivariant complex bordism ring is isomorphic to the $A$-equivariant Lazard ring for every abelian compact Lie group $A$, settling a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian $2$-groups. This generalizes classical theorems of Quillen on the connection between non-equivariant bordism rings and formal group laws, and extends the case $A=C_2$ due to Hanke--Wiemeler. We work in the framework of global homotopy theory, which is essential for our proof. Using this framework, we also give an algebraic characterization of the collection of equivariant complex bordism rings as the universal contravariant functor from abelian compact Lie groups to commutative rings that is equipped with a coordinate. More generally, the ring of $n$-fold cooperations of equivariant complex bordism is shown to be universal among such functors equipped with a strict $n$-tuple of coordinates.
{"title":"Global group laws and equivariant bordism rings","authors":"M. Hausmann","doi":"10.4007/annals.2022.195.3.2","DOIUrl":"https://doi.org/10.4007/annals.2022.195.3.2","url":null,"abstract":"We prove that the homotopical $A$-equivariant complex bordism ring is isomorphic to the $A$-equivariant Lazard ring for every abelian compact Lie group $A$, settling a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian $2$-groups. This generalizes classical theorems of Quillen on the connection between non-equivariant bordism rings and formal group laws, and extends the case $A=C_2$ due to Hanke--Wiemeler. \u0000We work in the framework of global homotopy theory, which is essential for our proof. Using this framework, we also give an algebraic characterization of the collection of equivariant complex bordism rings as the universal contravariant functor from abelian compact Lie groups to commutative rings that is equipped with a coordinate. More generally, the ring of $n$-fold cooperations of equivariant complex bordism is shown to be universal among such functors equipped with a strict $n$-tuple of coordinates.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43080531","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 : 2019-12-01DOI: 10.4007/annals.2020.192.3.7
Chenyang Xu, Ziquan Zhuang
In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations which can be used to test various K-stability notions of Fano varieties.
{"title":"On positivity of the CM line bundle on K-moduli spaces","authors":"Chenyang Xu, Ziquan Zhuang","doi":"10.4007/annals.2020.192.3.7","DOIUrl":"https://doi.org/10.4007/annals.2020.192.3.7","url":null,"abstract":"In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations which can be used to test various K-stability notions of Fano varieties.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46527886","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 : 2019-11-17DOI: 10.4007/annals.2021.194.2.3
Cole Hugelmeyer
We prove that for every smooth Jordan curve $gamma$, if $X$ is the set of all $r in [0,1]$ so that there is an inscribed rectangle in $gamma$ of aspect ratio $tan(rcdot pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study disjoint Mobius strips bounding a $(2n,n)$-torus link in the solid torus times an interval. We prove that any such set of Mobius strips can be equipped with a natural total ordering. We then combine this total ordering with some additive combinatorics to prove that $1/3$ is a sharp lower bound on the probability that a Mobius strip bounding the $(2,1)$-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.
证明了对于每一条光滑的约当曲线$gamma$，如果$X$是所有$r in [0,1]$的集合，使得在$gamma$中存在一个纵横比为$tan(rcdot pi/4)$的内切矩形，则$X$的勒贝格测度至少为$1/3$。为了做到这一点，我们研究了不相交的莫比乌斯带，该莫比乌斯带在实体环面乘以一个间隔内包围$(2n,n)$ -环面连接。我们证明了任何这样的莫比乌斯带集合都可以具有自然全序。然后，我们将这种总排序与一些加性组合结合起来，证明$1/3$是包围在实体环面中$(2,1)$ -环面结的莫比乌斯带乘以一个间隔将以均匀随机角度与其旋转相交的概率的一个明显下界。
{"title":"Inscribed rectangles in a smooth Jordan curve attain at least one\u0000 third of all aspect ratios","authors":"Cole Hugelmeyer","doi":"10.4007/annals.2021.194.2.3","DOIUrl":"https://doi.org/10.4007/annals.2021.194.2.3","url":null,"abstract":"We prove that for every smooth Jordan curve $gamma$, if $X$ is the set of all $r in [0,1]$ so that there is an inscribed rectangle in $gamma$ of aspect ratio $tan(rcdot pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study disjoint Mobius strips bounding a $(2n,n)$-torus link in the solid torus times an interval. We prove that any such set of Mobius strips can be equipped with a natural total ordering. We then combine this total ordering with some additive combinatorics to prove that $1/3$ is a sharp lower bound on the probability that a Mobius strip bounding the $(2,1)$-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2019-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44833876","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 : 2019-11-12DOI: 10.4007/annals.2022.195.2.3
Qian Wang
We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,varrho, fw) in H^stimes H^stimes H^{s'}$, $22$. Due to the works of Smith-Tataru and Wang for the irrotational isentropic case, the local well-posedness can be achieved if the data satisfy $v, varrho in H^{s}$, with $s>2$. In the incompressible case the solution is proven to be ill-posed for the datum $fwin H^frac{3}{2}$ by Bourgain-Li. The solution of the compressible Euler equations is not expected to be well-posed if the data merely satisfy $v, varrhoin H^{s}, s>2$ with a general rough vorticity. By decomposing the velocity into the term $(I-Delta_e)^{-1}curl fw$ and a wave function verifying an improved wave equation, with a series of cancellations for treating the latter, we achieve the $H^s$-energy bound and complete the linearization for the wave functions by using the $H^{s-f12}, , s>2$ norm for the vorticity. The propagation of energy for the vorticity typically requires $curl fwin C^{0, 0+}$ initially, stronger than our assumption by 1/2-derivative. We perform trilinear estimates to gain regularity by observing a div-curl structure when propagating the energy of the normalized double-curl of the vorticity, and also by spacetime integration by parts. To prove the Strichartz estimate for the linearized wave in the rough spacetime, we encounter a strong Ricci defect requiring the bound of $|curl fw|_{L_x^infty L_t^1}$ on null cones. This difficulty is solved by uncovering the cancellation structures due to the acoustic metric on the angular derivatives of Ricci and the second fundamental form.
{"title":"Rough solutions of the 3-D compressible Euler equations","authors":"Qian Wang","doi":"10.4007/annals.2022.195.2.3","DOIUrl":"https://doi.org/10.4007/annals.2022.195.2.3","url":null,"abstract":"We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,varrho, fw) in H^stimes H^stimes H^{s'}$, $22$. Due to the works of Smith-Tataru and Wang for the irrotational isentropic case, the local well-posedness can be achieved if the data satisfy $v, varrho in H^{s}$, with $s>2$. In the incompressible case the solution is proven to be ill-posed for the datum $fwin H^frac{3}{2}$ by Bourgain-Li. The solution of the compressible Euler equations is not expected to be well-posed if the data merely satisfy $v, varrhoin H^{s}, s>2$ with a general rough vorticity. \u0000By decomposing the velocity into the term $(I-Delta_e)^{-1}curl fw$ and a wave function verifying an improved wave equation, with a series of cancellations for treating the latter, we achieve the $H^s$-energy bound and complete the linearization for the wave functions by using the $H^{s-f12}, , s>2$ norm for the vorticity. The propagation of energy for the vorticity typically requires $curl fwin C^{0, 0+}$ initially, stronger than our assumption by 1/2-derivative. We perform trilinear estimates to gain regularity by observing a div-curl structure when propagating the energy of the normalized double-curl of the vorticity, and also by spacetime integration by parts. To prove the Strichartz estimate for the linearized wave in the rough spacetime, we encounter a strong Ricci defect requiring the bound of $|curl fw|_{L_x^infty L_t^1}$ on null cones. This difficulty is solved by uncovering the cancellation structures due to the acoustic metric on the angular derivatives of Ricci and the second fundamental form.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2019-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44289686","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 : 2019-10-19DOI: 10.4007/annals.2020.192.2.7
Olivier Brunat, O. Dudas, Jay Taylor
We show that the decomposition matrix of unipotent $ell$-blocks of a finite reductive group $mathbf{G}(mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $ell$ is very good for $mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.
{"title":"Unitriangular shape of decomposition matrices of unipotent\u0000 blocks","authors":"Olivier Brunat, O. Dudas, Jay Taylor","doi":"10.4007/annals.2020.192.2.7","DOIUrl":"https://doi.org/10.4007/annals.2020.192.2.7","url":null,"abstract":"We show that the decomposition matrix of unipotent $ell$-blocks of a finite reductive group $mathbf{G}(mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $ell$ is very good for $mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2019-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44856185","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 : 2019-10-11DOI: 10.4007/annals.2023.197.1.4
S. Møller, Nils R. Scheithauer
We prove a dimension formula for the weight-1 subspace of a vertex operator algebra $V^{operatorname{orb}(g)}$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with a finite order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $operatorname{Aut}(V)$. We then give a construction of all 70 strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space as orbifolds of the Leech lattice vertex operator algebra $V_Lambda$ associated with generalised deep holes. This provides the first uniform construction of these vertex operator algebras and naturally generalises the construction of the 23 Niemeier lattices with non-vanishing root system from the deep holes of the Leech lattice $Lambda$ by Conway, Parker and Sloane.
{"title":"Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra","authors":"S. Møller, Nils R. Scheithauer","doi":"10.4007/annals.2023.197.1.4","DOIUrl":"https://doi.org/10.4007/annals.2023.197.1.4","url":null,"abstract":"We prove a dimension formula for the weight-1 subspace of a vertex operator algebra $V^{operatorname{orb}(g)}$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with a finite order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $operatorname{Aut}(V)$. \u0000We then give a construction of all 70 strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space as orbifolds of the Leech lattice vertex operator algebra $V_Lambda$ associated with generalised deep holes. This provides the first uniform construction of these vertex operator algebras and naturally generalises the construction of the 23 Niemeier lattices with non-vanishing root system from the deep holes of the Leech lattice $Lambda$ by Conway, Parker and Sloane.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2019-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43860069","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 : 2019-09-18DOI: 10.4007/annals.2021.194.1.2
B. Jaye, X. Tolsa, Michele Villa
In this paper we provide a proof of the Carleson $varepsilon^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $varepsilon^2$-square function.
{"title":"A proof of Carleson's 𝜀2-conjecture","authors":"B. Jaye, X. Tolsa, Michele Villa","doi":"10.4007/annals.2021.194.1.2","DOIUrl":"https://doi.org/10.4007/annals.2021.194.1.2","url":null,"abstract":"In this paper we provide a proof of the Carleson $varepsilon^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $varepsilon^2$-square function.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2019-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43852231","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":"Weil representation and Arithmetic Fundamental Lemma","authors":"Wei Zhang","doi":"10.4007/ANNALS.2021.193.3.5","DOIUrl":"https://doi.org/10.4007/ANNALS.2021.193.3.5","url":null,"abstract":"By a global approach, we prove the arithmetic fundamental lemma conjecture for unitary groups in $n$ variables over $mathbb{Q}_p$ when $pgeq n$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2019-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41529540","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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