Pub Date : 2020-03-13DOI: 10.1215/00127094-2023-0005
M. Stolarski
In [Vel94], Velazquez constructed a countable collection of mean curvature flow solutions in $mathbb{R}^N$ in every dimension $N ge 8$. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. In contrast, we confirm here that, in every dimension $N ge 8$, a nontrivial subset of these solutions has uniformly bounded mean curvature.
在[Vel94]中，Velazquez构造了$mathbb{R}^N$中每维$N ge 8$的平均曲率流解的可数集合。这些解在有限时间内都是奇异的此时第二种基本形式就失效了。相反，我们在这里证实，在每个维度中，这些解的一个非平凡子集具有均匀有界的平均曲率。
{"title":"Existence of mean curvature flow singularities with bounded mean curvature","authors":"M. Stolarski","doi":"10.1215/00127094-2023-0005","DOIUrl":"https://doi.org/10.1215/00127094-2023-0005","url":null,"abstract":"In [Vel94], Velazquez constructed a countable collection of mean curvature flow solutions in $mathbb{R}^N$ in every dimension $N ge 8$. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. In contrast, we confirm here that, in every dimension $N ge 8$, a nontrivial subset of these solutions has uniformly bounded mean curvature.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44989454","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 : 2020-03-09DOI: 10.1215/00127094-2021-0044
Jun-Muk Hwang
Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulated question. When the submanifolds do have nonzero vector fields, we obtain an affirmative answer to the question under the additional assumption that submanifolds have certain nice deformation properties, which is applicable to free rational curves. As applications, we obtain a stronger version of the Cartan-Fubini type extension theorem for Fano manifolds of Picard number 1 and also prove that two linearly normal projective K3 surfaces in ${bf P}^g$ are projectively isomorphic if and only if the families of their general hyperplane sections trace the same locus in the moduli space of curves of genus $g >2$.
{"title":"Extending Nirenberg–Spencer’s question on holomorphic embeddings to families of holomorphic embeddings","authors":"Jun-Muk Hwang","doi":"10.1215/00127094-2021-0044","DOIUrl":"https://doi.org/10.1215/00127094-2021-0044","url":null,"abstract":"Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulated question. When the submanifolds do have nonzero vector fields, we obtain an affirmative answer to the question under the additional assumption that submanifolds have certain nice deformation properties, which is applicable to free rational curves. As applications, we obtain a stronger version of the Cartan-Fubini type extension theorem for Fano manifolds of Picard number 1 and also prove that two linearly normal projective K3 surfaces in ${bf P}^g$ are projectively isomorphic if and only if the families of their general hyperplane sections trace the same locus in the moduli space of curves of genus $g >2$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47671195","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 : 2020-03-09DOI: 10.1215/00127094-2022-0051
A. Figalli, Y. Zhang
Motivated by important applications to problems in the calculus of variations and evolution PDEs, in recent years there has been a growing interest around the understanding of quantitative stability for functional/geometric inequalities, see for instance [3, 2, 8, 27, 28, 21, 9, 22, 29, 18, 10, 6, 7, 11, 13, 19, 23, 35, 26, 5, 14, 16, 17, 20, 25, 30, 31, 24, 33, 34], as well as the survey papers [15, 26, 17]. Following this line of research, in this paper we shall investigate the stability of minimizers to the classical Sobolev inequality.
{"title":"Sharp gradient stability for the Sobolev inequality","authors":"A. Figalli, Y. Zhang","doi":"10.1215/00127094-2022-0051","DOIUrl":"https://doi.org/10.1215/00127094-2022-0051","url":null,"abstract":"Motivated by important applications to problems in the calculus of variations and evolution PDEs, in recent years there has been a growing interest around the understanding of quantitative stability for functional/geometric inequalities, see for instance [3, 2, 8, 27, 28, 21, 9, 22, 29, 18, 10, 6, 7, 11, 13, 19, 23, 35, 26, 5, 14, 16, 17, 20, 25, 30, 31, 24, 33, 34], as well as the survey papers [15, 26, 17]. Following this line of research, in this paper we shall investigate the stability of minimizers to the classical Sobolev inequality.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43569984","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 : 2020-03-09DOI: 10.1215/00127094-2020-0061
K. Goodearl, M. Yakimov
We prove a general theorem for constructing integral quantum cluster algebras over ${mathbb{Z}}[q^{pm 1/2}]$, namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over ${mathbb{Z}}[q^{pm 1/2}]$. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that for every symmetrizable Kac-Moody algebra ${mathfrak{g}}$ and Weyl group element $w$, the dual canonical form $A_q({mathfrak{n}}_+(w))_{mathbb{Z}[q^{pm 1}]}$ of the corresponding quantum unipotent cell has the property that $A_q( {mathfrak{n}}_+(w))_{mathbb{Z}[q^{pm 1}]} otimes_{mathbb{Z}[q^{ pm 1}]} {mathbb{Z}}[ q^{pm 1/2}]$ is isomorphic to a quantum cluster algebra over ${mathbb{Z}}[q^{pm 1/2}]$ and to the corresponding upper quantum cluster algebra over ${mathbb{Z}}[q^{pm 1/2}]$.
{"title":"Integral quantum cluster structures","authors":"K. Goodearl, M. Yakimov","doi":"10.1215/00127094-2020-0061","DOIUrl":"https://doi.org/10.1215/00127094-2020-0061","url":null,"abstract":"We prove a general theorem for constructing integral quantum cluster algebras over ${mathbb{Z}}[q^{pm 1/2}]$, namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over ${mathbb{Z}}[q^{pm 1/2}]$. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that for every symmetrizable Kac-Moody algebra ${mathfrak{g}}$ and Weyl group element $w$, the dual canonical form $A_q({mathfrak{n}}_+(w))_{mathbb{Z}[q^{pm 1}]}$ of the corresponding quantum unipotent cell has the property that $A_q( {mathfrak{n}}_+(w))_{mathbb{Z}[q^{pm 1}]} otimes_{mathbb{Z}[q^{ pm 1}]} {mathbb{Z}}[ q^{pm 1/2}]$ is isomorphic to a quantum cluster algebra over ${mathbb{Z}}[q^{pm 1/2}]$ and to the corresponding upper quantum cluster algebra over ${mathbb{Z}}[q^{pm 1/2}]$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45935598","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 : 2020-03-08DOI: 10.1215/00127094-2023-0031
Promit Ghosal, G. Remy, Xin Sun, Y. Sun
Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced by Polyakov in the context of string theory. Conformal blocks are objects underlying the integrable structure of CFT via the conformal bootstrap equation. The present work provides a probabilistic construction of the 1-point toric conformal block of Liouville theory in terms of a Gaussian multiplicative chaos measure corresponding to a one-dimensional log-correlated field. We prove that our probabilistic conformal block satisfies Zamolodchikov's recursion, and we relate it to the instanton part of Nekrasov's partition function by the Alday-Gaiotto-Tachikawa correspondence. Our proof rests upon an analysis of Belavin-Polyakov-Zamolodchikov differential equations, operator product expansions, and Dotsenko-Fateev type integrals.
{"title":"Probabilistic conformal blocks for Liouville CFT on the torus","authors":"Promit Ghosal, G. Remy, Xin Sun, Y. Sun","doi":"10.1215/00127094-2023-0031","DOIUrl":"https://doi.org/10.1215/00127094-2023-0031","url":null,"abstract":"Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced by Polyakov in the context of string theory. Conformal blocks are objects underlying the integrable structure of CFT via the conformal bootstrap equation. The present work provides a probabilistic construction of the 1-point toric conformal block of Liouville theory in terms of a Gaussian multiplicative chaos measure corresponding to a one-dimensional log-correlated field. We prove that our probabilistic conformal block satisfies Zamolodchikov's recursion, and we relate it to the instanton part of Nekrasov's partition function by the Alday-Gaiotto-Tachikawa correspondence. Our proof rests upon an analysis of Belavin-Polyakov-Zamolodchikov differential equations, operator product expansions, and Dotsenko-Fateev type integrals.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141224238","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 : 2020-02-18DOI: 10.1215/00127094-2022-0017
H. Duminil-Copin, Subhajit Goswami, Pierre-François Rodriguez, Franco Severo
We consider level-sets of the Gaussian free field on $mathbb Z^d$, for $dgeq 3$, above a given real-valued height parameter $h$. As $h$ varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated to this model, namely $h_{**}(d)$, $h_{*}(d)$ and $bar h(d)$, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide, i.e. $h_{**}(d)=h_{*}(d)= bar h(d)$ for any $d geq 3$. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. The successful implementation of this strategy relies extensively on certain novel renormalization techniques, in particular to control so-called large-field effects. This approach opens the way to a complete understanding of the off-critical phases of strongly correlated percolation models.
我们考虑$mathbb Z^d$上的高斯自由场的水平集，对于$dgeq 3$，高于给定的实值高度参数$h$。随着$h$的变化，这定义了一个具有强代数衰减相关性的规范渗透模型。我们证明了与该模型相关的三个自然临界参数，即$h_{**}(d)$, $h_{*}(d)$和$bar h(d)$，分别描述了有序的亚临界阶段，无限簇的出现和超临界阶段局部唯一性区域的开始，实际上是重合的，即$h_{**}(d)=h_{*}(d)= bar h(d)$对于任何$d geq 3$。我们证明的核心是一个新的插值方案，旨在积分出高斯自由场的远程依赖。这一策略的成功实施广泛依赖于某些新的重整化技术，特别是控制所谓的大场效应。这种方法为完全理解强相关渗流模型的非临界阶段开辟了道路。
{"title":"Equality of critical parameters for percolation of Gaussian free field level sets","authors":"H. Duminil-Copin, Subhajit Goswami, Pierre-François Rodriguez, Franco Severo","doi":"10.1215/00127094-2022-0017","DOIUrl":"https://doi.org/10.1215/00127094-2022-0017","url":null,"abstract":"We consider level-sets of the Gaussian free field on $mathbb Z^d$, for $dgeq 3$, above a given real-valued height parameter $h$. As $h$ varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated to this model, namely $h_{**}(d)$, $h_{*}(d)$ and $bar h(d)$, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide, i.e. $h_{**}(d)=h_{*}(d)= bar h(d)$ for any $d geq 3$. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. The successful implementation of this strategy relies extensively on certain novel renormalization techniques, in particular to control so-called large-field effects. This approach opens the way to a complete understanding of the off-critical phases of strongly correlated percolation models.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41930801","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 : 2020-02-17DOI: 10.1215/00127094-2022-0053
Florian Hanisch, A. Strohmaier, Alden Waters
We consider the case of scattering of several obstacles in $mathbb{R}^d$ for $d geq 2$. Then the absolutely continuous part of the Laplace operator $Delta$ with Dirichlet boundary conditions and the free Laplace operator $Delta_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast we have that the difference $g(Delta)-g(Delta_0)$ is a trace-class operator and its trace is described by the Krein spectral shift function. In this paper we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles we consider the Laplace operators $Delta_1$ and $Delta_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then $g(Delta) - g(Delta_1) - g(Delta_2) + g(Delta_0)$ is a trace class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman-Krein formula. In case $g(x)=x^frac{1}{2}$ the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using non-rigorous path integral derivations and our formula provides both a rigorous justification as well as a generalisation.
{"title":"A relative trace formula for obstacle scattering","authors":"Florian Hanisch, A. Strohmaier, Alden Waters","doi":"10.1215/00127094-2022-0053","DOIUrl":"https://doi.org/10.1215/00127094-2022-0053","url":null,"abstract":"We consider the case of scattering of several obstacles in $mathbb{R}^d$ for $d geq 2$. Then the absolutely continuous part of the Laplace operator $Delta$ with Dirichlet boundary conditions and the free Laplace operator $Delta_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast we have that the difference $g(Delta)-g(Delta_0)$ is a trace-class operator and its trace is described by the Krein spectral shift function. In this paper we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles we consider the Laplace operators $Delta_1$ and $Delta_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then $g(Delta) - g(Delta_1) - g(Delta_2) + g(Delta_0)$ is a trace class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman-Krein formula. In case $g(x)=x^frac{1}{2}$ the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using non-rigorous path integral derivations and our formula provides both a rigorous justification as well as a generalisation.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43445567","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 : 2020-02-10DOI: 10.1215/00127094-2022-0055
V. Bergelson, F. Richter
We establish two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg, a theorem of Erdős-Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Halasz. By building on the ideas behind our ergodic results, we recast Sarnak's Mobius disjointness conjecture in a new dynamical framework. This naturally leads to an extension of Sarnak's conjecture which focuses on the disjointness of additive and multiplicative semigroup actions. We substantiate this extension by providing proofs of several special cases thereof.
{"title":"Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions","authors":"V. Bergelson, F. Richter","doi":"10.1215/00127094-2022-0055","DOIUrl":"https://doi.org/10.1215/00127094-2022-0055","url":null,"abstract":"We establish two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg, a theorem of Erdős-Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Halasz. By building on the ideas behind our ergodic results, we recast Sarnak's Mobius disjointness conjecture in a new dynamical framework. This naturally leads to an extension of Sarnak's conjecture which focuses on the disjointness of additive and multiplicative semigroup actions. We substantiate this extension by providing proofs of several special cases thereof.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46991030","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 : 2020-02-04DOI: 10.1215/00127094-2021-0057
Danny Calegari, F. C. Marques, A. Neves
We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric.
{"title":"Counting minimal surfaces in negatively curved 3-manifolds","authors":"Danny Calegari, F. C. Marques, A. Neves","doi":"10.1215/00127094-2021-0057","DOIUrl":"https://doi.org/10.1215/00127094-2021-0057","url":null,"abstract":"We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47195096","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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