{"title":"Index to Volume 143 2021","authors":"","doi":"10.1353/ajm.2021.0051","DOIUrl":"https://doi.org/10.1353/ajm.2021.0051","url":null,"abstract":"","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66915033","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}
Abstract:We obtain compact orientable embedded surfaces with constant mean curvature $0
摘要：我们得到了具有常平均曲率$0
{"title":"Compact embedded surfaces with constant mean curvature in $Bbb{S}^2timesBbb{R}$","authors":"J. M. Manzano, Francisco Torralbo","doi":"10.1353/ajm.2020.0050","DOIUrl":"https://doi.org/10.1353/ajm.2020.0050","url":null,"abstract":"Abstract:We obtain compact orientable embedded surfaces with constant mean curvature $0<H<{1over 2}$ and arbitrary genus in $Bbb{S}^2timesBbb{R}$. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature ${1over 2}$ tangent along an equator. This is a particular case of a conjugate Plateau construction of doubly periodic surfaces with constant mean curvature in $Bbb{S}^2timesBbb{R}$, $Bbb{H}^2timesBbb{R}$, and $Bbb{R}^3$ with bounded height and enjoying the symmetries of certain tessellations of $Bbb{S}^2$, $Bbb{H}^2$, and $Bbb{R}^2$ by regular polygons.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0050","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46169856","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}
N. Gigli, C. Ketterer, Kazumasa Kuwada, Shin-ichi Ohta
Abstract:We consider a rigidity problem for the spectral gap of the Laplacian on an ${rm RCD}(K,infty)$-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive $K$. For a weighted Riemannian manifold, Cheng-Zhou showed that the sharp spectral gap is achieved only when a $1$-dimensional Gaussian space is split off. This can be regarded as an infinite-dimensional counterpart to Obata's rigidity theorem. Generalizing to ${rm RCD}(K,infty)$-spaces is not straightforward due to the lack of smooth structure and doubling condition. We employ the lift of an eigenfunction to the Wasserstein space and the theory of regular Lagrangian flows recently developed by Ambrosio-Trevisan to overcome this difficulty.
{"title":"Rigidity for the spectral gap on Rcd(K, ∞)-spaces","authors":"N. Gigli, C. Ketterer, Kazumasa Kuwada, Shin-ichi Ohta","doi":"10.1353/ajm.2020.0039","DOIUrl":"https://doi.org/10.1353/ajm.2020.0039","url":null,"abstract":"Abstract:We consider a rigidity problem for the spectral gap of the Laplacian on an ${rm RCD}(K,infty)$-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive $K$. For a weighted Riemannian manifold, Cheng-Zhou showed that the sharp spectral gap is achieved only when a $1$-dimensional Gaussian space is split off. This can be regarded as an infinite-dimensional counterpart to Obata's rigidity theorem. Generalizing to ${rm RCD}(K,infty)$-spaces is not straightforward due to the lack of smooth structure and doubling condition. We employ the lift of an eigenfunction to the Wasserstein space and the theory of regular Lagrangian flows recently developed by Ambrosio-Trevisan to overcome this difficulty.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0039","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42095026","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 study Lelong numbers of currents of full mass intersection on a compact Kaehler manifold in a mixed setting. Our main theorems cover some recent results due to Darvas-Di Nezza-Lu. One of the key ingredients in our approach is a new notion of products of pseudoeffective classes which captures some "pluripolar part" of the "total intersection" of given pseudoeffective classes.
{"title":"Lelong numbers of currents of full mass intersection","authors":"Duc-Viet Vu","doi":"10.1353/ajm.2023.0016","DOIUrl":"https://doi.org/10.1353/ajm.2023.0016","url":null,"abstract":"We study Lelong numbers of currents of full mass intersection on a compact Kaehler manifold in a mixed setting. Our main theorems cover some recent results due to Darvas-Di Nezza-Lu. One of the key ingredients in our approach is a new notion of products of pseudoeffective classes which captures some \"pluripolar part\" of the \"total intersection\" of given pseudoeffective classes.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47685159","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}
abstract:We extend work of Denef and Sperber and also Cluckers regarding a conjecture of Igusa in the two dimensional setting by no longer requiring the polynomial to be nondegenerate with respect to its Newton diagram. More precisely we establish sharp, uniform bounds for complete exponential sums and the number of polynomial congruences for general quasi-homogeneous polynomials in two variables.
{"title":"On a Conjecture of Igusa in Two Dimensions","authors":"James Wright","doi":"10.1353/ajm.2020.0026","DOIUrl":"https://doi.org/10.1353/ajm.2020.0026","url":null,"abstract":"abstract:We extend work of Denef and Sperber and also Cluckers regarding a conjecture of Igusa in the two dimensional setting by no longer requiring the polynomial to be nondegenerate with respect to its Newton diagram. More precisely we establish sharp, uniform bounds for complete exponential sums and the number of polynomial congruences for general quasi-homogeneous polynomials in two variables.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0026","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45036171","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}
abstract:We present new free-curvature one-cycle sweepout estimates in Riemannian geometry, both on surfaces and in higher dimension. More precisely, we derive upper bounds on the length of one-parameter families of one-cycles sweeping out essential surfaces in closed Riemannian manifolds. In particular, we show that there exists a homotopically substantial one-cycle sweepout of the essential sphere in the complex projective space, endowed with an arbitrary Riemannian metric, whose one-cycle length is bounded in terms of the volume (or diameter) of the manifold. This is the first estimate on sweepout volume in higher dimension without curvature assumption. We also give a detailed account of the situation for compact Riemannian surfaces with or without boundary, in relation with questions raised by P.~Buser and L.~Guth.
{"title":"One-Cycle Sweepout Estimates of Essential Surfaces in Closed Riemannian Manifolds","authors":"S. Sabourau","doi":"10.1353/ajm.2020.0031","DOIUrl":"https://doi.org/10.1353/ajm.2020.0031","url":null,"abstract":"abstract:We present new free-curvature one-cycle sweepout estimates in Riemannian geometry, both on surfaces and in higher dimension. More precisely, we derive upper bounds on the length of one-parameter families of one-cycles sweeping out essential surfaces in closed Riemannian manifolds. In particular, we show that there exists a homotopically substantial one-cycle sweepout of the essential sphere in the complex projective space, endowed with an arbitrary Riemannian metric, whose one-cycle length is bounded in terms of the volume (or diameter) of the manifold. This is the first estimate on sweepout volume in higher dimension without curvature assumption. We also give a detailed account of the situation for compact Riemannian surfaces with or without boundary, in relation with questions raised by P.~Buser and L.~Guth.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0031","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43288817","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}
abstract:We prove that any complete immersed two-sided mean convex translating soliton $Sigmasubset{Bbb R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in ${Bbb R}^3$ is the axisymmetric "bowl soliton". We also show that if the mean curvature of $Sigma$ tends to zero at infinity, then $Sigma$ can be represented as an entire graph and so is the "bowl soliton". Finally we classify the asymptotic behavior of all locally strictly convex graphical translating solitons defined over strip regions.
{"title":"Complete translating solitons to the mean curvature flow in ℝ3 with nonnegative mean curvature","authors":"J. Spruck, Ling Xiao","doi":"10.1353/ajm.2020.0023","DOIUrl":"https://doi.org/10.1353/ajm.2020.0023","url":null,"abstract":"abstract:We prove that any complete immersed two-sided mean convex translating soliton $Sigmasubset{Bbb R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in ${Bbb R}^3$ is the axisymmetric \"bowl soliton\". We also show that if the mean curvature of $Sigma$ tends to zero at infinity, then $Sigma$ can be represented as an entire graph and so is the \"bowl soliton\". Finally we classify the asymptotic behavior of all locally strictly convex graphical translating solitons defined over strip regions.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48158166","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 : 2020-04-02DOI: 10.1353/ajm.2023.a902955
J. Roos, A. Seeger
abstract:For the spherical mean operators $scr{A}_t$ in $Bbb{R}^d$, $dge 2$, we consider the maximal functions $M_Ef=sup_{tin E}|scr{A}_t f|$, with dilation sets $Esubset [1,2]$. In this paper we give a surprising characterization of the closed convex sets which can occur as closure of the sharp $L^p$ improving region of $M_E$ for some $E$. This region depends on the Minkowski dimension of $E$, but also other properties of the fractal geometry such as the Assouad spectrum of $E$ and subsets of $E$. A key ingredient is an essentially sharp result on $M_E$ for a class of sets called (quasi-)Assouad regular which is new in two dimensions.
{"title":"Spherical maximal functions and fractal dimensions of dilation sets","authors":"J. Roos, A. Seeger","doi":"10.1353/ajm.2023.a902955","DOIUrl":"https://doi.org/10.1353/ajm.2023.a902955","url":null,"abstract":"abstract:For the spherical mean operators $scr{A}_t$ in $Bbb{R}^d$, $dge 2$, we consider the maximal functions $M_Ef=sup_{tin E}|scr{A}_t f|$, with dilation sets $Esubset [1,2]$. In this paper we give a surprising characterization of the closed convex sets which can occur as closure of the sharp $L^p$ improving region of $M_E$ for some $E$. This region depends on the Minkowski dimension of $E$, but also other properties of the fractal geometry such as the Assouad spectrum of $E$ and subsets of $E$. A key ingredient is an essentially sharp result on $M_E$ for a class of sets called (quasi-)Assouad regular which is new in two dimensions.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66915761","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 : 2020-03-30DOI: 10.1353/ajm.2023.a902957
G. Fournodavlos, J. Luk
abstract:We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the form $$ {}^{(4)}g = -dt^2 + sum_{i,j=1}^3 a_{ij}t^{2 p_{max{i,j}}},{rm d} x^i,{rm d} x^j $$ on $(0,T]_ttimesBbb{T}^3_x$, where $a_{ij}(t,x)$ and $p_i(x)$ are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as $tto 0^+$. These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the ``singular hypersurface'' ${t=0}$. This is the first such result without imposing symmetry or analyticity. To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-$t$ hypersurfaces.
{"title":"Asymptotically Kasner-like singularities","authors":"G. Fournodavlos, J. Luk","doi":"10.1353/ajm.2023.a902957","DOIUrl":"https://doi.org/10.1353/ajm.2023.a902957","url":null,"abstract":"abstract:We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the form $$ {}^{(4)}g = -dt^2 + sum_{i,j=1}^3 a_{ij}t^{2 p_{max{i,j}}},{rm d} x^i,{rm d} x^j $$ on $(0,T]_ttimesBbb{T}^3_x$, where $a_{ij}(t,x)$ and $p_i(x)$ are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as $tto 0^+$. These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the ``singular hypersurface'' ${t=0}$. This is the first such result without imposing symmetry or analyticity. To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-$t$ hypersurfaces.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43651242","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}
abstract:We establish that if $dgeq 2k+6$ and $q$ is odd and sufficiently large with respect to $alphain (0,1)$, then every set $Asubseteq{bf F}_q^d$of size $|A|geqalpha q^d$ will contain an isometric copy of every spherical $(k+2)$-point configuration that spans $k$ dimensions.
{"title":"Spherical configurations over finite fields","authors":"N. Lyall, Á. Magyar, Hans Parshall","doi":"10.1353/ajm.2020.0010","DOIUrl":"https://doi.org/10.1353/ajm.2020.0010","url":null,"abstract":"abstract:We establish that if $dgeq 2k+6$ and $q$ is odd and sufficiently large with respect to $alphain (0,1)$, then every set $Asubseteq{bf F}_q^d$of size $|A|geqalpha q^d$ will contain an isometric copy of every spherical $(k+2)$-point configuration that spans $k$ dimensions.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48742730","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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