The original Riemann-Hilbert problem asks to find a Fuchsian ordinary differential equation with prescribed singularities and monodromy in the complex line. In the early 1980's Kashiwara solved a generalized version of the problem, valid on complex manifolds of any dimension. He presented it as a correspondence between regular holonomic D-modules and perverse sheaves. The analogous problem where one drops the regularity condition remained open for about thirty years. We solved it in the paper that received a 2024 Frontiers of Science Award. Our construction requires in particular an enhancement of the category of perverse sheaves. Here, using some examples in dimension one, we wish to convey the gist of the main ingredients used in our work. This is a written account of a talk given by the first named author at the International Congress of Basic Sciences on July 2024 in Beijing.
最初的黎曼-希尔伯特（Riemann-Hilbert）问题要求找到一个在复线上具有规定奇点和单色性的富奇异常微分方程。20 世纪 80 年代初，柏原（Kashiwara）解决了这个问题的一个广义版本，它在任何维度的复流形上都有效。他将其表述为正则整体 D 模块与反向剪切之间的对应关系。放弃正则性条件的类似问题，大约三十年来一直悬而未决。我们在获得 2024 年科学前沿奖的论文中解决了这个问题。我们的构造尤其需要加强反向剪切范畴。在这里，我们希望用一些一维的例子来表达我们工作中所使用的主要成分的要点。本文是第一作者于2024年7月在北京举行的国际基础科学大会上的演讲稿。
{"title":"On the irregular Riemann-Hilbert correspondence","authors":"Andrea D'Agnolo, Masaki Kashiwara","doi":"arxiv-2408.04260","DOIUrl":"https://doi.org/arxiv-2408.04260","url":null,"abstract":"The original Riemann-Hilbert problem asks to find a Fuchsian ordinary\u0000differential equation with prescribed singularities and monodromy in the\u0000complex line. In the early 1980's Kashiwara solved a generalized version of the\u0000problem, valid on complex manifolds of any dimension. He presented it as a\u0000correspondence between regular holonomic D-modules and perverse sheaves. The analogous problem where one drops the regularity condition remained open\u0000for about thirty years. We solved it in the paper that received a 2024\u0000Frontiers of Science Award. Our construction requires in particular an\u0000enhancement of the category of perverse sheaves. Here, using some examples in\u0000dimension one, we wish to convey the gist of the main ingredients used in our\u0000work. This is a written account of a talk given by the first named author at the\u0000International Congress of Basic Sciences on July 2024 in Beijing.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish a version of the Landen's transformation for Weierstrass functions and invariants that is applicable to general lattices in complex plane. Using it we present an effective method for computing Weierstrass functions, their periods, and elliptic integral in Weierstrass form given Weierstrass invariants $g_2$ and $g_3$ of an elliptic curve. Similarly to the classical Landen's method our algorithm has quadratic rate of convergence.
{"title":"A Landen-type method for computation of Weierstrass functions","authors":"Matvey Smirnov, Kirill Malkov, Sergey Rogovoy","doi":"arxiv-2408.05252","DOIUrl":"https://doi.org/arxiv-2408.05252","url":null,"abstract":"We establish a version of the Landen's transformation for Weierstrass\u0000functions and invariants that is applicable to general lattices in complex\u0000plane. Using it we present an effective method for computing Weierstrass\u0000functions, their periods, and elliptic integral in Weierstrass form given\u0000Weierstrass invariants $g_2$ and $g_3$ of an elliptic curve. Similarly to the\u0000classical Landen's method our algorithm has quadratic rate of convergence.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish general sufficient conditions for exact (and global) regularity in the $barpartial$-Neumann problem on $(p,q)$-forms, $0 leq p leq n$ and $1leq q leq n$, on a pseudoconvex domain $Omega$ with smooth boundary $bOmega$ in an $n$-dimensional complex manifold $M$. Our hypotheses include two assumptions: 1) $M$ admits a function that is strictly plurisubharmonic acting on $(p_0,q_0)$-forms in a neighborhood of $bOmega$ for some fixed $0 leq p_0 leq n$, $1 leq q_0 leq n$, or $M$ is a K"ahler metric whose holomorphic bisectional curvature acting $(p,q)$-forms is positive; and 2) there exists a family of vector fields $T_epsilon$ that are transverse to the boundary $bOmega$ and generate one forms, which when applied to $(p,q)$-forms, $0 leq p leq n$ and $q_0 leq q leq n$, satisfy a "weak form" of the compactness estimate. We also provide examples and applications of our main theorems.
{"title":"Global regularity for the $barpartial$-Neumann problem on pseudoconvex manifolds","authors":"Tran Vu Khanh, Andrew Raich","doi":"arxiv-2408.04512","DOIUrl":"https://doi.org/arxiv-2408.04512","url":null,"abstract":"We establish general sufficient conditions for exact (and global) regularity\u0000in the $barpartial$-Neumann problem on $(p,q)$-forms, $0 leq p leq n$ and\u0000$1leq q leq n$, on a pseudoconvex domain $Omega$ with smooth boundary\u0000$bOmega$ in an $n$-dimensional complex manifold $M$. Our hypotheses include\u0000two assumptions: 1) $M$ admits a function that is strictly plurisubharmonic\u0000acting on $(p_0,q_0)$-forms in a neighborhood of $bOmega$ for some fixed $0\u0000leq p_0 leq n$, $1 leq q_0 leq n$, or $M$ is a K\"ahler metric whose\u0000holomorphic bisectional curvature acting $(p,q)$-forms is positive; and 2)\u0000there exists a family of vector fields $T_epsilon$ that are transverse to the\u0000boundary $bOmega$ and generate one forms, which when applied to $(p,q)$-forms,\u0000$0 leq p leq n$ and $q_0 leq q leq n$, satisfy a \"weak form\" of the\u0000compactness estimate. We also provide examples and applications of our main theorems.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"93 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathcal{H}$ be the class of all analytic self-maps of the open unit disk $mathbb{D}$. Denote by $H^n f(z)$ the $n$-th order hyperbolic derivative of $fin mathcal H$ at $zin mathbb{D}$. For $z_0in mathbb{D}$ and $gamma = (gamma_0, gamma_1 , ldots , gamma_{n-1}) in {mathbb D}^{n}$, let ${mathcal H} (gamma) = {f in {mathcal H} : f (z_0) = gamma_0,H^1f (z_0) = gamma_1,ldots ,H^{n-1}f (z_0) = gamma_{n-1} }$. In this paper, we determine the variability region $V(z_0, gamma ) = { f^{(n)}(z_0) : f in {mathcal H} (gamma) }$, which can be called ``the generalized Schwarz-Pick Lemma of $n$-th derivative". We then apply the generalized Schwarz-Pick Lemma to establish a $n$-th order Dieudonn'e's Lemma, which provides an explicit description of the variability region ${h^{(n)}(z_0): hin mathcal{H}, h(0)=0,h(z_0) =w_0, h'(z_0)=w_1,ldots, h^{(n-1)}(z_0)=w_{n-1}}$ for given $z_0$, $w_0$, $w_1,dots,w_{n-1}$. Moreover, we determine the form of all extremal functions.
{"title":"Variability regions for the $n$-th derivative of bounded analytic functions","authors":"Gangqiang Chen","doi":"arxiv-2408.04030","DOIUrl":"https://doi.org/arxiv-2408.04030","url":null,"abstract":"Let $mathcal{H}$ be the class of all analytic self-maps of the open unit\u0000disk $mathbb{D}$. Denote by $H^n f(z)$ the $n$-th order hyperbolic derivative\u0000of $fin mathcal H$ at $zin mathbb{D}$. For $z_0in mathbb{D}$ and $gamma\u0000= (gamma_0, gamma_1 , ldots , gamma_{n-1}) in {mathbb D}^{n}$, let\u0000${mathcal H} (gamma) = {f in {mathcal H} : f (z_0) = gamma_0,H^1f (z_0) =\u0000gamma_1,ldots ,H^{n-1}f (z_0) = gamma_{n-1} }$. In this paper, we determine\u0000the variability region $V(z_0, gamma ) = { f^{(n)}(z_0) : f in {mathcal H}\u0000(gamma) }$, which can be called ``the generalized Schwarz-Pick Lemma of\u0000$n$-th derivative\". We then apply the generalized Schwarz-Pick Lemma to\u0000establish a $n$-th order Dieudonn'e's Lemma, which provides an explicit\u0000description of the variability region ${h^{(n)}(z_0): hin mathcal{H},\u0000h(0)=0,h(z_0) =w_0, h'(z_0)=w_1,ldots, h^{(n-1)}(z_0)=w_{n-1}}$ for given\u0000$z_0$, $w_0$, $w_1,dots,w_{n-1}$. Moreover, we determine the form of all\u0000extremal functions.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Koebe uniformization is a fundemental problem in complex analysis. In this paper, we use transboundary extremal length to show that every nondegenerate and uncountably connected domain with bounded gap-ratio is conformally homeomorphic to a circle domain.
{"title":"Koebe uniformization of nondegenerate domains with bounded gap-ratio","authors":"Yi Zhong","doi":"arxiv-2408.03484","DOIUrl":"https://doi.org/arxiv-2408.03484","url":null,"abstract":"Koebe uniformization is a fundemental problem in complex analysis. In this\u0000paper, we use transboundary extremal length to show that every nondegenerate\u0000and uncountably connected domain with bounded gap-ratio is conformally\u0000homeomorphic to a circle domain.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
begin{abstract} Let $P_+$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider the inequalities of the following form $$ |f|_{L^p(mathbb{T})}leq B_{p,s}|( |P_ + f | ^s + |P_- f |^s) ^{frac 1s}|_{L^p (mathbb{T})} $$ and prove them with sharp constant $B_{p,s}$ for $s in [p',+infty)$ and $1
{"title":"Best constants in reverse Riesz-type inequalities for analytoc and co-analytic projections","authors":"Petar Melentijević","doi":"arxiv-2408.02453","DOIUrl":"https://doi.org/arxiv-2408.02453","url":null,"abstract":"begin{abstract} Let $P_+$ be the Riesz's projection operator and let $P_-= I\u0000- P_+$. We consider the inequalities of the following form $$\u0000|f|_{L^p(mathbb{T})}leq B_{p,s}|( |P_ + f | ^s + |P_- f |^s) ^{frac\u00001s}|_{L^p (mathbb{T})} $$ and prove them with sharp constant $B_{p,s}$ for $s\u0000in [p',+infty)$ and $1<pleq 2$ and $pgeq 9,$ where\u0000$p':=min{p,frac{p}{p-1}}.$ end{abstract}","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a given entire function $f(z)=sum_{j=0}^{infty}a_{j}z^{j}$, we study the zero distribution of $f_{r}(z)=sum_{jequiv rpmod m}a_{j}z^{j}$ where $minmathbb{N}$ and $0le r
对于给定的全函数 $f(z)=sum_{j=0}^{infty}a_{j}z^{j}$，我们研究了 $f_{r}(z)=sum_{jequiv rpmod m}a_{j}z^{j}$ 的零点分布，其中 $minmathbb{N}$ 和 $0le r
{"title":"Entire functions with an arithmetic sequence of exponents","authors":"Dallas Ruth, Khang Tran","doi":"arxiv-2408.02096","DOIUrl":"https://doi.org/arxiv-2408.02096","url":null,"abstract":"For a given entire function $f(z)=sum_{j=0}^{infty}a_{j}z^{j}$, we study\u0000the zero distribution of $f_{r}(z)=sum_{jequiv rpmod m}a_{j}z^{j}$ where\u0000$minmathbb{N}$ and $0le r<m$. We find conditions under which the zeros of\u0000$f_{r}(z)$ lie on $m$ radial rays defined by $Im z^{m}=0$ and $Re z^{m}le0$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the problem of the lower bounds of the modulus of families of paths of order $p,$ $p>n-1,$ and their connection with the geometry of domains containing the specified families. Among other things, we have proved an analogue of N"akki's theorem on the positivity of the $p$-module of families of paths joining a pair of continua in the given domain. The geometry of domains with a strongly accessible boundary in the sense of the $p$-modulus of families of paths was also studied. We show that domains with a $p$-strongly accessible boundary with respect to a $p$-modulus, $p>n-1,$ are are finitely connected at their boundary. The mentioned result generalizes N"akki's result, which was proved for uniform domains in the case of a conformal modulus.
{"title":"On the lower bounds of the $p$-modulus of families","authors":"Evgeny Sevost'yanov, Zarina Kovba, Georgy Nosal","doi":"arxiv-2408.01771","DOIUrl":"https://doi.org/arxiv-2408.01771","url":null,"abstract":"We study the problem of the lower bounds of the modulus of families of paths\u0000of order $p,$ $p>n-1,$ and their connection with the geometry of domains\u0000containing the specified families. Among other things, we have proved an\u0000analogue of N\"akki's theorem on the positivity of the $p$-module of families\u0000of paths joining a pair of continua in the given domain. The geometry of\u0000domains with a strongly accessible boundary in the sense of the $p$-modulus of\u0000families of paths was also studied. We show that domains with a $p$-strongly\u0000accessible boundary with respect to a $p$-modulus, $p>n-1,$ are are finitely\u0000connected at their boundary. The mentioned result generalizes N\"akki's result,\u0000which was proved for uniform domains in the case of a conformal modulus.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish an upper estimate for the coefficient of quasiconformal reflection with respect to the boundary of an arbitrary isosceles trapezoid in terms of its geometric parameters; the estimate improve the result obtained in the recent paper by S.~Nasyrov, T.~Sugawa and M.~Vuorinen.
{"title":"Quasiconformal reflection with respect to the boundary of an isosceles trapezoid","authors":"A. Kushaeva, K. Kushaeva, S. Nasyrov","doi":"arxiv-2408.01821","DOIUrl":"https://doi.org/arxiv-2408.01821","url":null,"abstract":"We establish an upper estimate for the coefficient of quasiconformal\u0000reflection with respect to the boundary of an arbitrary isosceles trapezoid in\u0000terms of its geometric parameters; the estimate improve the result obtained in\u0000the recent paper by S.~Nasyrov, T.~Sugawa and M.~Vuorinen.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"103 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises Fueter-regular functions, slice-regular functions and a recently-discovered function class. In the special case of Clifford-valued functions of one paravector variable, it encompasses monogenic functions, slice-monogenic functions, generalized partial-slice monogenic functions, and a variety of function classes not yet considered in literature. For $T$-regular functions over an associative $*$-algebra, this work provides integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle and a Representation Formula. It also proves some foundational results about $T$-regular functions over an alternative but nonassociative $*$-algebra, such as the real algebra of octonions.
{"title":"A unified theory of regular functions of a hypercomplex variable","authors":"Riccardo Ghiloni, Caterina Stoppato","doi":"arxiv-2408.01523","DOIUrl":"https://doi.org/arxiv-2408.01523","url":null,"abstract":"This work proposes a unified theory of regularity in one hypercomplex\u0000variable: the theory of $T$-regular functions. In the special case of\u0000quaternion-valued functions of one quaternionic variable, this unified theory\u0000comprises Fueter-regular functions, slice-regular functions and a\u0000recently-discovered function class. In the special case of Clifford-valued\u0000functions of one paravector variable, it encompasses monogenic functions,\u0000slice-monogenic functions, generalized partial-slice monogenic functions, and a\u0000variety of function classes not yet considered in literature. For $T$-regular\u0000functions over an associative $*$-algebra, this work provides integral\u0000formulas, series expansions, an Identity Principle, a Maximum Modulus Principle\u0000and a Representation Formula. It also proves some foundational results about\u0000$T$-regular functions over an alternative but nonassociative $*$-algebra, such\u0000as the real algebra of octonions.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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