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On the lower bounds of the $p$-modulus of families 关于家系的 $p$ 模的下界
arXiv - MATH - Complex Variables Pub Date : 2024-08-03 DOI: arxiv-2408.01771
Evgeny Sevost'yanov, Zarina Kovba, Georgy Nosal
{"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}
引用次数: 0
Uniformization of intrinsic Gromov hyperbolic spaces with Busemann functions 具有布斯曼函数的本征格罗莫夫双曲空间的均匀化
arXiv - MATH - Complex Variables Pub Date : 2024-08-02 DOI: arxiv-2408.01412
Vasudevarao Allu, Alan P Jose
{"title":"Uniformization of intrinsic Gromov hyperbolic spaces with Busemann functions","authors":"Vasudevarao Allu, Alan P Jose","doi":"arxiv-2408.01412","DOIUrl":"https://doi.org/arxiv-2408.01412","url":null,"abstract":"For any intrinsic Gromov hyperbolic space we establish a Gehring-Hayman type\u0000theorem for conformally deformed spaces. As an application, we prove that any\u0000intrinsic hyperbolic space with atleast two points in the Gromov boundary can\u0000be uniformized by densities induced by Busemann functions.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941834","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}
引用次数: 0
Homeomorphic Sobolev extensions of parametrizations of Jordan curves 约旦曲线参数化的同态索波列夫扩展
arXiv - MATH - Complex Variables Pub Date : 2024-08-01 DOI: arxiv-2408.00506
Ondrěj Bouchala, Jarmo Jääskeläinen, Pekka Koskela, Haiqing Xu, Xilin Zhou
{"title":"Homeomorphic Sobolev extensions of parametrizations of Jordan curves","authors":"Ondrěj Bouchala, Jarmo Jääskeläinen, Pekka Koskela, Haiqing Xu, Xilin Zhou","doi":"arxiv-2408.00506","DOIUrl":"https://doi.org/arxiv-2408.00506","url":null,"abstract":"Each homeomorphic parametrization of a Jordan curve via the unit circle\u0000extends to a homeomorphism of the entire plane. It is a natural question to ask\u0000if such a homeomorphism can be chosen so as to have some Sobolev regularity.\u0000This prompts the simplified question: for a homeomorphic embedding of the unit\u0000circle into the plane, when can we find a homeomorphism from the unit disk that\u0000has the same boundary values and integrable first-order distributional\u0000derivatives? We give the optimal geometric criterion for the interior Jordan domain so\u0000that there exists a Sobolev homeomorphic extension for any homeomorphic\u0000parametrization of the Jordan curve. The problem is partially motivated by\u0000trying to understand which boundary values can correspond to deformations of\u0000finite energy.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"192 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884551","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}
引用次数: 0
On an Internal Characterization of Horocyclically Convex Domains in the Unit Disk 论单位盘中环状凸域的内部特征
arXiv - MATH - Complex Variables Pub Date : 2024-07-31 DOI: arxiv-2407.21271
Juan Arango, Hugo Arbeláez, Diego Mejía
{"title":"On an Internal Characterization of Horocyclically Convex Domains in the Unit Disk","authors":"Juan Arango, Hugo Arbeláez, Diego Mejía","doi":"arxiv-2407.21271","DOIUrl":"https://doi.org/arxiv-2407.21271","url":null,"abstract":"A proper subdomain $G$ of the unit disk $mathbb{D}$ is horocyclically convex\u0000(horo-convex) if, for every $omega in mathbb{D}cap partial G$, there\u0000exists a horodisk $H$ such that $omega in partial H$ and $Gcap\u0000H=emptyset$. In this paper we give an internal characterization of these\u0000domains, namely, that $G$ is horo-convex if and only if any two points can be\u0000joined inside $G$ by a $C^1$ curve composed with finitely many Jordan arcs with\u0000hyperbolic curvature in $(-2,2)$. We also give a lower bound for the hyperbolic\u0000metric of horo-convex regions and some consequences.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866078","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}
引用次数: 0
Observability of the heat equation from very small sets 从极小集合观察热方程的可观察性
arXiv - MATH - Complex Variables Pub Date : 2024-07-30 DOI: arxiv-2407.20954
A. Walton Green, Kévin Le Balc'h, Jérémy Martin, Marcu-Antone Orsoni
{"title":"Observability of the heat equation from very small sets","authors":"A. Walton Green, Kévin Le Balc'h, Jérémy Martin, Marcu-Antone Orsoni","doi":"arxiv-2407.20954","DOIUrl":"https://doi.org/arxiv-2407.20954","url":null,"abstract":"We consider the heat equation set on a bounded $C^1$ domain of $mathbb R^n$\u0000with Dirichlet boundary conditions. The first purpose of this paper is to prove\u0000that the heat equation is observable from any measurable set $omega$ with\u0000positive $(n-1+delta)$-Hausdorff content, for $delta >0$ arbitrary small. The\u0000proof relies on a new spectral estimate for linear combinations of Laplace\u0000eigenfunctions, obtained via a Remez type inequality, and the use of the\u0000so-called Lebeau-Robbiano's method. Even if this observability result is sharp\u0000with respect to the scale of Hausdorff dimension, our second goal is to\u0000construct families of sets $omega$ which have codimension greater than or\u0000equal to $1$ for which the heat equation remains observable.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866080","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}
引用次数: 0
On solutions of certain non-linear delay differential equations 论某些非线性延迟微分方程的解
arXiv - MATH - Complex Variables Pub Date : 2024-07-29 DOI: arxiv-2407.19855
Nidhi Gahlian
{"title":"On solutions of certain non-linear delay differential equations","authors":"Nidhi Gahlian","doi":"arxiv-2407.19855","DOIUrl":"https://doi.org/arxiv-2407.19855","url":null,"abstract":"In this paper, we study the existence and non-existence of entire solutions\u0000of certain non-linear delay-differential equations.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866082","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}
引用次数: 0
A sharp estimate of area for sublevel-set of Blaschke products 对布拉什克乘积子级集面积的精确估算
arXiv - MATH - Complex Variables Pub Date : 2024-07-28 DOI: arxiv-2407.19539
David Kalaj
{"title":"A sharp estimate of area for sublevel-set of Blaschke products","authors":"David Kalaj","doi":"arxiv-2407.19539","DOIUrl":"https://doi.org/arxiv-2407.19539","url":null,"abstract":"Let $mathbb{D}$ be the unit disk in the complex plane. Among other results,\u0000we prove the following curious result for a finite Blaschke product: $$B(z)=e\u0000^{is}prod_{k=1}^d frac{z-a_k}{1-z overline{a_k}}.$$ The Lebesgue measure of\u0000the sublevel set of $B$ satisfies the following sharp inequality for $t in\u0000[0,1]$: $$|{zin mathbb{D}:|B(z)|<t}|le pi t^{2/d},$$ with equality at a\u0000single point $tin(0,1)$ if and only if $a_k=0$ for every $k$. In that case the\u0000equality is attained for every $t$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866083","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}
引用次数: 0
A note on meromorphic functions on a compact Riemann surface having poles at a single point 关于紧凑黎曼曲面上有单点极点的微函数的说明
arXiv - MATH - Complex Variables Pub Date : 2024-07-25 DOI: arxiv-2407.18286
V V Hemasundar Gollakota
{"title":"A note on meromorphic functions on a compact Riemann surface having poles at a single point","authors":"V V Hemasundar Gollakota","doi":"arxiv-2407.18286","DOIUrl":"https://doi.org/arxiv-2407.18286","url":null,"abstract":"On a compact Riemann surface $X$ of genus $g$, one of the questions is the\u0000existence of meromorphic functions having poles at a point $P$ on $X$. One of\u0000the theorems is the Weierstrass gap theorem that determines a sequence of $g$\u0000numbers $1 < n_k < 2g$, $1 leq k leq g$ for which a meromorphic function with\u0000the order with $n_k$ fails to exist at $P$. In this note, we give proof of the\u0000Weierstrass gap theorem in cohomology terminology. We see that an interesting\u0000combinatorial problem may be formed as a byproduct from the statement of the\u0000Weierstrass gap theorem.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866086","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}
引用次数: 0
Parametric Symplectic Jet Interpolation 参数交映射流插值
arXiv - MATH - Complex Variables Pub Date : 2024-07-24 DOI: arxiv-2407.17581
Rafael B. Andrist, Gaofeng Huang, Frank Kutzschebauch, Josua Schott
{"title":"Parametric Symplectic Jet Interpolation","authors":"Rafael B. Andrist, Gaofeng Huang, Frank Kutzschebauch, Josua Schott","doi":"arxiv-2407.17581","DOIUrl":"https://doi.org/arxiv-2407.17581","url":null,"abstract":"We prove a parametric jet interpolation theorem for symplectic holomorphic\u0000automorphisms of $mathbb{C}^{2n}$ with parameters in a Stein space. Moreover,\u0000we provide an example of an unavoidable set for symplectic holomorphic maps.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772672","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}
引用次数: 0
Some variants of the generalized Borel Theorem and applications 广义伯勒定理的一些变体及其应用
arXiv - MATH - Complex Variables Pub Date : 2024-07-23 DOI: arxiv-2407.16163
Dinh Tuan Huynh
{"title":"Some variants of the generalized Borel Theorem and applications","authors":"Dinh Tuan Huynh","doi":"arxiv-2407.16163","DOIUrl":"https://doi.org/arxiv-2407.16163","url":null,"abstract":"In the first part of this paper, we establish some results around generalized\u0000Borel's Theorem. As an application, in the second part, we construct example of\u0000smooth surface of degree $dgeq 19$ in $mathbb{CP}^3$ whose complements is\u0000hyperbolically embedded in $mathbb{CP}^3$. This improves the previous\u0000construction of Shirosaki where the degree bound $d=31$ was gave. In the last\u0000part, for a Fermat-Waring type hypersurface $D$ in $mathbb{CP}^n$ defined by\u0000the homogeneous polynomial [ sum_{i=1}^m h_i^d, ] where $m,n,d$ are positive\u0000integers with $mgeq 3n-1$ and $dgeq m^2-m+1$, where $h_i$ are homogeneous\u0000generic linear forms on $mathbb{C}^{n+1}$, for a nonconstant holomorphic\u0000function $fcolonmathbb{C}rightarrowmathbb{CP}^n$ whose image is not\u0000contained in the support of $D$, we establish a Second Main Theorem type\u0000estimate: [ big(d-m(m-1)big),T_f(r)leq N_f^{[m-1]}(r,D)+S_f(r). ] This\u0000quantifies the hyperbolicity result due to Shiffman-Zaidenberg and Siu-Yeung.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772732","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}
引用次数: 0
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