In this note we show that the integral means spectrum of any univalent function admitting a quasiconformal extension to the extended complex plane is strictly less than the universal integral means spectrum. This gives an affirmative answer to a question raised in our recent paper.
{"title":"On the integral means spectrum of univalent functions with quasconformal extensions","authors":"Jianjun Jin","doi":"arxiv-2407.19240","DOIUrl":"https://doi.org/arxiv-2407.19240","url":null,"abstract":"In this note we show that the integral means spectrum of any univalent\u0000function admitting a quasiconformal extension to the extended complex plane is\u0000strictly less than the universal integral means spectrum. This gives an\u0000affirmative answer to a question raised in our recent paper.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866085","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}
On a compact Riemann surface $X$ of genus $g$, one of the questions is the existence of meromorphic functions having poles at a point $P$ on $X$. One of the theorems is the Weierstrass gap theorem that determines a sequence of $g$ numbers $1 < n_k < 2g$, $1 leq k leq g$ for which a meromorphic function with the order with $n_k$ fails to exist at $P$. In this note, we give proof of the Weierstrass gap theorem in cohomology terminology. We see that an interesting combinatorial problem may be formed as a byproduct from the statement of the Weierstrass gap theorem.
在属$g$的紧凑黎曼曲面$X$上，其中一个问题是在$X$上的点$P$上存在有极点的分形函数。其中一个定理是魏尔斯特拉斯间隙定理（Weierstrass gap theorem），该定理确定了一个$g$数序列：$1 < n_k < 2g$，$1 leq k leq g$，对于该序列，在$P$处不存在阶数为$n_k$的分垂函数。在本注中，我们用同调术语证明了韦尔斯特拉斯缺口定理。我们发现，从韦尔斯特拉斯间隙定理的陈述中可以得到一个有趣的组合问题作为副产品。
{"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}
This paper is the final step in solving the problem of starlikeness of Teichmuller spaces in Bers' embedding. This step concerns the case of finite dimensional Teichmuller spaces ${mathbf T}(g, n)$ of positive dimension (corresponding to punctured Riemann surfaces of finite conformal type $(g, n)$ with $2g - 2 + n > 0$).
{"title":"All Teichmuller spaces are not starlike","authors":"Samuel L. Krushkal","doi":"arxiv-2407.18239","DOIUrl":"https://doi.org/arxiv-2407.18239","url":null,"abstract":"This paper is the final step in solving the problem of starlikeness of\u0000Teichmuller spaces in Bers' embedding. This step concerns the case of finite\u0000dimensional Teichmuller spaces ${mathbf T}(g, n)$ of positive dimension\u0000(corresponding to punctured Riemann surfaces of finite conformal type $(g, n)$\u0000with $2g - 2 + n > 0$).","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772585","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}
In 1939 P'al Tur'an and J'anos ErH{o}d initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a matter of normalization they considered the family $mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given convex, compact subset $KSubset {mathbb C}$. While Tur'an obtained the first results for the interval $I:=[-1,1]$ and the disk $D:={ zin {mathbb C}~:~ |z|le 1}$, ErH{o}d extended investigations to other compact convex domains, too. The order of the optimal constant was found to be $sqrt{n}$ for $I$ and $n$ for $D$. It took until 2006 to clarify that all compact convex emph{domains} (with nonempty interior), follow the pattern of the disk, and admit an order $n$ inequality. For $L^q(partial K)$ norms with any $1le q
{"title":"Turán type oscillation inequalities in $L^q$ norm on the boundary of convex polygonal domains","authors":"Polina Glazyrina, Szilárd Gy. Révész","doi":"arxiv-2407.18404","DOIUrl":"https://doi.org/arxiv-2407.18404","url":null,"abstract":"In 1939 P'al Tur'an and J'anos ErH{o}d initiated the study of lower\u0000estimations of maximum norms of derivatives of polynomials, in terms of the\u0000maximum norms of the polynomials themselves, on convex domains of the complex\u0000plane. As a matter of normalization they considered the family\u0000$mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given\u0000convex, compact subset $KSubset {mathbb C}$. While Tur'an obtained the first\u0000results for the interval $I:=[-1,1]$ and the disk $D:={ zin {mathbb C}~:~\u0000|z|le 1}$, ErH{o}d extended investigations to other compact convex domains,\u0000too. The order of the optimal constant was found to be $sqrt{n}$ for $I$ and\u0000$n$ for $D$. It took until 2006 to clarify that all compact convex\u0000emph{domains} (with nonempty interior), follow the pattern of the disk, and\u0000admit an order $n$ inequality. For $L^q(partial K)$ norms with any $1le q <infty$ we obtained order $n$\u0000results for various classes of domains. Further, in the generality of all\u0000convex, compact domains we could show a $c n/log n$ lower bound together with\u0000an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all\u0000compact convex domains admit an order $n$ Tur'an type inequality. Here we\u0000prove this for all emph{polygonal} convex domains and any $0< q <infty$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866087","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}
Rafael B. Andrist, Gaofeng Huang, Frank Kutzschebauch, Josua Schott
We prove a parametric jet interpolation theorem for symplectic holomorphic automorphisms of $mathbb{C}^{2n}$ with parameters in a Stein space. Moreover, we provide an example of an unavoidable set for symplectic holomorphic maps.
{"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}
In the first part of this paper, we establish some results around generalized Borel's Theorem. As an application, in the second part, we construct example of smooth surface of degree $dgeq 19$ in $mathbb{CP}^3$ whose complements is hyperbolically embedded in $mathbb{CP}^3$. This improves the previous construction of Shirosaki where the degree bound $d=31$ was gave. In the last part, for a Fermat-Waring type hypersurface $D$ in $mathbb{CP}^n$ defined by the homogeneous polynomial [ sum_{i=1}^m h_i^d, ] where $m,n,d$ are positive integers with $mgeq 3n-1$ and $dgeq m^2-m+1$, where $h_i$ are homogeneous generic linear forms on $mathbb{C}^{n+1}$, for a nonconstant holomorphic function $fcolonmathbb{C}rightarrowmathbb{CP}^n$ whose image is not contained in the support of $D$, we establish a Second Main Theorem type estimate: [ big(d-m(m-1)big),T_f(r)leq N_f^{[m-1]}(r,D)+S_f(r). ] This quantifies the hyperbolicity result due to Shiffman-Zaidenberg and Siu-Yeung.
{"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}
We introduce the novel family of probability distributions on hyperbolic disc. The distinctive property of the proposed family is invariance under the actions of the group of disc-preserving conformal mappings. The group-invariance property renders it a convenient and tractable model for encoding uncertainties in hyperbolic data. Potential applications in Geometric Deep Learning and bioinformatics are numerous, some of them are briefly discussed. We also emphasize analogies with hyperbolic coherent states in quantum physics.
{"title":"Conformally Natural Families of Probability Distributions on Hyperbolic Disc with a View on Geometric Deep Learning","authors":"Vladimir Jacimovic, Marijan Markovic","doi":"arxiv-2407.16733","DOIUrl":"https://doi.org/arxiv-2407.16733","url":null,"abstract":"We introduce the novel family of probability distributions on hyperbolic\u0000disc. The distinctive property of the proposed family is invariance under the\u0000actions of the group of disc-preserving conformal mappings. The\u0000group-invariance property renders it a convenient and tractable model for\u0000encoding uncertainties in hyperbolic data. Potential applications in Geometric\u0000Deep Learning and bioinformatics are numerous, some of them are briefly\u0000discussed. We also emphasize analogies with hyperbolic coherent states in\u0000quantum physics.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772602","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 boundary behavior of the Kobayashi-Royden metric and the Kobayashi hyperbolicity of domains in Riemannian manifolds.
我们研究了小林-罗伊登公设的边界行为和黎曼流形中域的小林双曲性。
{"title":"Kobayashi hyperbolicity in Riemannian manifolds","authors":"Hervé Gaussier, Alexandre Sukhov","doi":"arxiv-2407.15976","DOIUrl":"https://doi.org/arxiv-2407.15976","url":null,"abstract":"We study the boundary behavior of the Kobayashi-Royden metric and the\u0000Kobayashi hyperbolicity of domains in Riemannian manifolds.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"101 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772648","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 a variant of the uncertainty principle in terms of the annihilation and creation operator on generalized Segal Bargmann spaces, which are used for the FBI-Bargmann transform. In addition, we compute the Berezin transform of these operators and indicate how to use spaces of entire functions in one variable to study the SzegH{o} kernel for hypersurfaces in $mathbb C^2.$
{"title":"Unbounded operators and the uncertainty principle","authors":"Friedrich Haslinger","doi":"arxiv-2407.15803","DOIUrl":"https://doi.org/arxiv-2407.15803","url":null,"abstract":"We study a variant of the uncertainty principle in terms of the annihilation\u0000and creation operator on generalized Segal Bargmann spaces, which are used for\u0000the FBI-Bargmann transform. In addition, we compute the Berezin transform of\u0000these operators and indicate how to use spaces of entire functions in one\u0000variable to study the SzegH{o} kernel for hypersurfaces in $mathbb C^2.$","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772593","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}
Consider the family of locally univalent analytic functions $h$ in the unit disk $|z|<1$ with the normalization $h(0)=0$, $h'(0)=1$ and satisfying the condition $${real} left( frac{z h''(z)}{alpha h'(z)}right)
{"title":"Geometric subfamily of functions convex in some direction and Blaschke products","authors":"Liulan Li, Saminthan Ponnusamy","doi":"arxiv-2407.14922","DOIUrl":"https://doi.org/arxiv-2407.14922","url":null,"abstract":"Consider the family of locally univalent analytic functions $h$ in the unit\u0000disk $|z|<1$ with the normalization $h(0)=0$, $h'(0)=1$ and satisfying the\u0000condition $${real} left( frac{z h''(z)}{alpha h'(z)}right) <frac{1}{2}\u0000~mbox{ for $zin ID$,} $$ where $0<alphaleq1$. The aim of this article is\u0000to show that this family has several elegant properties such as involving\u0000Blaschke products, Schwarzian derivative and univalent harmonic mappings.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772587","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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