{"title":"黎曼球分支覆盖的离散周期矩阵和离散全纯积分的收敛性","authors":"Alexander I. Bobenko, Ulrike Bücking","doi":"10.1007/s11040-021-09394-2","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere <span>\\(\\hat {\\mathbb {C}}\\)</span>. Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 3","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09394-2","citationCount":"2","resultStr":"{\"title\":\"Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere\",\"authors\":\"Alexander I. Bobenko, Ulrike Bücking\",\"doi\":\"10.1007/s11040-021-09394-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere <span>\\\\(\\\\hat {\\\\mathbb {C}}\\\\)</span>. Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.</p></div>\",\"PeriodicalId\":694,\"journal\":{\"name\":\"Mathematical Physics, Analysis and Geometry\",\"volume\":\"24 3\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2021-07-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s11040-021-09394-2\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Physics, Analysis and Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11040-021-09394-2\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-021-09394-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere
We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere \(\hat {\mathbb {C}}\). Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.
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