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The method extends Weber and Wolf’s orthodisk approach and introduces partial symmetry to control global geometry while solving the period problem, generalizing earlier numerically constructed examples.",{"@graph":14,"@context":66},[15,34,49],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/presentations/","Presentations",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/higher-genus-angel-surfaces-abstract-and-contents/567273/",4,{"url":32,"name":10,"@type":35,"author":36,"headline":10,"publisher":39,"fileFormat":42,"inLanguage":8,"description":12,"dateModified":43,"datePublished":43,"encodingFormat":42,"isAccessibleForFree":44,"interactionStatistic":45},"DigitalDocument",{"name":37,"@type":38},"\tCallum ","Person",{"url":19,"name":40,"@type":41},"DocShare","Organization","application/pdf","2026-10-13",true,{"@type":46,"interactionType":47,"userInteractionCount":4},"InteractionCounter",{"@type":48},"ViewAction",{"@type":50,"mainEntity":51},"FAQPage",[52,58,62],{"name":53,"@type":54,"acceptedAnswer":55},"What is the main result about minimal surfaces of genus p?","Question",{"text":56,"@type":57},"For every p ≥ 1, the article constructs complete minimal immersions in R3 of genus p with exactly two ends that attain the least total absolute curvature.","Answer",{"name":59,"@type":54,"acceptedAnswer":60},"How are the two ends of an Angel surface characterized?",{"text":61,"@type":57},"One end is catenoidal and the other is of Enneper type, with the surface generalizing earlier examples and using controlled symmetry to manage the period problem.",{"name":63,"@type":54,"acceptedAnswer":64},"What method is used to solve the period problem for these surfaces?",{"text":65,"@type":57},"The construction generalizes Weber and Wolf’s orthodisk approach by translating the period problem into Teichmüller-theory questions and using partial symmetry to introduce controlled symmetry into the surface.","https://schema.org",{"og:url":32,"og:type":68,"og:title":10,"og:site_name":40,"og:description":12},"article",{"robots":70,"canonical":32},"index,follow",{"doc_id":72,"site_id":7},567273,1791865765,{"code":4,"msg":75,"data":76},"success",[77,81,86,91,96,101,106,111,116],{"id":78,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":79,"slug":80},11,90,"presentations",{"id":82,"doc_module":22,"doc_module_name":25,"category_name":83,"show_sort_weight":84,"slug":85},12,"Resumes",80,"resumes",{"id":87,"doc_module":22,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},14,"Invoices",70,"invoices",{"id":92,"doc_module":22,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},15,"Posters",60,"posters",{"id":97,"doc_module":22,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},16,"Social Media",50,"social-media",{"id":102,"doc_module":22,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},17,"Forms",40,"forms",{"id":107,"doc_module":22,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},18,"Letters",30,"letters",{"id":112,"doc_module":22,"doc_module_name":25,"category_name":113,"show_sort_weight":114,"slug":115},21,"Paper Templates",5,"papers-templates",{"id":117,"doc_module":22,"doc_module_name":25,"category_name":118,"show_sort_weight":4,"slug":119},158,"General","general-158",{"code":4,"msg":75,"data":121},{"doc_id":72,"user_id":122,"nickname":37,"user_avatar":123,"doc_module":22,"category_id":78,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":124,"file_id":125,"file_url":126,"file_type":127,"file_size":128,"view_count":4,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":129,"language":130,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":131,"faqs":132,"seo_title":133,"seo_description":12,"update_tm":73,"read_time":134},137451211410,"https://ap-avatar.wpscdn.com/avatar/2000bb0a9246f588df?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786362646172706240","HIGHER GENUS ANGEL SURFACES  \nRIVU BARDHAN, INDRANIL BISWAS, SHOICHI FUJIMORI, AND PRADIP KUMAR  \narXiv :2509 .03925v2 [math .DG] 6 Apr 2026  \nAbstract. We prove the existence of complete minimal surfaces in R3 of arbitrary genus p ≥ 1 and least total absolute curvature with precisely two ends—one catenoidaland one Enneper-type — thereby solving, affirmatively, a problem posed by Fujimori and Shoda. These surfaces, which are called Angel surfaces, generalize some examples numerically constructed earlier by Weber. The construction of these minimal surfaces involves extending the orthodisk method developed by Weber and Wolf [9] . A central idea in our construction is the notion of partial symmetry, which enables us to introduce controlled symmetry into the surface.  \nContents  \n1. Introduction 1  \n2. Preliminaries 6  \n3. Generalized orthodisk and minimal surface 9  \n4. Higher genus angel surface 13  \n5. Generalized orthodisk for the genus 1 case 14  \n6. Proposed data for genus-p Angel surfaces 17  \n7. E-conjugate orthodisks with partial symmetry 19  \n8. Space of e-conjugate pair of orthodisks with partial symmetry 27  \n9. Height function on the space Tλ0 ,p 29  \n10. Tangent space Tζ (Tλ0 ,p) 30  \n11. Existence of e-reflexive generalized orthodisks of genus p 31  \nReferences 35  \n1. Introduction  \nComplete minimal surfaces in R3 with finite total curvature have long been one of the central objects of study in differential geometry. Two topological invariants associated with such surfaces are the genus and the number of ends, which are related to the total curvature of the surface by the Osserman inequality [3] . It is natural to ask whether there exists a complete minimal surface, with given genus and number of ends, that minimizes the total absolute curvature.  \n2020 Mathematics Subject Classification. 53A10, 49Q05, 53C42, 30F60 .  \nKey words and phrases . Minimal surface, finite total curvature, two ends, angel surface.  \n2 R. BARDHAN, I. BISWAS, S. FUJIMORI, AND P. KUMAR  \nThis question remains open in general, and no complete classification is currently known. In this work, we restrict our attention to the simplest nontrivial cases, that is to surfaces with one or two ends. Classical examples illustrate the genus-zero case: the plane, with one planar end, and the catenoid, with two catenoidal ends. A celebrated result of Schoen, [5], establishes that the catenoid is the unique complete minimal surface of genus zero with two ends and least total absolute curvature.  \nFor higher genus surfaces with one end, Weber and Wolf, [8], and independently Sato,[4], constructed examples realizing the least total absolute curvature. The two-ended case, however, remained largely open. Motivated by the known examples and the structure of the Weierstrass representation, Weber provided numerical evidence that for each p ≥ 1 there exists a complete minimal surface of genus p with the least total absolute curvature and exactly two ends—one asymptotic to an end of the catenoid and the other to the end of Enneper surface [2, Example 4.1] . Such surfaces are referred to as Angel surfaces.  \nProgress toward this problem was made by Fujimori and Shoda [2], who constructed explicit examples of genus-one, with a catenoidal end and an Enneper end, and also of even genus, with two twice-wrapped catenoidal ends, surfaces which attain the least total absolute curvature. Their construction crucially relies on the presence of symmetry in order to reduce the complexity of the period problem and also to control the global geometry. However, to date, no such construction is known for any odd genus greater than one.  \nThe purpose of this article is to construct these surfaces for all genera p ≥ 1, thus affirmatively solving the problem (Problem 4.1) in general. In particular, we prove that for each p ≥ 1 there exists a complete minimal immersion of genus p with least absolute curvature with one Enneper end and one catenoid end; see Figure 1 .  \nFor genus one, in","cbCaidhGbpvQVXUD","https://ap.wps.com/l/cbCaidhGbpvQVXUD","pdf",1570719,36,"English","# Introduction\n# Preliminaries\n# Generalized orthodisk and minimal surface\n# Higher genus angel surface\n# Generalized orthodisk for the genus 1 case\n# Proposed data for genus-p Angel surfaces\n# E-conjugate orthodisks with partial symmetry\n# Space of e-conjugate pair of orthodisks with partial symmetry\n# Height function on the space Tλ0,p\n# Tangent space Tζ(Tλ0,p)\n# Existence of e-reflexive generalized orthodisks of genus p\n# References","[{\"question\":\"What is the main result about minimal surfaces of genus p?\",\"answer\":\"For every p ≥ 1, the article constructs complete minimal immersions in R3 of genus p with exactly two ends that attain the least total absolute curvature.\"},{\"question\":\"How are the two ends of an Angel surface characterized?\",\"answer\":\"One end is catenoidal and the other is of Enneper type, with the surface generalizing earlier examples and using controlled symmetry to manage the period problem.\"},{\"question\":\"What method is used to solve the period problem for these surfaces?\",\"answer\":\"The construction generalizes Weber and Wolf’s orthodisk approach by translating the period problem into Teichmüller-theory questions and using partial symmetry to introduce controlled symmetry into the surface.\"}]","HIGHER GENUS ANGEL SURFACES - Abstract and Contents | PDF",13]