Weingarten Surfaces Associated to Laguerre Minimal Surfaces

Weingarten Surfaces Associated to Laguerre Minimal Surfaces | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Weingarten Surfaces Associated to Laguerre Minimal Surfaces Laredo Rennan Pereira Santos, Armando Mauro Vasquez Corro This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4214419/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In the work \cite{Laredo}, the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function $R$ is a geometric invariant of hypersurface. In this paper, we define the spherical mean curvature $H_S$ for any surface $\Sigma$, which depends on the principal curvatures of $\Sigma$ and the radius function $R$. We then explore two classes of surfaces: those with $H_S = 0$, referred to as $H_1$-surfaces, and the surfaces with spherical mean curvature of harmonic type, denoted as $H_2$-surfaces. We provide a Weierstrass-type representation for each of these classes depending on three holomorphic functions. We prove that the $H_1$-surfaces are associated to the minimal surfaces, whereas the $H_2$-surfaces are related to Laguerre minimal surfaces. As an application, we present a new Weierstrass-type representation for Laguerre minimal surfaces, and specifically for minimal surfaces. In this way, the same holomorphic data can be used to provide examples in $H_1$-surface/minimal surface classes or in $H_2$-surface/Laguerre minimal surface classes. We also characterize the rotational cases, allowing us to find a complete rotational Laguerre minimal surface. Sphere Congruence Generalized Weingarten Surfaces Laguerre Minimal Surfaces Weierstrass-Type Representation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4214419","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":287740533,"identity":"b75d2cb1-dd58-4362-91b7-3d20cb390e07","order_by":0,"name":"Laredo Rennan Pereira Santos","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBElEQVRIie3OPUvDQBjA8SccXJeLtz4h4H2FhAOxKKQfxRCIo2sGKQdCHV0Fg59Bl84tAad0D5jBWzJlcCotLl5FLA5n7SZ4f7g3uB93AC7XH+xgM7GvY4F8syDAzErolhAzagzUfsSbwC/IYKFf+gISfncz1+v7YxQjMn9m0F5YCTuXcVlDettWRPpTxHhGsxMG3VBZP5bT0J/AGTQZDb0pjmPFjkIGVWR9hXeDN0MSYUiwLs0rii9/JphTYoj3YAj6ClEAoztIR4KyxvSxyaRkT4gRoXJYRp2VcJ57r31xmhw2qdaryzGK6yvd9EVrJZ/hdhtVH/MO8C2h9rntcrlc/6F3krRIaM3fp7kAAAAASUVORK5CYII=","orcid":"","institution":"Instituto Federal de Goiás","correspondingAuthor":true,"prefix":"","firstName":"Laredo","middleName":"Rennan Pereira","lastName":"Santos","suffix":""},{"id":287740534,"identity":"1a2585a8-24e8-4902-ae29-9c5223c7c4bb","order_by":1,"name":"Armando Mauro Vasquez Corro","email":"","orcid":"","institution":"Universidade Federal de Goiás","correspondingAuthor":false,"prefix":"","firstName":"Armando","middleName":"Mauro Vasquez","lastName":"Corro","suffix":""}],"badges":[],"createdAt":"2024-04-03 18:44:26","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4214419/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4214419/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":55264141,"identity":"55373078-c534-48d3-b47a-81a3b5308971","added_by":"auto","created_at":"2024-04-25 01:38:24","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":17045540,"visible":true,"origin":"","legend":"","description":"","filename":"Artigo2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4214419/v1_covered_d5d17521-f30f-45af-9cb8-944c56ced43a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Weingarten Surfaces Associated to Laguerre Minimal Surfaces","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Sphere Congruence, Generalized Weingarten Surfaces, Laguerre Minimal Surfaces, Weierstrass-Type Representation","lastPublishedDoi":"10.21203/rs.3.rs-4214419/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4214419/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In the work \\cite{Laredo}, the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function $R$ is a geometric invariant of hypersurface. In this paper, we define the spherical mean curvature $H_S$ for any surface $\\Sigma$, which depends on the principal curvatures of $\\Sigma$ and the radius function $R$. We then explore two classes of surfaces: those with $H_S = 0$, referred to as $H_1$-surfaces, and the surfaces with spherical mean curvature of harmonic type, denoted as $H_2$-surfaces.\nWe provide a Weierstrass-type representation for each of these classes depending on three holomorphic functions. We prove that the $H_1$-surfaces are associated to the minimal surfaces, whereas the $H_2$-surfaces are related to Laguerre minimal surfaces. As an application, we present a new Weierstrass-type representation for Laguerre minimal surfaces, and specifically for minimal surfaces. In this way, the same holomorphic data can be used to provide examples in $H_1$-surface/minimal surface classes or in $H_2$-surface/Laguerre minimal surface classes. 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