{"id":"https://openalex.org/W7164856048","doi":"https://doi.org/10.48550/arxiv.2606.14080","title":"Geodesic nets on the Euclidean plane and closed geodesic nets on Euclidean surfaces","display_name":"Geodesic nets on the Euclidean plane and closed geodesic nets on Euclidean surfaces","publication_year":2026,"publication_date":"2026-06-12","ids":{"openalex":"https://openalex.org/W7164856048","doi":"https://doi.org/10.48550/arxiv.2606.14080"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2606.14080","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.14080","pdf_url":"https://arxiv.org/pdf/2606.14080","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2606.14080","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120094930","display_name":"Ivan Il'ich Frolov","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Frolov, Ivan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5028425216","display_name":"Denis Gorodkov","orcid":"https://orcid.org/0000-0002-5728-4396","observed_orcids":["https://orcid.org/0000-0002-5728-4396"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gorodkov, Denis","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5137355732","display_name":"Dmitrii Korshunov","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Korshunov, Dmitrii","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5035125984","display_name":"Alexander Nabutovsky","orcid":"https://orcid.org/0000-0002-4506-9664","observed_orcids":["https://orcid.org/0000-0002-4506-9664"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nabutovsky, Alexander","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9276999831199646,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9276999831199646,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.04089999943971634,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.014299999922513962,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/euclidean-plane","display_name":"Euclidean plane","score":0.6959999799728394},{"id":"https://openalex.org/keywords/riemannian-surfaces","display_name":"Riemannian surfaces","score":0.8629999756813049},{"id":"https://openalex.org/keywords/branch-points","display_name":"branch points","score":0.7580000162124634},{"id":"https://openalex.org/keywords/diameter","display_name":"diameter","score":0.38199999928474426},{"id":"https://openalex.org/keywords/sectional-curvature","display_name":"sectional curvature","score":0.6380000114440918}],"concepts":[{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.8553000092506409},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.79339998960495},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.651199996471405},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6399000287055969},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.5629000067710876},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.5232999920845032},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.5038999915122986},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4636000096797943},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.4620000123977661},{"id":"https://openalex.org/C14166107","wikidata":"https://www.wikidata.org/wiki/Q253829","display_name":"Net (polyhedron)","level":2,"score":0.43130001425743103},{"id":"https://openalex.org/C162269530","wikidata":"https://www.wikidata.org/wiki/Q17100560","display_name":"Geodesic map","level":3,"score":0.3864000141620636},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.31619998812675476},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.28839999437332153},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28279998898506165},{"id":"https://openalex.org/C43809302","wikidata":"https://www.wikidata.org/wiki/Q1060874","display_name":"Plane curve","level":2,"score":0.2732999920845032},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2671000063419342}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2606.14080","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.14080","pdf_url":"https://arxiv.org/pdf/2606.14080","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2606.14080","doi":"https://doi.org/10.48550/arxiv.2606.14080","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.14080","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2606.14080","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.14080","pdf_url":"https://arxiv.org/pdf/2606.14080","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7164856048.pdf","grobid_xml":"https://content.openalex.org/works/W7164856048.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3156167623","https://openalex.org/W2791610653","https://openalex.org/W2918101206","https://openalex.org/W4416189867","https://openalex.org/W7134210481","https://openalex.org/W7095338758","https://openalex.org/W2945095489","https://openalex.org/W1606310105","https://openalex.org/W2036303766","https://openalex.org/W2078905101"],"abstract_inverted_index":{"We":[0,52],"prove":[1,54],"that":[2,55,72],"if":[3],"$M$":[4],"is":[5],"a":[6,24,46,97],"closed":[7,25],"Riemannian":[8],"surface":[9],"of":[10,28,82,85],"diameter":[11],"$d$":[12],"and":[13,102],"area":[14],"$v$":[15],"with":[16,64],"sectional":[17],"curvature":[18],"in":[19,60,100],"the":[20,61,80],"$[-1,1]$":[21],"interval,":[22],"then":[23],"geodesic":[26,58],"net":[27,59],"length":[29],"$l$":[30],"has":[31],"at":[32,65],"most":[33,66],"$f(l,d,v)$":[34],"branch":[35],"points,":[36],"where":[37],"$f(l,d,v)=(400\\bar{l})^{(180\\bar{l})^4}$":[38],"for":[39,56],"$\\bar{l}=":[40],"\\max\\{l,":[41],"\\frac{\\exp(d)}{\\min\\{1,":[42],"\\frac{v}{4}\\}}\\}$":[43],"This":[44,95],"answers":[45,96],"question":[47,98],"posed":[48,99],"by":[49],"S.":[50],"Becker-Kahn.":[51],"also":[53],"each":[57],"Euclidean":[62],"plane":[63],"n":[67],"unbalanced":[68,75],"(boundary)":[69],"vertices":[70,76,84],"such":[71],"all":[73],"its":[74],"have":[77],"degree":[78,86],"1,":[79],"number":[81],"balanced":[83],"$\\ge":[87],"3$":[88],"(=branch":[89],"points)":[90],"does":[91],"not":[92],"exceed":[93],"$(25n)^{2n^2}$.":[94],"[GM]":[101],"[NP].":[103]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-06-16T00:00:00"}