{"id":"https://openalex.org/W7214729874","doi":"https://doi.org/10.48550/arxiv.2609.32001","title":"Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds","display_name":"Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds","publication_year":2026,"publication_date":"2026-09-25","ids":{"openalex":"https://openalex.org/W7214729874","doi":"https://doi.org/10.48550/arxiv.2609.32001"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.32001","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32001","pdf_url":"https://arxiv.org/pdf/2609.32001","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":"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/2609.32001","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5054713548","display_name":"Riccardo Caniato","orcid":"https://orcid.org/0000-0002-6035-7816","observed_orcids":["https://orcid.org/0000-0002-6035-7816"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Caniato, Riccardo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5054713548"],"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":0.9961000084877014,"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":0.9961000084877014,"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":0.002899999963119626,"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":0.0006000000284984708,"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/minimal-surfaces","display_name":"minimal surfaces","score":0.9020000100135803},{"id":"https://openalex.org/keywords/3-manifolds","display_name":"3-manifolds","score":0.8669999837875366},{"id":"https://openalex.org/keywords/morse-index","display_name":"Morse index","score":0.9919999837875366},{"id":"https://openalex.org/keywords/genus-bounds","display_name":"genus bounds","score":0.4359999895095825},{"id":"https://openalex.org/keywords/universal-inequalities","display_name":"universal inequalities","score":0.6819999814033508},{"id":"https://openalex.org/keywords/positive-ricci-curvature","display_name":"positive Ricci curvature","score":0.9279999732971191},{"id":"https://openalex.org/keywords/positive-scalar-curvature","display_name":"positive scalar curvature","score":0.9990000128746033},{"id":"https://openalex.org/keywords/total-curvature","display_name":"total curvature","score":0.9359999895095825},{"id":"https://openalex.org/keywords/area-estimates","display_name":"area estimates","score":0.531000018119812}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8432999849319458},{"id":"https://openalex.org/C30707913","wikidata":"https://www.wikidata.org/wiki/Q1545397","display_name":"Minimal surface","level":2,"score":0.7178000211715698},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5482000112533569},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5321000218391418},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5309000015258789},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.5238000154495239},{"id":"https://openalex.org/C140031139","wikidata":"https://www.wikidata.org/wiki/Q79897","display_name":"Morse code","level":2,"score":0.4961000084877014},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.48510000109672546},{"id":"https://openalex.org/C157369684","wikidata":"https://www.wikidata.org/wiki/Q34740","display_name":"Genus","level":2,"score":0.4268999993801117},{"id":"https://openalex.org/C2780032228","wikidata":"https://www.wikidata.org/wiki/Q6865324","display_name":"Minimal model","level":2,"score":0.4032999873161316},{"id":"https://openalex.org/C2778681612","wikidata":"https://www.wikidata.org/wiki/Q15427593","display_name":"Minimal models","level":2,"score":0.382999986410141},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37229999899864197},{"id":"https://openalex.org/C156730297","wikidata":"https://www.wikidata.org/wiki/Q2296951","display_name":"Morse theory","level":2,"score":0.3564000129699707},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31929999589920044},{"id":"https://openalex.org/C167204820","wikidata":"https://www.wikidata.org/wiki/Q912058","display_name":"Simply connected space","level":2,"score":0.28529998660087585},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.27570000290870667},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.2721000015735626},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.260699987411499}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.32001","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32001","pdf_url":"https://arxiv.org/pdf/2609.32001","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.32001","doi":"https://doi.org/10.48550/arxiv.2609.32001","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.32001","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.32001","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32001","pdf_url":"https://arxiv.org/pdf/2609.32001","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":"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/W7214729874.pdf","grobid_xml":"https://content.openalex.org/works/W7214729874.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W4416258246","https://openalex.org/W7171920177"],"related_works":["https://openalex.org/W2908334430","https://openalex.org/W4399210754","https://openalex.org/W4372238319","https://openalex.org/W4283716438","https://openalex.org/W7099442456","https://openalex.org/W4415192208","https://openalex.org/W2766950964","https://openalex.org/W2949936240","https://openalex.org/W7118844369","https://openalex.org/W3093075711"],"abstract_inverted_index":{"We":[0],"prove":[1,42],"that,":[2],"in":[3,47,57],"every":[4,18],"closed":[5,19,35],"Riemannian":[6],"three-manifold":[7],"$(M^3,\\bar":[8],"g)$,":[9],"there":[10],"exists":[11],"a":[12],"constant":[13],"$C=C(M,\\bar":[14],"g)>0$":[15],"such":[16],"that":[17],"smoothly":[20],"embedded":[21],"minimal":[22,39],"surface":[23],"$\u03a3\\subset":[24],"M$":[25],"satisfies":[26],"$b_1(\u03a3;\\mathbb{Z}_2)\\le":[27],"C\\bigl(\\operatorname{Ind}(\u03a3)+\\operatorname{Area}(\u03a3)\\bigr)$,":[28],"establishing":[29],"Song's":[30],"additive":[31],"genus-index-area":[32],"conjecture.":[33],"For":[34],"connected":[36],"orientable":[37],"two-sided":[38],"immersions,":[40],"we":[41,65],"the":[43,52,62],"universal":[44],"inequality":[45],"$\u03b3(\u03a3)\\le8\\operatorname{Ind}(\u03a3)$":[46],"positive":[48,58],"Ricci":[49],"curvature":[50,72],"and":[51,70],"estimate":[53],"$\u03b3(\u03a3)\\le":[54],"C(M,\\bar":[55],"g)\\operatorname{Ind}(\u03a3)$":[56],"scalar":[59],"curvature.":[60],"In":[61],"latter":[63],"case,":[64],"also":[66],"bound":[67],"both":[68],"area":[69],"total":[71],"by":[73],"$C(M,\\bar":[74],"g)(1+\\operatorname{Ind}(\u03a3))$.":[75]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-30T00:00:00"}
