{"id":"https://openalex.org/W7204698706","doi":"https://doi.org/10.5281/zenodo.22171949","title":"Meagerness of the realizable set for hypersurfaces of Euclidean four-space","display_name":"Meagerness of the realizable set for hypersurfaces of Euclidean four-space","publication_year":2026,"publication_date":"2026-08-30","ids":{"openalex":"https://openalex.org/W7204698706","doi":"https://doi.org/10.5281/zenodo.22171949"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.22171949","doi":"https://doi.org/10.5281/zenodo.22171949","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22171949","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.22171949","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5149201535","display_name":"Jay Thrussell","orcid":"https://orcid.org/0009-0004-7707-7871","observed_orcids":["https://orcid.org/0009-0004-7707-7871"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Thrussell, Jay","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0004-7707-7871","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5149201535"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.8734999895095825,"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"}},"topics":[{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.8734999895095825,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.12020000070333481,"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":0.00279999990016222,"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/isometric-embedding","display_name":"isometric embedding","score":0.8640000224113464},{"id":"https://openalex.org/keywords/nowhere-dense-sets","display_name":"nowhere dense sets","score":0.8090000152587891},{"id":"https://openalex.org/keywords/baire-category","display_name":"Baire category","score":0.765999972820282},{"id":"https://openalex.org/keywords/total-absolute-curvature","display_name":"total absolute curvature","score":0.7829999923706055},{"id":"https://openalex.org/keywords/second-variation","display_name":"second variation","score":0.8130000233650208},{"id":"https://openalex.org/keywords/round-metric","display_name":"round metric","score":0.4959999918937683}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8769000172615051},{"id":"https://openalex.org/C114410712","wikidata":"https://www.wikidata.org/wiki/Q973321","display_name":"Hypersurface","level":2,"score":0.7563999891281128},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.5005000233650208},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4569000005722046},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.44929999113082886},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4422000050544739},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.374099999666214},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.366100013256073},{"id":"https://openalex.org/C175017881","wikidata":"https://www.wikidata.org/wiki/Q1318998","display_name":"Mean curvature","level":3,"score":0.33180001378059387},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.3165999948978424},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.31299999356269836},{"id":"https://openalex.org/C146846114","wikidata":"https://www.wikidata.org/wiki/Q901448","display_name":"Einstein","level":2,"score":0.31279999017715454},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.311599999666214},{"id":"https://openalex.org/C119830904","wikidata":"https://www.wikidata.org/wiki/Q233220","display_name":"Second fundamental form","level":4,"score":0.30889999866485596},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.299699991941452},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.2937000095844269},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.29269999265670776},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2915000021457672},{"id":"https://openalex.org/C12089564","wikidata":"https://www.wikidata.org/wiki/Q1195879","display_name":"Ricci curvature","level":3,"score":0.2824999988079071},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.27219998836517334},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.2709999978542328},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.2703000009059906},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.26989999413490295},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.2597000002861023},{"id":"https://openalex.org/C42357961","wikidata":"https://www.wikidata.org/wiki/Q213363","display_name":"Open set","level":2,"score":0.25380000472068787}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.5281/zenodo.22171949","doi":"https://doi.org/10.5281/zenodo.22171949","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22171949","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.22171950","doi":"https://doi.org/10.5281/zenodo.22171950","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22171950","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"article-journal"},{"id":"doi:10.5281/zenodo.22177620","doi":"https://doi.org/10.5281/zenodo.22177620","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22177620","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.22777315","doi":"https://doi.org/10.5281/zenodo.22777315","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22777315","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"doi:10.5281/zenodo.22171949","doi":"https://doi.org/10.5281/zenodo.22171949","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22171949","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7203630594","https://openalex.org/W7213624631","https://openalex.org/W7204712579","https://openalex.org/W7213670143","https://openalex.org/W1521506135","https://openalex.org/W2014280773","https://openalex.org/W1599830584","https://openalex.org/W1606440516","https://openalex.org/W1976400340","https://openalex.org/W3139847702"],"abstract_inverted_index":{"**Abstract.**":[0],"Let":[1],"$\\mathcal":[2,43,62],"U$":[3],"be":[4],"the":[5,24,48,73,141,147,151,156,167],"set":[6,49],"of":[7,50,57,72,112,150,166],"$C^{k,\\alpha}$":[8],"Riemannian":[9],"metrics":[10,51],"on":[11,96],"$S^3$":[12],"with":[13,34,125],"$\\det\\mathbb":[14,115,126],"M":[15,20,116,127],">":[16,117],"0$,":[17],"where":[18],"$\\mathbb":[19],"=":[21,31,37,45],"-G$":[22],"is":[23,47,64,78,130],"negative":[25],"Einstein":[26],"endomorphism,":[27],"and":[28,159],"let":[29],"$F(g)":[30],"\\mathrm{Cod}(\\tilde":[32],"K(g))$":[33],"$\\tilde":[35],"K":[36],"\\sqrt{\\det\\mathbb":[38],"M}\\,\\mathbb":[39],"M^{-1}$,":[40],"so":[41,120],"that":[42,61,108,121],"R":[44],"F^{-1}(0)$":[46],"realizable":[52],"as":[53],"isometrically":[54],"immersed":[55],"hypersurfaces":[56],"$E^4$.":[58],"We":[59,144],"prove":[60],"R$":[63],"nowhere":[65],"dense,":[66],"hence":[67],"meager,":[68],"in":[69,163],"a":[70,83,90,97,109],"neighbourhood":[71],"round":[74,157],"metric.":[75],"The":[76],"proof":[77],"unconditional:":[79],"it":[80],"uses":[81],"neither":[82],"slice":[84],"theorem":[85],"nor":[86],"pseudodifferential":[87],"machinery,":[88],"only":[89],"uniform":[91],"lower":[92],"bound":[93],"for":[94],"$DF$":[95],"fixed":[98],"transverse-traceless":[99],"subspace.":[100],"Separately,":[101],"we":[102],"show":[103],"by":[104],"an":[105],"elementary":[106],"argument":[107],"closed":[110],"hypersurface":[111],"$E^4$":[113],"satisfies":[114],"0$":[118,129],"somewhere,":[119],"no":[122],"compact":[123],"three-manifold":[124],"\\equiv":[128],"realizable;":[131],"this":[132],"excludes":[133],"six":[134],"homogeneous":[135],"geometries":[136],"at":[137,155],"once,":[138],"among":[139],"them":[140],"flat":[142],"three-torus.":[143],"also":[145],"compute":[146],"second":[148],"variation":[149],"total":[152,185],"absolute":[153,186],"curvature":[154],"sphere":[158],"determine":[160],"its":[161],"sign":[162],"each":[164],"sector":[165],"York":[168],"decomposition.":[169],"**MSC":[170],"2020.**":[171],"53C24":[172],"(primary);":[173],"53C42,":[174],"53A07,":[175],"58D17.":[176],"**Keywords.**":[177],"isometric":[178],"embedding,":[179],"hypersurfaces,":[180],"Codazzi":[181],"equation,":[182],"Baire":[183],"category,":[184],"curvature.":[187]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-08-31T00:00:00"}
