{"id":"https://openalex.org/W7204600036","doi":"https://doi.org/10.48550/arxiv.2608.26598","title":"Pinching cones for positive isotropic curvature in dimensions seven and eight","display_name":"Pinching cones for positive isotropic curvature in dimensions seven and eight","publication_year":2026,"publication_date":"2026-08-27","ids":{"openalex":"https://openalex.org/W7204600036","doi":"https://doi.org/10.48550/arxiv.2608.26598"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.26598","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.26598","pdf_url":"https://arxiv.org/pdf/2608.26598","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.26598","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5159855072","display_name":"Jae Ho Cho","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Cho, Jae Ho","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5159855072"],"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":0.9980000257492065,"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":0.9980000257492065,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.0012000000569969416,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12351","display_name":"Geometry and complex manifolds","score":0.00039999998989515007,"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/curvature-operator","display_name":"curvature operator","score":0.6000000238418579},{"id":"https://openalex.org/keywords/space-forms","display_name":"space forms","score":0.7699999809265137},{"id":"https://openalex.org/keywords/ricci-flow","display_name":"Ricci flow","score":0.7409999966621399},{"id":"https://openalex.org/keywords/ancient-solutions","display_name":"ancient solutions","score":0.9649999737739563},{"id":"https://openalex.org/keywords/7-manifolds","display_name":"seven-dimensional manifolds","score":0.6899999976158142}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7946000099182129},{"id":"https://openalex.org/C30014739","wikidata":"https://www.wikidata.org/wiki/Q5159445","display_name":"Cone (formal languages)","level":2,"score":0.6347000002861023},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5928999781608582},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5655999779701233},{"id":"https://openalex.org/C184050105","wikidata":"https://www.wikidata.org/wiki/Q273163","display_name":"Isotropy","level":2,"score":0.5375000238418579},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.48170000314712524},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3917999863624573},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.38909998536109924},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.37549999356269836},{"id":"https://openalex.org/C16977076","wikidata":"https://www.wikidata.org/wiki/Q1589551","display_name":"Principal curvature","level":4,"score":0.36079999804496765},{"id":"https://openalex.org/C149944404","wikidata":"https://www.wikidata.org/wiki/Q2392464","display_name":"Tangent cone","level":3,"score":0.3531000018119812},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.33489999175071716},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.33379998803138733},{"id":"https://openalex.org/C127813850","wikidata":"https://www.wikidata.org/wiki/Q6803617","display_name":"Mean curvature flow","level":4,"score":0.31790000200271606},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.3109000027179718},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.2935999929904938},{"id":"https://openalex.org/C84655787","wikidata":"https://www.wikidata.org/wiki/Q8067817","display_name":"Compressibility","level":2,"score":0.2831999957561493},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.2754000127315521},{"id":"https://openalex.org/C2776341046","wikidata":"https://www.wikidata.org/wiki/Q4381552","display_name":"Space form","level":3,"score":0.26330000162124634},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.26170000433921814},{"id":"https://openalex.org/C131584629","wikidata":"https://www.wikidata.org/wiki/Q4308705","display_name":"Coupling (piping)","level":2,"score":0.25519999861717224},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.25189998745918274}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.26598","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.26598","pdf_url":"https://arxiv.org/pdf/2608.26598","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.2608.26598","doi":"https://doi.org/10.48550/arxiv.2608.26598","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.26598","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:2608.26598","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.26598","pdf_url":"https://arxiv.org/pdf/2608.26598","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204600036.pdf","grobid_xml":"https://content.openalex.org/works/W7204600036.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7220428771","https://openalex.org/W2969808372","https://openalex.org/W2780400340","https://openalex.org/W4414973342","https://openalex.org/W7140228852","https://openalex.org/W2580815225","https://openalex.org/W2735664388","https://openalex.org/W4415317664","https://openalex.org/W4324325372","https://openalex.org/W1972427397"],"abstract_inverted_index":{"We":[0,90],"construct":[1],"and":[2,32,114,150,201,206],"prove":[3,115],"the":[4,14,23,54,66,72,75,99,107,129,137,145,169,211,219],"transversal":[5],"invariance":[6],"of":[7,10,74,109,123,166,175,193,204,213,228],"two":[8,42,76,130],"families":[9],"pinching":[11,24,148,160],"cones":[12],"for":[13,29,35,53,79],"Hamilton":[15],"ODE":[16],"$\\frac{\\mathrm{d}}{\\mathrm{d}t}R=Q(R)$":[17],"in":[18,44,65,172],"dimensions":[19],"$n=7,8$,":[20],"thereby":[21],"extending":[22,190,225],"estimate":[25,52,82,149],"established":[26,142],"by":[27,33,60,86,118],"Brendle":[28,194],"$n\\geq":[30,196,233],"12$":[31],"Chen":[34],"$9\\leq":[36],"n\\leq":[37],"11$.":[38],"In":[39,126],"dimension":[40,127,156,173],"$n=8$,":[41],"steps":[43],"Chen's":[45,81],"construction":[46],"genuinely":[47],"fail.":[48],"The":[49,69,158],"first,":[50],"an":[51,119],"second":[55,70],"cone":[56,77,108],"family,":[57],"is":[58,71,140],"repaired":[59],"retaining":[61],"a":[62,87,94,102,191,226],"coupling":[63],"discarded":[64],"dimension-general":[67],"argument.":[68],"gluing":[73,100,138],"families,":[78],"which":[80],"fails":[83],"at":[84],"$n=8$":[85,159],"definite":[88],"margin.":[89],"replace":[91],"it":[92,208],"with":[93,163,178,199,222],"new":[95],"argument":[96],"that":[97,182],"reduces":[98],"to":[101,155,218],"single":[103],"linear":[104],"inequality":[105,117],"on":[106],"weakly":[110],"PIC":[111],"curvature":[112,181],"operators,":[113],"this":[116],"explicit":[120],"finite":[121],"combination":[122],"frame":[124],"inequalities.":[125],"$n=7$,":[128],"separate":[131],"cone-invariance":[132],"results":[133,203],"are":[134,153],"proved,":[135],"but":[136],"step":[139],"not":[141],"here.":[143],"Accordingly,":[144],"resulting":[146],"full":[147],"classification":[151,171,202,212],"statements":[152],"restricted":[154],"$n=8$.":[157],"estimate,":[161],"together":[162],"dimension-free":[164],"arguments":[165],"Brendle,":[167],"yields":[168,210],"topological":[170],"eight":[174],"compact":[176],"manifolds":[177],"positive":[179],"isotropic":[180],"contain":[183],"no":[184],"nontrivial":[185],"incompressible":[186],"$(n-1)$-dimensional":[187],"space":[188],"forms,":[189],"theorem":[192,227],"from":[195,230],"12$.":[197,234],"Together":[198],"curvature-improvement":[200],"Cho--Li":[205,229],"Brendle--Naff,":[207],"also":[209],"noncompact":[214],"$\u03ba$-noncollapsed":[215],"ancient":[216],"solutions":[217],"Ricci":[220],"flow":[221],"uniformly":[223],"PIC,":[224],"$n=4$":[231],"or":[232]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-08-29T00:00:00"}