{"id":"https://openalex.org/W7208715006","doi":"https://doi.org/10.48550/arxiv.2609.03332","title":"Singularity Models of Finite-Time K\u00e4hler-Ricci Flows","display_name":"Singularity Models of Finite-Time K\u00e4hler-Ricci Flows","publication_year":2026,"publication_date":"2026-09-03","ids":{"openalex":"https://openalex.org/W7208715006","doi":"https://doi.org/10.48550/arxiv.2609.03332"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.03332","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03332","pdf_url":"https://arxiv.org/pdf/2609.03332","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":"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/2609.03332","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025969451","display_name":"Frederick Fong","orcid":"https://orcid.org/0000-0002-4310-9386","observed_orcids":["https://orcid.org/0000-0002-4310-9386"]},"institutions":[{"id":"https://openalex.org/I200769079","display_name":"Hong Kong University of Science and Technology","ror":"https://ror.org/00q4vv597","country_code":"HK","type":"education","lineage":["https://openalex.org/I200769079"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Fong, Frederick Tsz-Ho","raw_affiliation_strings":["Department of Mathematics , Hong Kong University of Science and Technology , Clear Water Bay , Kowloon , Hong Kong SAR"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Hong Kong University of Science and Technology , Clear Water Bay , Kowloon , Hong Kong SAR","institution_ids":["https://openalex.org/I200769079"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5135417641","display_name":"Hung Tran","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I12315562","display_name":"Texas Tech University","ror":"https://ror.org/0405mnx93","country_code":"US","type":"education","lineage":["https://openalex.org/I12315562"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Tran, Hung","raw_affiliation_strings":["Department of Mathematics and Statistics , Texas Tech University , Lubbock , TX 79409"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics , Texas Tech University , Lubbock , TX 79409","institution_ids":["https://openalex.org/I12315562"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":3,"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/T12351","display_name":"Geometry and complex manifolds","score":0.9970999956130981,"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/T12351","display_name":"Geometry and complex manifolds","score":0.9970999956130981,"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":0.0024999999441206455,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.00019999999494757503,"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/kahler-ricci-flow","display_name":"K\u00e4hler-Ricci flow","score":0.9760000109672546},{"id":"https://openalex.org/keywords/blow-up","display_name":"blow-up","score":0.875},{"id":"https://openalex.org/keywords/singularity-models","display_name":"singularity models","score":0.8159999847412109},{"id":"https://openalex.org/keywords/type-i-singularities","display_name":"Type I singularities","score":0.7630000114440918},{"id":"https://openalex.org/keywords/circle-bundles","display_name":"circle bundles","score":0.6650000214576721}],"concepts":[{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.8227999806404114},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.680899977684021},{"id":"https://openalex.org/C2779165322","wikidata":"https://www.wikidata.org/wiki/Q546145","display_name":"Compactification (mathematics)","level":2,"score":0.5884000062942505},{"id":"https://openalex.org/C130979935","wikidata":"https://www.wikidata.org/wiki/Q568954","display_name":"Ansatz","level":2,"score":0.567300021648407},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5131999850273132},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4975999891757965},{"id":"https://openalex.org/C168425004","wikidata":"https://www.wikidata.org/wiki/Q525802","display_name":"Foliation (geology)","level":3,"score":0.44999998807907104},{"id":"https://openalex.org/C71287196","wikidata":"https://www.wikidata.org/wiki/Q847204","display_name":"Limit of a sequence","level":3,"score":0.42660000920295715},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.39750000834465027},{"id":"https://openalex.org/C71383303","wikidata":"https://www.wikidata.org/wiki/Q1876526","display_name":"Essential singularity","level":3,"score":0.39590001106262207},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.39489999413490295},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.38760000467300415},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.38179999589920044},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3700000047683716},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.36739999055862427},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.35569998621940613},{"id":"https://openalex.org/C2778134712","wikidata":"https://www.wikidata.org/wiki/Q1047307","display_name":"Bundle","level":2,"score":0.3522999882698059},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.34880000352859497},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.310699999332428},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2863999903202057},{"id":"https://openalex.org/C207769094","wikidata":"https://www.wikidata.org/wiki/Q4245771","display_name":"Singularity theory","level":3,"score":0.27959999442100525},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.2556000053882599},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.25369998812675476}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.03332","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03332","pdf_url":"https://arxiv.org/pdf/2609.03332","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.03332","doi":"https://doi.org/10.48550/arxiv.2609.03332","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.03332","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:2609.03332","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03332","pdf_url":"https://arxiv.org/pdf/2609.03332","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":"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/W7208715006.pdf","grobid_xml":"https://content.openalex.org/works/W7208715006.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1970564715","https://openalex.org/W2040052719","https://openalex.org/W2057489153","https://openalex.org/W2074985705","https://openalex.org/W2129013141","https://openalex.org/W2322367106","https://openalex.org/W2913507782","https://openalex.org/W2964272579","https://openalex.org/W3088829178","https://openalex.org/W4387635059","https://openalex.org/W4400867252","https://openalex.org/W4416149846","https://openalex.org/W7125934898","https://openalex.org/W7203763573","https://openalex.org/W7206179186","https://openalex.org/W7207531830"],"related_works":["https://openalex.org/W7212005226","https://openalex.org/W2177089878","https://openalex.org/W4240998417","https://openalex.org/W7203945479","https://openalex.org/W4416149846","https://openalex.org/W2944245107","https://openalex.org/W3104508346","https://openalex.org/W7213507500","https://openalex.org/W2963162531","https://openalex.org/W7125934898"],"abstract_inverted_index":{"We":[0,156],"study":[1,175],"the":[2,8,16,39,48,53,57,64,68,79,92,108,139,147,166,176,181,187,197,217,240,245,258],"singularity":[3,167],"type":[4],"and":[5,83,107,144,146,163,183,194,233],"models":[6],"of":[7,19,25,63,91,138,170,180,186,191,216,242,244],"K\u00e4hler--Ricci":[9],"flow":[10,188],"on":[11,78],"compact":[12],"manifolds":[13,27],"constructed":[14,37],"from":[15],"1-parameter":[17],"foliation":[18,65],"a":[20,23,96,111,120,151,222,236,254],"circle-bundle":[21,259],"over":[22,122,153,239],"product":[24,241],"K\u00e4hler--Einstein":[26],"$N":[28],":=":[29,101],"N_1":[30],"\\times":[31,33,103,105,204],"\\cdots":[32,104],"N_r$,":[34],"with":[35,67,95,110,133,207,227],"metric":[36,251],"using":[38],"ansatz":[40],"considered":[41,56],"in":[42,159,189],"\\cite{DW2011},":[43],"\\cite{WW}":[44],"et.":[45],"al.":[46],"In":[47,71],"earlier":[49],"work":[50,77],"\\cite{FT}":[51],"by":[52],"authors,":[54],"we":[55,74,174],"``two-bolt''":[58],"case":[59,126],"where":[60,212],"both":[61,130],"ends":[62,131],"close":[66,132],"``bolt''":[69],"$N$.":[70],"this":[72],"article,":[73],"continue":[75],"our":[76],"more":[80],"subtle":[81],"``nut-bolt''":[82,162],"``two-nut''":[84,125,164],"cases.":[85],"The":[86,115,124,250],"former":[87],"has":[88],"one":[89,128,215],"end":[90],"interval":[93],"closes":[94],"nut-type":[97,134],"collapse":[98],"(i.e.":[99,113],"$N'":[100],"N_2":[102],"N_r$)":[106],"other":[109],"bolt":[112],"$N$).":[114],"compactification":[116,148],"$\\widehat{M}$":[117,149],"is":[118,127,150,214,235,253],"then":[119],"$\\mathbb{CP}^{m+1}$-bundle":[121],"$N'$.":[123],"that":[129,158,196],"collapses,":[135],"necessarily":[136],"two":[137],"$N_i$'s":[140],"must":[141,168,200],"be":[142,169,201],"$\\mathbb{CP}^{m_0}$":[143],"$\\mathbb{CP}^{m_\\ell}$,":[145],"$\\mathbb{CP}^{m_0+m_\\ell+1}$-bundle":[152],"$\\prod_{k\\geq":[154],"3}N_k$.":[155],"proved":[157],"all":[160,190],"``two-bolt'',":[161],"caess":[165],"Type":[171],"I.":[172],"Furthermore,":[173],"pointed":[177],"Cheeger-Gromov":[178],"limit":[179,198],"rescaled":[182],"dilated":[184],"sequence":[185],"three":[192],"cases,":[193],"prove":[195],"model":[199],"$(\u03a3^{m+1},":[202],"g_\u03a3(t))":[203],"(\\mathbb{C}^{k},":[205],"\\textrm{flat})$":[206],"$m,":[208],"k":[209],"\\geq":[210],"0$,":[211],"$\u03a3$":[213],"following:":[218],"$\\mathbb{CP}^{m+1}$,":[219],"$\\textrm{Tot}(\\mathcal{L}^{\\oplus(m+1)})$,":[220],"or":[221],"projectivization":[223],"$\\mathbb{P}\\big(\\mathcal{O}^{\\oplus(m_0+1)}":[224],"\\oplus":[225],"\\mathcal{L}^{\\oplus(m_\\ell+1)}\\big)$":[226],"$m_0":[228],"+":[229],"m_\\ell":[230],"=":[231],"m$,":[232],"$\\mathcal{L}$":[234],"line":[237],"bundle":[238],"\\emph{some}":[243],"$N_1,":[246],"\\cdots,":[247],"N_r$":[248],"factors.":[249],"$g_\u03a3(t)$":[252],"K\u00e4hler-Ricci":[255],"shrinker":[256],"satisfying":[257],"ansatz.":[260]},"counts_by_year":[{"year":2026,"cited_by_count":3}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-05T00:00:00"}
