{"id":"https://openalex.org/W7225148755","doi":"https://doi.org/10.48550/arxiv.2611.03249","title":"Entropy and round-point convergence for the $\u03c3_k$-curvature flow","display_name":"Entropy and round-point convergence for the $\u03c3_k$-curvature flow","publication_year":2026,"publication_date":"2026-11-02","ids":{"openalex":"https://openalex.org/W7225148755","doi":"https://doi.org/10.48550/arxiv.2611.03249"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2611.03249","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2611.03249","pdf_url":"https://arxiv.org/pdf/2611.03249","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/2611.03249","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5113233283","display_name":"Tianci Luo","orcid":"https://orcid.org/0009-0000-8157-6471","observed_orcids":["https://orcid.org/0009-0000-8157-6471"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Luo, Tianci","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5150890886","display_name":"Yong Wei","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wei, Yong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5160431592","display_name":"Ruijia Zhang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhang, Ruijia","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5101899533","display_name":"Rong Zhou","orcid":"https://orcid.org/0009-0007-9328-3001","observed_orcids":["https://orcid.org/0009-0007-9328-3001"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhou, Rong","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":0.9990000128746033,"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":0.9990000128746033,"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":0.0003000000142492354,"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/T11830","display_name":"Point processes and geometric inequalities","score":9.999999747378752e-05,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/entropy","display_name":"entropy","score":0.6169999837875366},{"id":"https://openalex.org/keywords/strictly-convex-hypersurfaces","display_name":"strictly convex hypersurfaces","score":0.8930000066757202},{"id":"https://openalex.org/keywords/mean-width","display_name":"mean width","score":0.8809999823570251},{"id":"https://openalex.org/keywords/inradius","display_name":"inradius","score":0.6549999713897705},{"id":"https://openalex.org/keywords/curvature-bounds","display_name":"curvature bounds","score":0.6650000214576721},{"id":"https://openalex.org/keywords/mean-curvature-flow","display_name":"mean curvature flow","score":0.503000020980835}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8687000274658203},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5760999917984009},{"id":"https://openalex.org/C114410712","wikidata":"https://www.wikidata.org/wiki/Q973321","display_name":"Hypersurface","level":2,"score":0.5396999716758728},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.526199996471405},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5184000134468079},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.491100013256073},{"id":"https://openalex.org/C127813850","wikidata":"https://www.wikidata.org/wiki/Q6803617","display_name":"Mean curvature flow","level":4,"score":0.4871000051498413},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.39559999108314514},{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.3765999972820282},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.373199999332428},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.37220001220703125},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.36579999327659607},{"id":"https://openalex.org/C175017881","wikidata":"https://www.wikidata.org/wiki/Q1318998","display_name":"Mean curvature","level":3,"score":0.35019999742507935},{"id":"https://openalex.org/C171752962","wikidata":"https://www.wikidata.org/wiki/Q255166","display_name":"Kullback\u2013Leibler divergence","level":2,"score":0.34869998693466187},{"id":"https://openalex.org/C2780695682","wikidata":"https://www.wikidata.org/wiki/Q4005959","display_name":"Jump","level":2,"score":0.32749998569488525},{"id":"https://openalex.org/C140142295","wikidata":"https://www.wikidata.org/wiki/Q617417","display_name":"Isoperimetric inequality","level":2,"score":0.30869999527931213},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.2847000062465668},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.26969999074935913},{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.2687999904155731},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.267300009727478},{"id":"https://openalex.org/C135402231","wikidata":"https://www.wikidata.org/wiki/Q898440","display_name":"Dissipation","level":2,"score":0.2669000029563904},{"id":"https://openalex.org/C2780016784","wikidata":"https://www.wikidata.org/wiki/Q2714507","display_name":"Orthant","level":2,"score":0.26510000228881836},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2639000117778778},{"id":"https://openalex.org/C201779956","wikidata":"https://www.wikidata.org/wiki/Q5426803","display_name":"Ergodicity","level":2,"score":0.25929999351501465},{"id":"https://openalex.org/C24495805","wikidata":"https://www.wikidata.org/wiki/Q7269091","display_name":"Quantum relative entropy","level":5,"score":0.2590999901294708},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.25459998846054077},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2533000111579895},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.2502000033855438}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2611.03249","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2611.03249","pdf_url":"https://arxiv.org/pdf/2611.03249","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.2611.03249","doi":"https://doi.org/10.48550/arxiv.2611.03249","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2611.03249","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2611.03249","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2611.03249","pdf_url":"https://arxiv.org/pdf/2611.03249","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":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321543","display_name":"China Postdoctoral Science Foundation","ror":"https://ror.org/0426zh255"},{"id":"https://openalex.org/F4320335777","display_name":"National Key Research and Development Program of China","ror":null},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7225148755.pdf","grobid_xml":"https://content.openalex.org/works/W7225148755.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2920750376","https://openalex.org/W2193432315","https://openalex.org/W2763037810","https://openalex.org/W3007856367","https://openalex.org/W4409728426","https://openalex.org/W4362679568","https://openalex.org/W3101511752","https://openalex.org/W3133995951","https://openalex.org/W2549150306","https://openalex.org/W1510725409"],"abstract_inverted_index":{"We":[0,111],"prove":[1],"that":[2],"every":[3],"smooth":[4],"closed":[5],"strictly":[6],"convex":[7],"hypersurface":[8],"in":[9,47],"$\\mathbb":[10],"R^{n+1}$":[11],"contracts":[12],"to":[13,37,50,96],"a":[14,70],"round":[15],"point":[16,36],"under":[17,64],"the":[18,34,41,51,90,93,97,104,108,118],"$\u03c3_k$-curvature":[19],"flow":[20],"for":[21,89],"$2\\leq":[22],"k\\leq":[23],"n-1$,":[24],"without":[25],"additional":[26],"pinching":[27],"or":[28],"symmetry":[29],"assumptions.":[30],"After":[31],"rescaling":[32],"about":[33],"extinction":[35,120],"mean":[38,98],"width":[39],"two,":[40],"hypersurfaces":[42],"converge":[43],"smoothly":[44],"and":[45,66,68],"exponentially":[46],"normalized":[48,109],"time":[49],"unit":[52],"sphere.":[53],"The":[54],"proof":[55],"is":[56,62],"based":[57],"on":[58],"an":[59],"entropy":[60,91],"which":[61],"invariant":[63],"translations":[65],"dilations":[67],"admits":[69],"representation":[71],"by":[72],"weighted":[73],"boundary":[74],"areas":[75],"of":[76,107],"orthogonal":[77],"projections.":[78],"A":[79],"strengthened":[80],"Codazzi":[81],"inequality":[82],"yields":[83],"its":[84,101],"monotonicity.":[85],"An":[86],"upper":[87],"bound":[88],"controls":[92,103],"inradius":[94],"relative":[95],"width,":[99],"while":[100],"dissipation":[102],"spatial":[105],"variation":[106],"speed.":[110],"also":[112],"obtain":[113],"two-sided":[114],"curvature":[115],"bounds":[116],"at":[117],"natural":[119],"scale.":[121]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-11-04T00:00:00"}
