{"id":"https://openalex.org/W7213607884","doi":"https://doi.org/10.48550/arxiv.2609.20517","title":"The isoperimetric inequality and CMC hypersurfaces in Cartan-Hadamard manifolds","display_name":"The isoperimetric inequality and CMC hypersurfaces in Cartan-Hadamard manifolds","publication_year":2026,"publication_date":"2026-09-17","ids":{"openalex":"https://openalex.org/W7213607884","doi":"https://doi.org/10.48550/arxiv.2609.20517"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.20517","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.20517","pdf_url":"https://arxiv.org/pdf/2609.20517","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/2609.20517","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151334454","display_name":"Shibing Chen","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I126520041","display_name":"University of Science and Technology of China","ror":"https://ror.org/04c4dkn09","country_code":"CN","type":"education","lineage":["https://openalex.org/I126520041","https://openalex.org/I19820366"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Chen, Shibing","raw_affiliation_strings":["School of Mathematical Sciences , University of Science and Technology of China , Hefei , Anhui 230026 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , University of Science and Technology of China , Hefei , Anhui 230026 , China","institution_ids":["https://openalex.org/I126520041"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019994337","display_name":"Mohammad Ghomi","orcid":"https://orcid.org/0000-0003-3767-9347","observed_orcids":["https://orcid.org/0000-0003-3767-9347"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ghomi, Mohammad","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5159901912","display_name":"Peng Wang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I126520041","display_name":"University of Science and Technology of China","ror":"https://ror.org/04c4dkn09","country_code":"CN","type":"education","lineage":["https://openalex.org/I126520041","https://openalex.org/I19820366"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wang, Peng","raw_affiliation_strings":["School of Mathematical Sciences , University of Science and Technology of China , Hefei , Anhui 230026 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , University of Science and Technology of China , Hefei , Anhui 230026 , China","institution_ids":["https://openalex.org/I126520041"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"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":0.000699999975040555,"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/T11830","display_name":"Point processes and geometric inequalities","score":0.0,"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/isoperimetric-inequality","display_name":"isoperimetric inequality","score":0.9940000176429749},{"id":"https://openalex.org/keywords/constant-mean-curvature-hypersurfaces","display_name":"constant mean curvature hypersurfaces","score":0.9599999785423279},{"id":"https://openalex.org/keywords/cartan-hadamard-manifolds","display_name":"Cartan-Hadamard manifolds","score":0.9350000023841858},{"id":"https://openalex.org/keywords/nonpositive-sectional-curvature","display_name":"nonpositive sectional curvature","score":0.9440000057220459},{"id":"https://openalex.org/keywords/isoperimetric-regions","display_name":"isoperimetric regions","score":0.9409999847412109},{"id":"https://openalex.org/keywords/geodesic-balls","display_name":"geodesic balls","score":0.8040000200271606},{"id":"https://openalex.org/keywords/jacobi-fields","display_name":"Jacobi fields","score":0.843999981880188},{"id":"https://openalex.org/keywords/isoperimetric-profile","display_name":"isoperimetric profile","score":0.8479999899864197}],"concepts":[{"id":"https://openalex.org/C140142295","wikidata":"https://www.wikidata.org/wiki/Q617417","display_name":"Isoperimetric inequality","level":2,"score":0.9549000263214111},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8288999795913696},{"id":"https://openalex.org/C70266181","wikidata":"https://www.wikidata.org/wiki/Q3707726","display_name":"Isoperimetric dimension","level":3,"score":0.7271999716758728},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.6794999837875366},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5891000032424927},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5715000033378601},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5475999712944031},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.5083000063896179},{"id":"https://openalex.org/C2779668893","wikidata":"https://www.wikidata.org/wiki/Q2250422","display_name":"Constant curvature","level":3,"score":0.4885999858379364},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4884999990463257},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.47369998693466187},{"id":"https://openalex.org/C167204820","wikidata":"https://www.wikidata.org/wiki/Q912058","display_name":"Simply connected space","level":2,"score":0.4690000116825104},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.4449000060558319},{"id":"https://openalex.org/C175017881","wikidata":"https://www.wikidata.org/wiki/Q1318998","display_name":"Mean curvature","level":3,"score":0.4438000023365021},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.44290000200271606},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.43860000371932983},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3797999918460846},{"id":"https://openalex.org/C194147245","wikidata":"https://www.wikidata.org/wiki/Q1076368","display_name":"Chord (peer-to-peer)","level":2,"score":0.35989999771118164},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.35120001435279846},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.33649998903274536},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.3264000117778778},{"id":"https://openalex.org/C30707913","wikidata":"https://www.wikidata.org/wiki/Q1545397","display_name":"Minimal surface","level":2,"score":0.32519999146461487},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.31290000677108765},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.27720001339912415},{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.27000001072883606},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.25450000166893005}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.20517","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.20517","pdf_url":"https://arxiv.org/pdf/2609.20517","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.2609.20517","doi":"https://doi.org/10.48550/arxiv.2609.20517","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.20517","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:2609.20517","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.20517","pdf_url":"https://arxiv.org/pdf/2609.20517","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/W7213607884.pdf","grobid_xml":"https://content.openalex.org/works/W7213607884.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7212237171","https://openalex.org/W2785547653","https://openalex.org/W3211462470","https://openalex.org/W4387415047","https://openalex.org/W7162409737","https://openalex.org/W7097626006","https://openalex.org/W7213263486","https://openalex.org/W1976334161","https://openalex.org/W1974173281","https://openalex.org/W2317142650"],"abstract_inverted_index":{"We":[0],"show":[1],"that":[2],"the":[3,25,59,68,71,77,109,125],"sharp":[4,36],"Euclidean":[5],"isoperimetric":[6,96],"inequality":[7,37,91],"holds":[8],"for":[9,38,52,93],"domains":[10],"in":[11,28,63,99],"complete":[12],"simply":[13],"connected":[14],"Riemannian":[15],"$n$-manifolds":[16],"of":[17,46,58,70,95,120],"nonpositive":[18],"sectional":[19],"curvature,":[20],"$3\\leq":[21],"n\\leq9$,":[22],"which":[23,113],"establishes":[24],"Cartan-Hadamard":[26],"conjecture":[27],"these":[29,64],"dimensions.":[30],"The":[31,61,90,117],"main":[32],"step":[33],"is":[34,105],"a":[35,86],"constant-mean-curvature":[39],"hypersurfaces,":[40],"proved":[41],"via":[42],"integrals":[43,65],"over":[44],"pairs":[45],"boundary":[47],"points,":[48],"together":[49],"with":[50,76,85],"estimates":[51],"Jacobi":[53],"fields":[54],"along":[55],"geodesic":[56,100],"chords":[57],"boundary.":[60],"weights":[62],"depend":[66],"on":[67,108],"length":[69],"chord":[72],"and":[73,79,112,122],"its":[74],"angles":[75],"boundary,":[78],"are":[80],"chosen":[81],"dimension":[82],"by":[83],"dimension,":[84],"Green":[87],"function":[88],"pole.":[89],"persists":[92],"boundaries":[94],"regions":[97],"trapped":[98],"balls,":[101],"whose":[102],"mean":[103],"curvature":[104],"constant":[106],"only":[107],"free":[110],"part":[111],"may":[114],"have":[115],"singularities.":[116],"isoperimetric-profile":[118],"argument":[119],"Kleiner":[121],"Ghomi-Spruck":[123],"completes":[124],"proof.":[126]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-19T00:00:00"}