{"id":"https://openalex.org/W7214671200","doi":"https://doi.org/10.48550/arxiv.2609.30740","title":"Energy convexity and uniformity of the $H$-surface flow in $\\mathbb{R}^{3}$ with Dirichlet boundary condition","display_name":"Energy convexity and uniformity of the $H$-surface flow in $\\mathbb{R}^{3}$ with Dirichlet boundary condition","publication_year":2026,"publication_date":"2026-09-25","ids":{"openalex":"https://openalex.org/W7214671200","doi":"https://doi.org/10.48550/arxiv.2609.30740"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.30740","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30740","pdf_url":"https://arxiv.org/pdf/2609.30740","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","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.30740","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076266672","display_name":"Da Rong Cheng","orcid":"https://orcid.org/0000-0002-4675-3839","observed_orcids":["https://orcid.org/0000-0002-4675-3839"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cheng, Da Rong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025371544","display_name":"Longzhi Lin","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Lin, Longzhi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5008210639","display_name":"Xin Zhou","orcid":"https://orcid.org/0000-0001-8133-259X","observed_orcids":["https://orcid.org/0000-0001-8133-259X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhou, Xin","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":0.9994999766349792,"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":0.9994999766349792,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.00019999999494757503,"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":0.0,"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/h-surfaces","display_name":"H-surfaces","score":0.9169999957084656},{"id":"https://openalex.org/keywords/dirichlet-energy","display_name":"Dirichlet energy","score":0.7099999785423279},{"id":"https://openalex.org/keywords/weak-solutions","display_name":"weak solutions","score":0.7379999756813049},{"id":"https://openalex.org/keywords/uniform-convergence","display_name":"uniform convergence","score":0.7789999842643738},{"id":"https://openalex.org/keywords/dirichlet-boundary-conditions","display_name":"Dirichlet boundary conditions","score":0.7760000228881836},{"id":"https://openalex.org/keywords/small-energy","display_name":"small energy","score":0.3140000104904175}],"concepts":[{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.8339999914169312},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8108999729156494},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.727400004863739},{"id":"https://openalex.org/C96234433","wikidata":"https://www.wikidata.org/wiki/Q5280760","display_name":"Dirichlet's energy","level":4,"score":0.6960999965667725},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6926000118255615},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5297999978065491},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.4952000081539154},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.48350000381469727},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4772999882698059},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.4571000039577484},{"id":"https://openalex.org/C127813850","wikidata":"https://www.wikidata.org/wiki/Q6803617","display_name":"Mean curvature flow","level":4,"score":0.42489999532699585},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.40790000557899475},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.38989999890327454},{"id":"https://openalex.org/C2779560616","wikidata":"https://www.wikidata.org/wiki/Q1192869","display_name":"Dirichlet problem","level":3,"score":0.375},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.3619000017642975},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.34630000591278076},{"id":"https://openalex.org/C191640071","wikidata":"https://www.wikidata.org/wiki/Q5377056","display_name":"Energy functional","level":2,"score":0.33970001339912415},{"id":"https://openalex.org/C2985596519","wikidata":"https://www.wikidata.org/wiki/Q179635","display_name":"Heat flow","level":3,"score":0.3310999870300293},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.32749998569488525},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.32120001316070557},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.32010000944137573},{"id":"https://openalex.org/C148158259","wikidata":"https://www.wikidata.org/wiki/Q975727","display_name":"Dirichlet's principle","level":4,"score":0.3068000078201294},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.26980000734329224},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.26080000400543213},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.25920000672340393},{"id":"https://openalex.org/C6270764","wikidata":"https://www.wikidata.org/wiki/Q5659257","display_name":"Harmonic map","level":2,"score":0.2565999925136566}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.30740","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30740","pdf_url":"https://arxiv.org/pdf/2609.30740","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.2609.30740","doi":"https://doi.org/10.48550/arxiv.2609.30740","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.30740","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:2609.30740","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30740","pdf_url":"https://arxiv.org/pdf/2609.30740","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214671200.pdf","grobid_xml":"https://content.openalex.org/works/W7214671200.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7099881338","https://openalex.org/W4386982901","https://openalex.org/W7042312817","https://openalex.org/W1505106535","https://openalex.org/W7213499688","https://openalex.org/W3176838514","https://openalex.org/W1987029984","https://openalex.org/W1967984121","https://openalex.org/W4387587926","https://openalex.org/W2002925081"],"abstract_inverted_index":{"We":[0],"prove":[1],"that":[2,37,58,121],"the":[3,24,38,65,69,75,107,116,122,136,150,158],"energy":[4,30,78],"functional":[5,70],"associated":[6],"with":[7,125,157],"surfaces":[8],"of":[9,68,115,129],"prescribed":[10],"mean":[11],"curvature":[12],"in":[13],"$\\mathbb{R}^3$":[14],"exhibits":[15],"a":[16,32,44,98,113,142],"convexity":[17,61,117],"property":[18],"when":[19,74],"restricted":[20],"to":[21,94,141],"maps":[22],"from":[23,103],"unit":[25,137],"$2$-disk":[26,138],"having":[27],"small":[28],"Dirichlet":[29,77],"and":[31,89],"fixed":[33],"boundary":[34,160],"value,":[35],"provided":[36],"$3$-form":[39],"$H":[40],"\\cdot":[41],"\\text{vol}_{\\mathbb{R}^3}$":[42],"has":[43],"primitive":[45],"satisfying":[46],"certain":[47],"bounds.":[48],"Under":[49],"milder":[50],"assumptions":[51],"on":[52,106,135],"$H:\\mathbb{R}^3":[53],"\\to":[54],"\\mathbb{R}$,":[55],"we":[56,119],"show":[57,120],"an":[59],"analogous":[60],"estimate":[62],"holds":[63],"along":[64],"heat":[66,110],"flow":[67,124],"(the":[71,154],"$H$-surface":[72,123,155],"flow)":[73],"initial":[76,127],"is":[79,83],"sufficiently":[80],"small.":[81],"This":[82],"done":[84],"first":[85],"for":[86],"classical":[87],"solutions,":[88],"then":[90],"extended":[91],"by":[92],"approximation":[93],"weak":[95],"solutions":[96],"using":[97],"quantitative":[99],"uniqueness":[100],"result":[101],"adapted":[102],"previous":[104],"work":[105],"harmonic":[108],"map":[109,128],"flow.":[111],"As":[112],"consequence":[114],"estimate,":[118],"small-energy":[126],"class":[130],"$C^0":[131],"\\cap":[132],"W^{1,":[133],"2}$":[134],"converges":[139],"uniformly":[140],"unique":[143],"limit":[144],"at":[145],"infinite":[146],"time,":[147],"which":[148],"solves":[149],"corresponding":[151],"stationary":[152],"problem":[153],"system)":[156],"same":[159],"value.":[161]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-29T00:00:00"}
