{"id":"https://openalex.org/W7214801555","doi":"https://doi.org/10.48550/arxiv.2609.33789","title":"Instability and Morse Index of Exponentially Subelliptic Harmonic Maps","display_name":"Instability and Morse Index of Exponentially Subelliptic Harmonic Maps","publication_year":2026,"publication_date":"2026-09-27","ids":{"openalex":"https://openalex.org/W7214801555","doi":"https://doi.org/10.48550/arxiv.2609.33789"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.33789","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33789","pdf_url":"https://arxiv.org/pdf/2609.33789","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/2609.33789","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101587748","display_name":"Xin Huang","orcid":"https://orcid.org/0000-0003-0967-3909","observed_orcids":["https://orcid.org/0000-0003-0967-3909","https://orcid.org/0000-0002-5625-0338"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Huang, Xin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5101587748"],"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":0.9423999786376953,"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":0.9423999786376953,"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":0.04610000178217888,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.005100000184029341,"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/morse-index","display_name":"Morse index","score":0.7879999876022339},{"id":"https://openalex.org/keywords/index-form","display_name":"index form","score":0.7960000038146973},{"id":"https://openalex.org/keywords/round-spheres","display_name":"round spheres","score":0.6579999923706055},{"id":"https://openalex.org/keywords/convex-hypersurfaces","display_name":"convex hypersurfaces","score":0.9470000267028809}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8069000244140625},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6327000260353088},{"id":"https://openalex.org/C130432447","wikidata":"https://www.wikidata.org/wiki/Q11420049","display_name":"Quartic function","level":2,"score":0.5228999853134155},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4961000084877014},{"id":"https://openalex.org/C54207081","wikidata":"https://www.wikidata.org/wiki/Q937750","display_name":"Nabla symbol","level":3,"score":0.44020000100135803},{"id":"https://openalex.org/C140031139","wikidata":"https://www.wikidata.org/wiki/Q79897","display_name":"Morse code","level":2,"score":0.4375},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.4323999881744385},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.39559999108314514},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.38909998536109924},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.3828999996185303},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.3817000091075897},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3698999881744385},{"id":"https://openalex.org/C207821765","wikidata":"https://www.wikidata.org/wiki/Q405372","display_name":"Instability","level":2,"score":0.34700000286102295},{"id":"https://openalex.org/C55350006","wikidata":"https://www.wikidata.org/wiki/Q237193","display_name":"Exponential distribution","level":2,"score":0.3458999991416931},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.34470000863075256},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.3292999863624573},{"id":"https://openalex.org/C156004811","wikidata":"https://www.wikidata.org/wiki/Q2228257","display_name":"Multiplicity (mathematics)","level":2,"score":0.320499986410141},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.3005000054836273},{"id":"https://openalex.org/C2777382242","wikidata":"https://www.wikidata.org/wiki/Q6017816","display_name":"Index (typography)","level":2,"score":0.29809999465942383},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.29490000009536743},{"id":"https://openalex.org/C2777680557","wikidata":"https://www.wikidata.org/wiki/Q7259417","display_name":"Pullback","level":2,"score":0.2867000102996826},{"id":"https://openalex.org/C85856720","wikidata":"https://www.wikidata.org/wiki/Q900653","display_name":"Heisenberg picture","level":3,"score":0.2793999910354614},{"id":"https://openalex.org/C627467","wikidata":"https://www.wikidata.org/wiki/Q599027","display_name":"Harmonic function","level":2,"score":0.2766000032424927},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.27250000834465027},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.2685999870300293},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.2680000066757202},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.26080000400543213},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.25999999046325684},{"id":"https://openalex.org/C11302649","wikidata":"https://www.wikidata.org/wiki/Q6927157","display_name":"Moving frame","level":3,"score":0.2554999887943268},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2542000114917755}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.33789","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33789","pdf_url":"https://arxiv.org/pdf/2609.33789","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.2609.33789","doi":"https://doi.org/10.48550/arxiv.2609.33789","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.33789","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:2609.33789","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33789","pdf_url":"https://arxiv.org/pdf/2609.33789","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/W7214801555.pdf","grobid_xml":"https://content.openalex.org/works/W7214801555.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2968455162","https://openalex.org/W2953089661","https://openalex.org/W6950837344","https://openalex.org/W754628839","https://openalex.org/W4406705024","https://openalex.org/W1179216587","https://openalex.org/W2465881596","https://openalex.org/W2549308176","https://openalex.org/W2956145090","https://openalex.org/W4388342063"],"abstract_inverted_index":{"Let":[0],"$M$":[1],"be":[2],"a":[3,7,48,56,77,85,103,111],"closed":[4,104],"manifold":[5],"with":[6,55],"bracket-generating":[8],"distribution":[9],"$H$.":[10],"We":[11],"study":[12],"the":[13,26,36,45,67,72,107,115],"Morse":[14],"index":[15,27,37,73],"of":[16,66,97],"exponentially":[17],"subelliptic":[18],"harmonic":[19],"maps.":[20],"After":[21],"establishing":[22],"finiteness,":[23],"we":[24],"estimate":[25,46],"through":[28],"two":[29],"test":[30],"spaces.":[31],"The":[32,60,90],"extrinsic":[33,53],"one":[34,62],"averages":[35],"form":[38,74,80],"over":[39],"projected":[40],"ambient":[41],"parallel":[42,64,87],"frames,":[43],"reducing":[44],"to":[47,76,93],"trace":[49],"identity":[50],"involving":[51],"an":[52],"scalar":[54],"quartic":[57],"exponential":[58],"term.":[59],"intrinsic":[61,108],"uses":[63],"sections":[65],"pullback":[68],"bundle,":[69],"on":[70],"which":[71],"descends":[75],"coupled":[78],"quadratic":[79],"that":[81],"is":[82],"exact":[83],"when":[84],"global":[86],"frame":[88],"exists.":[89],"bounds":[91],"apply":[92],"round":[94],"spheres,":[95,98],"products":[96],"and":[99],"convex":[100],"hypersurfaces.":[101],"On":[102],"Heisenberg":[105],"nilmanifold,":[106],"reduction":[109],"yields":[110],"spectral":[112],"formula":[113],"for":[114],"index.":[116]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-30T00:00:00"}