{"id":"https://openalex.org/W7128366151","doi":"https://doi.org/10.48550/arxiv.2602.06468","title":"Extremal Alexandrov estimates: singularities, obstacles, and stability","display_name":"Extremal Alexandrov estimates: singularities, obstacles, and stability","publication_year":2026,"publication_date":"2026-02-06","ids":{"openalex":"https://openalex.org/W7128366151","doi":"https://doi.org/10.48550/arxiv.2602.06468"},"language":null,"primary_location":{"id":"pmh:oai:arXiv.org:2602.06468","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2602.06468","pdf_url":"https://arxiv.org/pdf/2602.06468","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/2602.06468","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5071379037","display_name":"Tianling Jin","orcid":"https://orcid.org/0000-0002-6739-1101","observed_orcids":["https://orcid.org/0000-0002-6739-1101"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jin, Tianling","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014215121","display_name":"Xushan Tu","orcid":"https://orcid.org/0000-0001-8284-7077","observed_orcids":["https://orcid.org/0000-0001-8284-7077"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tu, Xushan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5148935051","display_name":"Jingang Xiong","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xiong, Jingang","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":false,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.4616999924182892,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.4616999924182892,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.37619999051094055,"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/T11822","display_name":"Analytic and geometric function theory","score":0.03020000085234642,"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/monge-ampere-measure","display_name":"Monge-Amp\u00e8re measure","score":0.8939999938011169},{"id":"https://openalex.org/keywords/convex-functions","display_name":"convex functions","score":0.949999988079071},{"id":"https://openalex.org/keywords/extremal-functions","display_name":"extremal functions","score":0.5699999928474426},{"id":"https://openalex.org/keywords/monge-ampere-equation","display_name":"Monge-Amp\u00e8re equation","score":0.7710000276565552},{"id":"https://openalex.org/keywords/isolated-singularities","display_name":"isolated singularities","score":0.8799999952316284},{"id":"https://openalex.org/keywords/total-variation","display_name":"total variation","score":0.550000011920929},{"id":"https://openalex.org/keywords/sharp-asymptotics","display_name":"sharp asymptotics","score":0.43799999356269836},{"id":"https://openalex.org/keywords/rigidity","display_name":"rigidity","score":0.5},{"id":"https://openalex.org/keywords/logarithmic-corrections","display_name":"logarithmic corrections","score":0.5619999766349792}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8271999955177307},{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.8156999945640564},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6097999811172485},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5764999985694885},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5515999794006348},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.5062999725341797},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.46790000796318054},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.429500013589859},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4293000102043152},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3871999979019165},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.38350000977516174},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.37929999828338623},{"id":"https://openalex.org/C27851653","wikidata":"https://www.wikidata.org/wiki/Q829653","display_name":"Bounded variation","level":2,"score":0.36959999799728394},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3668999969959259},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.3589000105857849},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.3424000144004822},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.33160001039505005},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.31369999051094055},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.31139999628067017},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3003999888896942},{"id":"https://openalex.org/C66690126","wikidata":"https://www.wikidata.org/wiki/Q2914302","display_name":"Concave function","level":3,"score":0.2969000041484833},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.2944999933242798},{"id":"https://openalex.org/C157709441","wikidata":"https://www.wikidata.org/wiki/Q1411887","display_name":"Uniform convergence","level":3,"score":0.2912999987602234},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.2741999924182892},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.26969999074935913},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.2694000005722046},{"id":"https://openalex.org/C75438885","wikidata":"https://www.wikidata.org/wiki/Q3403615","display_name":"Large deviations theory","level":2,"score":0.2671000063419342},{"id":"https://openalex.org/C15964574","wikidata":"https://www.wikidata.org/wiki/Q7295534","display_name":"Rate function","level":3,"score":0.26019999384880066},{"id":"https://openalex.org/C2777634741","wikidata":"https://www.wikidata.org/wiki/Q768993","display_name":"Wasserstein metric","level":2,"score":0.25859999656677246},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.25189998745918274}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2602.06468","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2602.06468","pdf_url":"https://arxiv.org/pdf/2602.06468","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.2602.06468","doi":"https://doi.org/10.48550/arxiv.2602.06468","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2602.06468","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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2602.06468","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2602.06468","pdf_url":"https://arxiv.org/pdf/2602.06468","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":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"The":[0],"classical":[1],"Alexandrov":[2],"estimate":[3,237],"controls":[4],"the":[5,12,29,40,52,58,63,87,91,101,107,115,144,150,183,199,202,235,242,248,251],"oscillation":[6],"of":[7,14,25,57,65,90,93,140,191,256],"a":[8,33,119,138,176,187,231],"convex":[9,23,84],"function":[10,205],"by":[11,73,86,160,170],"mass":[13,109],"its":[15],"associated":[16],"Monge-Amp\u00e8re":[17,67,95,163,173],"measure":[18,41,243],"and":[19,71,147,153,169,201,208],"yields,":[20],"for":[21,129],"two":[22,74,83,193],"functions":[24,59,85],"$n$":[26],"variables":[27],"with":[28,36,165,175,220],"same":[30],"boundary":[31],"values,":[32],"sup-norm":[34],"bound":[35],"exponent":[37,47],"$1/n$":[38],"in":[39,51,62,97,106,112,149,225],"discrepancy.":[42],"We":[43,77,142],"show":[44],"that":[45,155,198],"this":[46],"is":[48,60,104,137,238],"not":[49],"optimal":[50,102,116,128],"small-discrepancy":[53],"regime":[54],"once":[55],"one":[56],"non-degenerate":[61,203],"sense":[64],"having":[66],"density":[68],"bounded":[69],"above":[70],"below":[72],"positive":[75],"constants.":[76,223],"prove":[78,154],"sharp":[79,184,213],"quantitative":[80],"estimates":[81,185],"comparing":[82],"total":[88],"variation":[89],"difference":[92],"their":[94],"measures:":[96],"dimensions":[98,226],"$n\\ge":[99,227],"3$":[100,228],"dependence":[103,117],"quadratic":[105],"natural":[108,252],"scale,":[110,253],"while":[111],"dimension":[113],"$n=2$":[114],"contains":[118],"logarithmic":[120],"correction.":[121],"These":[122],"rates":[123],"are":[124,157,206],"shown":[125],"to":[126,162,172,186],"be":[127],"all":[130],"small":[131],"discrepancies.":[132],"A":[133],"key":[134],"structural":[135],"ingredient":[136],"characterization":[139],"extremizers.":[141],"identify":[143],"pointwise":[145,214,236],"minimizers":[146],"maximizers":[148],"admissible":[151],"class":[152],"they":[156],"realized,":[158],"respectively,":[159],"solutions":[161,171],"equations":[164,174],"an":[166],"isolated":[167],"singularity":[168],"linear":[177],"obstacle.":[178],"This":[179],"extremal":[180],"description":[181],"reduces":[182],"precise":[188],"asymptotic":[189],"analysis":[190],"these":[192],"model":[194],"configurations.":[195,258],"Assuming":[196],"further":[197],"domain":[200],"reference":[204],"$C^{2,\u03b1}$":[207],"uniformly":[209],"convex,":[210],"we":[211,229],"obtain":[212],"two-sided":[215],"asymptotics":[216],"at":[217,250],"interior":[218],"points":[219],"explicit":[221],"leading":[222],"Finally,":[224],"establish":[230],"stability":[232],"phenomenon:":[233],"if":[234],"nearly":[239],"saturated,":[240],"then":[241],"discrepancy":[244],"must":[245],"concentrate":[246],"near":[247],"point":[249],"quantifying":[254],"rigidity":[255],"almost-extremal":[257]},"counts_by_year":[],"updated_date":"2026-09-26T08:19:03.415552","created_date":"2026-02-10T00:00:00"}