Deriving Perelman's entropy from Colding's monotonic volume
Resumen:
In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop-Gromov volume comparison applied to a suitably constructed -space, which becomes Ricci-flat as , Perelman did not provide a corresponding explanation for the origin of the entropy. In this article, we demonstrate that Perelman's entropy emerges as the limit of Colding's monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman's -space.
| 2025 | |
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DIFFERENTIAL GEOMETRY ANALYSIS OF PDES |
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| Inglés | |
| Universidad de la República | |
| COLIBRI | |
| https://hdl.handle.net/20.500.12008/54970 | |
| Acceso abierto | |
| Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0) |
| _version_ | 1872864706561048576 |
|---|---|
| author | Bustamante, Ignacio |
| author2 | Reiris Ithurralde, Martín |
| author2_role | author |
| author_facet | Bustamante, Ignacio Reiris Ithurralde, Martín |
| author_role | author |
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| collection | COLIBRI |
| dc.contributor.filiacion.none.fl_str_mv | Bustamante Ignacio, Universidad de la República (Uruguay). Facultad de Ingeniería. Reiris Ithurralde Martín, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática. |
| dc.creator.none.fl_str_mv | Bustamante, Ignacio Reiris Ithurralde, Martín |
| dc.date.accessioned.none.fl_str_mv | 2026-05-13T12:03:26Z |
| dc.date.available.none.fl_str_mv | 2026-05-13T12:03:26Z |
| dc.date.issued.none.fl_str_mv | 2025 |
| dc.description.abstract.none.fl_txt_mv | In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop-Gromov volume comparison applied to a suitably constructed -space, which becomes Ricci-flat as , Perelman did not provide a corresponding explanation for the origin of the entropy. In this article, we demonstrate that Perelman's entropy emerges as the limit of Colding's monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman's -space. |
| dc.description.es.fl_txt_mv | Publicado también en: Journal für die reine und angewandte Mathematik. 2025, 826. DOI: 10.1515/crelle-2025-0034 |
| dc.format.extent.es.fl_str_mv | 26 h. |
| dc.format.mimetype.es.fl_str_mv | application/pdf |
| dc.identifier.citation.es.fl_str_mv | Bustamante, I y Reiris Ithurralde, M. "Deriving Perelman's entropy from Colding's monotonic volume". [Preprint Publicado en: Mathematics (Differential Geometry). 2025, arXiv:2501.12949, jul. 2025, pp. 1-26. DOI: 10.48550/arXiv.2501.12949 |
| dc.identifier.doi.none.fl_str_mv | 10.48550/arXiv.2501.12949 |
| dc.identifier.uri.none.fl_str_mv | https://hdl.handle.net/20.500.12008/54970 |
| dc.language.iso.none.fl_str_mv | en eng |
| dc.publisher.es.fl_str_mv | arXiv |
| dc.relation.none.fl_str_mv | Mathematics (Differential Geometry), arXiv:2501.12949, jul. 2025, pp. 1-26 |
| dc.rights.license.none.fl_str_mv | Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0) |
| dc.rights.none.fl_str_mv | info:eu-repo/semantics/openAccess |
| dc.source.none.fl_str_mv | reponame:COLIBRI instname:Universidad de la República instacron:Universidad de la República |
| dc.subject.other.es.fl_str_mv | DIFFERENTIAL GEOMETRY ANALYSIS OF PDES |
| dc.title.none.fl_str_mv | Deriving Perelman's entropy from Colding's monotonic volume |
| dc.type.es.fl_str_mv | Preprint |
| dc.type.none.fl_str_mv | info:eu-repo/semantics/preprint |
| dc.type.version.none.fl_str_mv | info:eu-repo/semantics/submittedVersion |
| description | Publicado también en: Journal für die reine und angewandte Mathematik. 2025, 826. DOI: 10.1515/crelle-2025-0034 |
| eu_rights_str_mv | openAccess |
| format | preprint |
| id | COLIBRI_4199cf09176ba60bd72aa2245d021787 |
| identifier_str_mv | Bustamante, I y Reiris Ithurralde, M. "Deriving Perelman's entropy from Colding's monotonic volume". [Preprint Publicado en: Mathematics (Differential Geometry). 2025, arXiv:2501.12949, jul. 2025, pp. 1-26. DOI: 10.48550/arXiv.2501.12949 10.48550/arXiv.2501.12949 |
| instacron_str | Universidad de la República |
| institution | Universidad de la República |
| instname_str | Universidad de la República |
| language | eng |
| language_invalid_str_mv | en |
| network_acronym_str | COLIBRI |
| network_name_str | COLIBRI |
| oai_identifier_str | oai:colibri.udelar.edu.uy:20.500.12008/54970 |
| publishDate | 2025 |
| reponame_str | COLIBRI |
| repository.mail.fl_str_mv | karina.camps@seciu.edu.uy |
| repository.name.fl_str_mv | COLIBRI - Universidad de la República |
| repository_id_str | 4771 |
| rights_invalid_str_mv | Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0) |
| spelling | Bustamante Ignacio, Universidad de la República (Uruguay). Facultad de Ingeniería.Reiris Ithurralde Martín, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática.2026-05-13T12:03:26Z2026-05-13T12:03:26Z2025Bustamante, I y Reiris Ithurralde, M. "Deriving Perelman's entropy from Colding's monotonic volume". [Preprint Publicado en: Mathematics (Differential Geometry). 2025, arXiv:2501.12949, jul. 2025, pp. 1-26. DOI: 10.48550/arXiv.2501.12949https://hdl.handle.net/20.500.12008/5497010.48550/arXiv.2501.12949Publicado también en: Journal für die reine und angewandte Mathematik. 2025, 826. DOI: 10.1515/crelle-2025-0034In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop-Gromov volume comparison applied to a suitably constructed -space, which becomes Ricci-flat as , Perelman did not provide a corresponding explanation for the origin of the entropy. In this article, we demonstrate that Perelman's entropy emerges as the limit of Colding's monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman's -space.Submitted by Egaña Lachaga Florencia (florencia.egana@fic.edu.uy) on 2026-05-11T19:10:02Z No. of bitstreams: 2 license_rdf: 27293 bytes, checksum: d62648cf14c1e37917d392ac87012955 (MD5) 2501.12949v3.pdf: 396668 bytes, checksum: 0f89a3d58e8de40c44c2bbd4827d8a02 (MD5)Approved for entry into archive by Faget Cecilia (lfaget@fcien.edu.uy) on 2026-05-13T11:47:35Z (GMT) No. of bitstreams: 2 license_rdf: 27293 bytes, checksum: d62648cf14c1e37917d392ac87012955 (MD5) 2501.12949v3.pdf: 396668 bytes, checksum: 0f89a3d58e8de40c44c2bbd4827d8a02 (MD5)Made available in DSpace by Luna Fabiana (fabiana.luna@seciu.edu.uy) on 2026-05-13T12:03:26Z (GMT). No. of bitstreams: 2 license_rdf: 27293 bytes, checksum: d62648cf14c1e37917d392ac87012955 (MD5) 2501.12949v3.pdf: 396668 bytes, checksum: 0f89a3d58e8de40c44c2bbd4827d8a02 (MD5) Previous issue date: 202526 h.application/pdfenengarXivMathematics (Differential Geometry), arXiv:2501.12949, jul. 2025, pp. 1-26Las obras depositadas en el Repositorio se rigen por la Ordenanza de los Derechos de la Propiedad Intelectual de la Universidad de la República.(Res. Nº 91 de C.D.C. de 8/III/1994 – D.O. 7/IV/1994) y por la Ordenanza del Repositorio Abierto de la Universidad de la República (Res. Nº 16 de C.D.C. de 07/10/2014)info:eu-repo/semantics/openAccessLicencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0)DIFFERENTIAL GEOMETRYANALYSIS OF PDESDeriving Perelman's entropy from Colding's monotonic volumePreprintinfo:eu-repo/semantics/preprintinfo:eu-repo/semantics/submittedVersionreponame:COLIBRIinstname:Universidad de la Repúblicainstacron:Universidad de la RepúblicaBustamante, IgnacioReiris Ithurralde, MartínLICENSElicense.txtlicense.txttext/plain; charset=utf-84267http://localhost:8080/xmlui/bitstream/20.500.12008/54970/5/license.txt6429389a7df7277b72b7924fdc7d47a9MD55CC-LICENSElicense_urllicense_urltext/plain; charset=utf-850http://localhost:8080/xmlui/bitstream/20.500.12008/54970/2/license_urla006180e3f5b2ad0b88185d14284c0e0MD52license_textlicense_texttext/html; charset=utf-835935http://localhost:8080/xmlui/bitstream/20.500.12008/54970/3/license_text00c6a163e62314ee9d2a181a90bf9e84MD53license_rdflicense_rdfapplication/rdf+xml; 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públicahttps://udelar.edu.uy/https://www.colibri.udelar.edu.uy/oai/requestkarina.camps@seciu.edu.uyUruguayopendoar:47712026-05-13T12:03:26COLIBRI - Universidad de la Repúblicafalse |
| spellingShingle | Deriving Perelman's entropy from Colding's monotonic volume Bustamante, Ignacio DIFFERENTIAL GEOMETRY ANALYSIS OF PDES |
| status_str | submittedVersion |
| title | Deriving Perelman's entropy from Colding's monotonic volume |
| title_full | Deriving Perelman's entropy from Colding's monotonic volume |
| title_fullStr | Deriving Perelman's entropy from Colding's monotonic volume |
| title_full_unstemmed | Deriving Perelman's entropy from Colding's monotonic volume |
| title_short | Deriving Perelman's entropy from Colding's monotonic volume |
| title_sort | Deriving Perelman's entropy from Colding's monotonic volume |
| topic | DIFFERENTIAL GEOMETRY ANALYSIS OF PDES |
| url | https://hdl.handle.net/20.500.12008/54970 |
