Deriving Perelman's entropy from Colding's monotonic volume
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908451109601280 |
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| author | Bustamante, Ignacio Reiris, Martin |
| author_facet | Bustamante, Ignacio Reiris, Martin |
| contents | 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 $N$-space, which becomes Ricci-flat as $N\rightarrow \infty$, 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 $N$-space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deriving Perelman's entropy from Colding's monotonic volume Bustamante, Ignacio Reiris, Martin Differential Geometry Analysis of PDEs 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 $N$-space, which becomes Ricci-flat as $N\rightarrow \infty$, 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 $N$-space. |
| title | Deriving Perelman's entropy from Colding's monotonic volume |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2501.12949 |