Volume entropy and rigidity for RCD-spaces
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913573193646080 |
|---|---|
| author | Connell, Chris Dai, Xianzhe Núñez-Zimbrón, Jesús Perales, Raquel Suárez-Serrato, Pablo Wei, Guofang |
| author_facet | Connell, Chris Dai, Xianzhe Núñez-Zimbrón, Jesús Perales, Raquel Suárez-Serrato, Pablo Wei, Guofang |
| contents | We develop the barycenter technique of Besson--Courtois--Gallot so that it can be applied on RCD metric measure spaces. Given a continuous map $f$ from a non-collapsed RCD$(-(N-1),N)$ space $X$ without boundary to a locally symmetric $N$-manifold we show a version of BCG's entropy-volume inequality. The lower bound involves homological and homotopical indices which we introduce. We prove that when equality holds and these indices coincide $X$ is a locally symmetric manifold, and $f$ is homotopic to a Riemannian covering whose degree equals the indices. Moreover, we show a measured Gromov--Hausdorff stability of $X$ and $Y$ involving the homotopical invariant. As a byproduct, we extend a Lipschitz volume rigidity result of Li--Wang to RCD$(K,N)$ spaces without boundary. Finally, we include an application of these methods to the study of Einstein metrics on $4$-orbifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04327 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Volume entropy and rigidity for RCD-spaces Connell, Chris Dai, Xianzhe Núñez-Zimbrón, Jesús Perales, Raquel Suárez-Serrato, Pablo Wei, Guofang Differential Geometry Geometric Topology Metric Geometry 53C23, 46E36 We develop the barycenter technique of Besson--Courtois--Gallot so that it can be applied on RCD metric measure spaces. Given a continuous map $f$ from a non-collapsed RCD$(-(N-1),N)$ space $X$ without boundary to a locally symmetric $N$-manifold we show a version of BCG's entropy-volume inequality. The lower bound involves homological and homotopical indices which we introduce. We prove that when equality holds and these indices coincide $X$ is a locally symmetric manifold, and $f$ is homotopic to a Riemannian covering whose degree equals the indices. Moreover, we show a measured Gromov--Hausdorff stability of $X$ and $Y$ involving the homotopical invariant. As a byproduct, we extend a Lipschitz volume rigidity result of Li--Wang to RCD$(K,N)$ spaces without boundary. Finally, we include an application of these methods to the study of Einstein metrics on $4$-orbifolds. |
| title | Volume entropy and rigidity for RCD-spaces |
| topic | Differential Geometry Geometric Topology Metric Geometry 53C23, 46E36 |
| url | https://arxiv.org/abs/2411.04327 |