New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929751717838848 |
|---|---|
| author | Antonelli, Gioacchino Xu, Kai |
| author_facet | Antonelli, Gioacchino Xu, Kai |
| contents | We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold $(M,g)$ of dimension $n\geq3$ satisfies $$ λ_1\left(-\frac{n-1}{n-2}Δ+\mathrm{Ric}\right)\geq n-1, $$ then $\operatorname{vol}(M)\leq\operatorname{vol}(\mathbb S^{n})$, and $π_1(M)$ is finite. The constant $\frac{n-1}{n-2}$ cannot be improved, and if $\mathrm{vol}(M)=\mathrm{vol}(\mathbb S^n)$ holds, then $M\cong \mathbb S^{n}$. A sharp generalization of the Bonnet--Myers theorem is also shown under the same spectral condition.
The proofs involve the use of a new unequally weighted isoperimetric problem, and unequally warped $μ$-bubbles. As an application, in dimensions $3\leq n\leq 5$, we infer sharp results on the isoperimetric structure at infinity of complete manifolds with nonnegative Ricci curvature and uniformly positive spectral biRicci curvature.
Furthermore, the main result of this paper is applied in Mazet's recent solution of the stable Bernstein problem in $\mathbb R^6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08918 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry Antonelli, Gioacchino Xu, Kai Differential Geometry Analysis of PDEs We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold $(M,g)$ of dimension $n\geq3$ satisfies $$ λ_1\left(-\frac{n-1}{n-2}Δ+\mathrm{Ric}\right)\geq n-1, $$ then $\operatorname{vol}(M)\leq\operatorname{vol}(\mathbb S^{n})$, and $π_1(M)$ is finite. The constant $\frac{n-1}{n-2}$ cannot be improved, and if $\mathrm{vol}(M)=\mathrm{vol}(\mathbb S^n)$ holds, then $M\cong \mathbb S^{n}$. A sharp generalization of the Bonnet--Myers theorem is also shown under the same spectral condition. The proofs involve the use of a new unequally weighted isoperimetric problem, and unequally warped $μ$-bubbles. As an application, in dimensions $3\leq n\leq 5$, we infer sharp results on the isoperimetric structure at infinity of complete manifolds with nonnegative Ricci curvature and uniformly positive spectral biRicci curvature. Furthermore, the main result of this paper is applied in Mazet's recent solution of the stable Bernstein problem in $\mathbb R^6$. |
| title | New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2405.08918 |