A pinching theorem for the first eigenvalue of the laplacian on hypersurface of the euclidean space
| dc.creator | Colbois, Bruno | |
| dc.creator | Grosjean, Jean-Francois | |
| dc.date | 2006-09-18 | |
| dc.date.accessioned | 2026-07-07T07:39:43Z | |
| dc.date.available | 2026-07-07T07:39:43Z | |
| dc.description | In this paper, we give pinching Theorems for the first nonzero eigenvalue $λ$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is 1 then, for any $ε>0$, there exists a constant $C\_ε$ depending on the dimension $n$ of $M$ and the $L\_{\infty}$-norm of the mean curvature $H$, so that if the $L\_{2p}$-norm $\|H\|\_{2p}$ ($p\geq 2$) of $H$ satisfies $n\|H\|\_{2p}-C\_ε<λ$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/λ)^{1/2}$ is smaller than $ε$. Furthermore, we prove that if $C$ is a small enough constant depending on $n$ and the $L\_{\infty}$-norm of the second fundamental form, then the pinching condition $n\|H\|\_{2p}-C<\la$ implies that $M$ is diffeomorphic to an $n$-dimensional sphere. | |
| dc.identifier | https://arxiv.org/abs/math/0609494 | |
| dc.identifier | http://arxiv.org/abs/math/0609494 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/121559 | |
| dc.subject | Differential Geometry | |
| dc.subject | 53A07, 53C21 | |
| dc.title | A pinching theorem for the first eigenvalue of the laplacian on hypersurface of the euclidean space | |
| dc.type | text |