The smallest Laplace eigenvalue of homogeneous 3-spheres
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2018
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916654654423040 |
|---|---|
| author | Lauret, Emilio A. |
| author_facet | Lauret, Emilio A. |
| contents | We establish an explicit expression for the smallest non-zero eigenvalue of the Laplace--Beltrami operator on every homogeneous metric on the 3-sphere, or equivalently, on SU(2) endowed with left-invariant metric. For the subfamily of 3-dimensional Berger spheres, we obtain a full description of their spectra. We also give several consequences of the mentioned expression. One of them improves known estimates for the smallest non-zero eigenvalue in terms of the diameter for homogeneous 3-spheres. Another application shows that the spectrum of the Laplace--Beltrami operator distinguishes up to isometry any left-invariant metric on SU(2). It is also proved the non-existence of constant scalar curvature metrics conformal and arbitrarily close to any non-round homogeneous metric on the 3-sphere. All of the above results are extended to left-invariant metrics on SO(3), that is, homogeneous metrics on the 3-dimensional real projective space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1801_04259 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | The smallest Laplace eigenvalue of homogeneous 3-spheres Lauret, Emilio A. Differential Geometry Spectral Theory 58J50 (Primary) 35P15, 22C05, 58J53, 53C30, 58C40 (Secondary) We establish an explicit expression for the smallest non-zero eigenvalue of the Laplace--Beltrami operator on every homogeneous metric on the 3-sphere, or equivalently, on SU(2) endowed with left-invariant metric. For the subfamily of 3-dimensional Berger spheres, we obtain a full description of their spectra. We also give several consequences of the mentioned expression. One of them improves known estimates for the smallest non-zero eigenvalue in terms of the diameter for homogeneous 3-spheres. Another application shows that the spectrum of the Laplace--Beltrami operator distinguishes up to isometry any left-invariant metric on SU(2). It is also proved the non-existence of constant scalar curvature metrics conformal and arbitrarily close to any non-round homogeneous metric on the 3-sphere. All of the above results are extended to left-invariant metrics on SO(3), that is, homogeneous metrics on the 3-dimensional real projective space. |
| title | The smallest Laplace eigenvalue of homogeneous 3-spheres |
| topic | Differential Geometry Spectral Theory 58J50 (Primary) 35P15, 22C05, 58J53, 53C30, 58C40 (Secondary) |
| url | https://arxiv.org/abs/1801.04259 |