Geometric Spectral Theory

Responsable du projet |
Bruno Colbois |

Résumé |
The main topic of this proposal is spectral geometry on Riemannian
manifolds. The Laplacian on a Riemannian manifold and the study of the relationship between the spectrum of the Laplacian and the geometry/topology of the manifold are now classical and well-studied subjects. However, for other Laplace-type (or related) operators, the relationship between the spectrum and the geometry/topology of the underlying manifold is much less clear, and the aim of this proposal is to investigate this relationship for two operators: the Steklov (or Dirichlet-to-Neumann) operator and the magnetic Laplacian. The Steklov problem is defined on a manifold with boundary, and what we want to understand better is the influence of the geometry of the boundary and the geometry of the whole manifold on the Steklov spectrum. The two main projects of this proposal regarding the Steklov spectrum are: - the Ph.D thesis of Léonard Tschanz for whom I am asking for a fellowship. One topic of this thesis is the relationship between the spectrum on graphs and that on manifolds: in some situations, in order to obtain results for the spectrum on graphs, it seems crucial to pass via Riemannian manifolds. Another topic is the study of the spectrum of hypersurfaces of revolution in Euclidean space. - a collaboration with A. Girouard and J. Brisson, for whom I am asking for a fellowship. We want to investigate the Steklov spectrum on manifolds/domains with boundary for a particular normalization . This normalization allows us to find upper bounds in the conformal class. This is not the case with a normalization with respect to the volume of the boundary or of the manifold. The main project of this proposal regarding the magnetic Laplacian (with Neumann magnetic boundary condition) is the continuation of a collaboration with A. Savo and, more recently, with L. Provenzano and C. Léna. We will mainly focus on lower and upper bounds for 2-dimensional domains (of Euclidean space and possible generalizations to 2-dimensional surfaces) for two different magnetic Laplacians. The first has a magnetic potential with a singularity and curvature 0 (the Aharonov-Bohm potential) the second has a magnetic potential with constant, nonzero curvature. Initially, my research topics were often studied from an analytical point of view: this is due to the fact that they were (and still are) dealt with in Euclidean space where powerful analytical methods are available. The common theme of all my research work is that I seek to understand the different problems that interest me from the geometric and metric points of view. On the one hand, this allows me to deal with them in a more general framework, but also to have a point of view that differs from that of an analytical approach. |

Mots-clés |
spectral theory on Riemannian manifolds, upper and lower bounds for the spectrum, Laplace-type operators, Steklov operator, magnetic Laplacian with constant magnetic field, Aharonov-Bohm potential. |

Type de projet | Recherche fondamentale |

Domaine de recherche | Mathematics |

Etat | En cours |

Début de projet | 1-2-2023 |

Fin du projet | 31-1-2025 |

Contact | Bruno Colbois |