Stable eigenvalues of compact anti-de Sitter 3-manifolds
Geometric objects that have been investigated in detail so far, such as closed Riemann surfaces, are sometimes locally homogeneous. Loosely speaking, their infinitesimal behavior is the same at each point. In this talk, I would like to explain the idea of investigating such objects using the Lie group theory.In the first part of the talk, I will recall the notions of Lie group actions and their quotient spaces with examples, and then explain the definitions of locally homogeneous spaces and their deformations (Teichmüller spaces). In the second part of the talk, I will consider anti-de Sitter manifolds as a special case, i.e., Lorentzian manifolds of negative constant curvature. As in the Riemannian case, a differential operator called the Laplacian (or the Klein-Gordon operator) is defined on Lorentzian manifolds. Unlike the Riemannian case, it is no longer an elliptic differential operator but a hyperbolic differential operator. In its spectral analysis, new phenomena different from those in the Riemannian case have been discovered in recent years, following pioneering works by Toshiyuki Kobayashi and Fanny Kassel. I would like to explain stable eigenvalues of the hyperbolic Laplacian of anti-de Sitter 3-manifolds with recent progress.
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