Laplacian and spectral gap in regular Hilbert geometries - Barthelmé, Thomas; Colbois, Bruno; Crampon, Mickael; Verovic, Patrick (My University, 2014)
We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with C2 boundaries. We show that for an n-dimensional geometry, the spectral gap is bounded above by (n−1)2/4, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues.