Tohoku Mathematical Journal
2014
September
SECOND SERIES VOL. 66, NO. 3

Tohoku Math. J.
66 (2014), 377-407

Title LAPLACIAN AND SPECTRAL GAP IN REGULAR HILBERT GEOMETRIES

Author Thomas Barthelmé, Bruno Colbois, Mickaël Crampon and Patrick Verovic

(Received November 29, 2012, revised July 2, 2013)
Abstract. We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ 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.

Mathematics Subject Classification. Primary 53C60; Secondary 58J60.
Key words and phrases. Hilbert geometries, Laplace operator, spectral gap.

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