Revista Matemática Iberoamericana


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Volume 35, Issue 6, 2019, pp. 1835–1858
DOI: 10.4171/rmi/1103

Published online: 2019-08-02

Upper bounds for the spectral function on homogeneous spaces via volume growth

Chris Judge[1] and Russell Lyons[2]

(1) Indiana University, Bloomington, USA
(2) Indiana University, Bloomington, USA

We use spectral embeddings to give upper bounds on the spectral function of the Laplace–Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower bound of Peter Li for the smallest positive eigenvalue to all eigenvalues. We also improve Li’s bound itself. Our bounds translate to explicit upper bounds on the heat kernel for both compact and noncompact homogeneous spaces.

Keywords: Eigenvalues, Laplacian, spectral embedding, compact, noncompact, heat kernel

Judge Chris, Lyons Russell: Upper bounds for the spectral function on homogeneous spaces via volume growth. Rev. Mat. Iberoam. 35 (2019), 1835-1858. doi: 10.4171/rmi/1103