Year: 2014
Type: preprint
Abstract: On a closed weighted Riemannian manifold with nonnegative Bakry-Émery Ricci curvature, it is shown that the ratio of the $k$-th to first eigenvalues of the weighted Laplacian is dominated by $641k^2$, using an argument via the Cheeger constant. While improving the previous exponential upper bound, the order of $k$ here is optimal, and hence answers an open question of Funano. This approach works still on a compact finite-dimensional Alexandrov space of nonnegative curvature and proves affirmatively a conjecture of Funano and Shioya asserting a dimension free upper bound for eigenvalue ratios in that setting.
Source: arXiv (Cornell University)
Author: Liu, Shiping
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G6087763000
OpenAlex ID: W170146992