New features of the first eigenvalue on negatively curved spaces
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Alexandru Kristály
Abstract
The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model n-dimensional hyperbolic space, complementing the results of Borisov and Freitas (2017), Hurtado, Markvorsen and Palmer (2016) and Savo (2008); in odd dimensions, such eigenvalues appear as roots of an inductively constructed transcendental equation. We then give a synthetic proof of Cheng’s sharp eigenvalue comparison theorem in metric measure spaces satisfying a Bishop–Gromov-type volume monotonicity hypothesis. As a byproduct, we provide an example of simply connected, non-compact Finsler manifold with constant negative flag curvature whose first eigenvalue is zero; this result is in a sharp contrast with its celebrated Riemannian counterpart due to McKean (1970). Our proofs are based on specific properties of the Gaussian hypergeometric function combined with intrinsic aspects of the negatively curved smooth/non-smooth spaces.
Funding source: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal
Award Identifier / Grant number: 127926
Funding statement: Research supported by the National Research, Development and Innovation Fund of Hungary, financed under the K18 funding scheme, Project No. 127926.
Acknowledgements
The author is grateful to Denis Borisov and Pedro Freitas for the conversations concerning their paper [4]. He also thanks the anonymous Referees for their valuable comments as well as Árpád Baricz, Csaba Farkas, Mihai Mihăilescu and Tibor Pogány for their help in special functions and eigenvalue problems.
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Articles in the same Issue
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Articles in the same Issue
- Frontmatter
- Michell truss type theories as a Γ-limit of optimal design in linear elasticity
- The local structure of the free boundary in the fractional obstacle problem
- Homogenization of quadratic convolution energies in periodically perforated domains
- Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow
- Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth
- On a comparison principle for Trudinger’s equation
- Equivalence of ray monotonicity properties and classification of optimal transport maps for strictly convex norms
- An extensive study of the regularity of solutions to doubly singular equations
- New features of the first eigenvalue on negatively curved spaces
- High order curvature flows of plane curves with generalised Neumann boundary conditions
- (BV,L p )-decomposition, p = 1,2, of functions in metric random walk spaces
- Causal variational principles in the σ-locally compact setting: Existence of minimizers
- On sub-Riemannian geodesic curvature in dimension three
- Existence for singular critical exponential (𝑝, 𝑄) equations in the Heisenberg group
