2020
DOI: 10.1016/j.aim.2020.107113
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Fundamental tones of clamped plates in nonpositively curved spaces

Abstract: We study Lord Rayleigh's problem for clamped plates on an arbitrary n-dimensional (n ≥ 2) Cartan-Hadamard manifold (M, g) with sectional curvature K ≤ −κ 2 for some κ ≥ 0. We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in (M, g) is universally bounded from below by (n−1) 4 16 κ 4 whenever the κ-Cartan-Hadamard conjecture holds on (M, g), e.g. in 2-and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2-and 3-dimensions we prove sharp isoperimetric … Show more

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Cited by 8 publications
(6 citation statements)
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“…Let a, b ≥ 0 be real numbers that verify (4.29). We note that the infimum in (4.30) is a minimum; this fact can be stated, by using a similar argument as in the flat and negatively curved cases studied by Ashbaugh and Benguria [AB95] and Kristály [Kri20], respectively. Accordingly, the value…”
Section: Coupled Minimization On Spherical Capssupporting
confidence: 56%
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“…Let a, b ≥ 0 be real numbers that verify (4.29). We note that the infimum in (4.30) is a minimum; this fact can be stated, by using a similar argument as in the flat and negatively curved cases studied by Ashbaugh and Benguria [AB95] and Kristály [Kri20], respectively. Accordingly, the value…”
Section: Coupled Minimization On Spherical Capssupporting
confidence: 56%
“…The paper is devoted to the analogue of Lord Rayleigh's conjecture, concerning the lowest principal frequency of vibrating clamped plates on positively curved spaces. Our results can be viewed as the concluding piece in the theory of clamped plates after the seminal works of Ashbaugh and Benguria [AB95] and Nadirashvili [Nad95] in Euclidean spaces, and the recent paper by the author [Kri20] on non-positively curved spaces, all valid in dimensions 2 and 3. To our surprise, positively curved spaces provide an appropriate geometric setting for the validity of Lord Rayleigh's conjecture not only in dimensions 2 and 3 for any clamped plate, but also in dimensions beyond 3 for sufficiently large domains.…”
Section: Introductionmentioning
confidence: 69%
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“…This problem has been settled later, in the 1990s, by Nadirashvili [19] and Ashbaugh and Banguria [3], in dimensions two and three, in the Euclidean setting, and Ashbaugh and Laugesen [4] for higher dimensions. Furthermore, very recently, Kristaly [18] handled the problem of clamped plates on Riemannian manifolds with negative curvature. For the eigenvalue problem of the clamped plate problem, some interesting inequalities have been established at works [5]- [7], [15], [16], [21], and [25], we present some of them in more detail below.…”
Section: Introduction and The Main Resultsmentioning
confidence: 99%