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Theoretical analysis demonstrates lower bounds for reciprocal Laplacian eigenvalues across domains and surfaces, expanding classical spectral geometry bounds via conformal volume.
We establish lower bounds for sums of reciprocals of Laplacian eigenvalues for bounded Euclidean domains, spheres and surfaces. For bounded Euclidean domains, we recover a sharp inequality that extends Bucur and Henrot’s upper bound on the second Neumann eigenvalue. For spheres and surfaces, we obtain new Li- and Yau-type bounds and we discuss them in terms of conformal volume.
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Mehdi Eddaoudi (2026) studied this question.
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