Abstract
We prove explicit upper and lower bounds for the Poisson hierarchy, the averaged L1-moment spectra \(\left \{\frac {\mathcal {A}_{k}\left ({B_{R}^{M}}\right )}{\text {vol}\left ({S_{R}^{M}}\right )}\right \}_{k=1}^{\infty }\), and the torsional rigidity \(\mathcal {A}_{1}({B^{M}_{R}})\) of a geodesic ball \({B^{M}_{R}}\) in a Riemannian manifold Mn which satisfies that the mean curvatures of the geodesic spheres \({S^{M}_{r}}\) included in it, (up to the boundary \({S^{M}_{R}}\)), are controlled by the radial mean curvature of the geodesic spheres \(S^{\omega }_{r}(o_{\omega })\) with same radius centered at the center oω of a rotationally symmetric model space \(M^{n}_{\omega }\). As a consecuence, we prove a first Dirichlet eigenvalue \(\lambda _{1}({B^{M}_{R}})\) comparison theorem and show that equality with the bound \(\lambda _{1}(B^{\omega }_{R}(o_{\omega }))\), (where \(B^{\omega }_{r}(o_{\omega })\) is the geodesic r-ball in \(M^{n}_{\omega }\)), characterizes the L1-moment spectrum \(\left \{\mathcal {A}_{k}({B^{M}_{R}})\right \}_{k=1}^{\infty }\) as the sequence \(\left \{\mathcal {A}_{k}(B^{\omega }_{R})\right \}_{k=1}^{\infty }\) and vice-versa.
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Acknowledgements
We thanks V. Gimeno his useful help. We also thanks the referee for the careful review of errata and his/her enlightening remarks.
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Work partially supported by the Research Program of University Jaume I Project UJI-B2021-08, AEI grant (FEDER) PID2020-115930GA-I00 and AICO grant AICO/2021/252.
Work partially supported by the Research Program of University Jaume I Project UJI-B2021-08, AEI grant (FEDER) PID2020-115930GA-I00, predoctoral fellowship GVA-ESF ACIF/2019/096 and AICO grant AICO/2021/252.
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Palmer, V., Sarrión-Pedralva, E. First Dirichlet Eigenvalue and Exit Time Moment Spectra Comparisons. Potential Anal 60, 489–531 (2024). https://doi.org/10.1007/s11118-022-10058-1
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DOI: https://doi.org/10.1007/s11118-022-10058-1