First eigenvalue of the Laplacian of a geodesic ball and area-based symmetrization of its metric tensor
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comunitat-uji-handle2:10234/7037
comunitat-uji-handle3:10234/8635
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Title
First eigenvalue of the Laplacian of a geodesic ball and area-based symmetrization of its metric tensorDate
2022Publisher
ElementISSN
1846-579XBibliographic citation
Gimeno, V., & Sarrion-Pedralva, E. (2021). First Eigenvalue of the Laplacian of a Geodesic Ball and Area-Based Symmetrization of its Metric Tensor. arXiv preprint arXiv:2103.17134.Type
info:eu-repo/semantics/articleVersion
info:eu-repo/semantics/publishedVersionAbstract
Given a Riemmanian manifold, we provide a new method to compute a sharp upper
bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of
radius less than the injectivity radius ... [+]
Given a Riemmanian manifold, we provide a new method to compute a sharp upper
bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of
radius less than the injectivity radius of the manifold. This upper bound is obtained by transforming the metric tensor into a rotationally symmetric metric tensor that preserves the area of
the geodesic spheres. The provided upper bound can be computed using only the area function
of the geodesic spheres contained in the geodesic ball and it is sharp in the sense that the first
eigenvalue of geodesic ball coincides with our upper bound if and only if the mean curvature
pointed inward of each geodesic sphere is a radial function. [-]
Is part of
Journal of Mathematical Inequalities. Volume 16, Number 1 (2022), 371–391Funder Name
Universitat Jaume I | Ministerio de Economía, Industria y Competitividad | Generalitat Valenciana
Project code
UJI-B2018-35 | MTM2017-84851-C2-2-P | PID2020-115930GA-I00 | MCIN/ AEI /10.13039/501100011033 | ACIF-2019-096
Rights
© Element, Zagreb
Paper JMI-16-28
info:eu-repo/semantics/openAccess
info:eu-repo/semantics/openAccess
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