Abstract
In this article, M is a convex domain of a regular geodesic ball of a smooth Riemannian manifold (N, g). We prove that, when the radius of the ball is small enough, M contains an unique ellipsoid of maximal volume. That is a generalization of John’s theorem to Riemannian manifolds. Then, we use these results to obtain an upper bound and a lower bound for the first non-zero eigenvalue of the Hodge Laplacian acting on differential p-forms defined on M.
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Kouassy, W.G., Kourouma, M. Estimates for the first non-zero eigenvalue of the Hodge Laplacian acting on differential forms defined on a Riemannian manifold. Afr. Mat. 31, 333–366 (2020). https://doi.org/10.1007/s13370-019-00727-7
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DOI: https://doi.org/10.1007/s13370-019-00727-7
Keywords
- John’s ellipsoid
- Injectivity radius
- Regular domain
- Parallel vector fields
- Convexity
- Orthogonal section
- Hodge Laplacian
- Differential forms
- Eigenvalue estimates