Hodge-Laplace eigenvalues of convex bodies

Savo, Alessandro

doi:10.1090/S0002-9947-2010-04844-5

Hodge-Laplace eigenvalues of convex bodies
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Trans. Amer. Math. Soc. 363 (2011), 1789-1804 Request permission

Abstract:

We give upper and lower bounds of the first eigenvalue of the Hodge Laplacian acting on smooth $p$-forms on a convex Euclidean domain for the absolute and relative boundary conditions. In particular, for the absolute conditions we show that it behaves like the squared inverse of the $p$-th longest principal axis of the ellipsoid of maximal volume included in the domain (the John ellipsoid). Using John’s theorem, we then give a spectral geometric interpretation of the bounds and relate the eigenvalues with the largest volume of a $p$-dimensional section of the domain.

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