Cheng's eigenvalue comparison theorem
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Cheng's eigenvalue comparison theorem generally states that the larger the domain (or measure), the smaller the first eigenvalue.[1] The 1975 theorem is named after S.Y. Cheng and, using geodesic balls, can be generalized to certain tubular domains. [2]
Theorem
S.Y. Cheng used Barta's theorem to derive the eigenvalue comparison theorem.
Let N be a Riemannian n-manifold and BN(p, r) be a geodesic ball centered at p with radius r < inj(p). Let c be the least upper bound for all sectional curvatures at BN(p, r) and let Nn(c) be the simply connected n-space form of constant sectional curvature c. Then λ1(BN(p, r)) ≥ λ1(BNn(c)(r)) is Cheng's inequality. In particular, when c = −1 and inj(p) = ∞, Cheng’s inequality becomes λ*(N) ≥ λ*(Hn(−1)) which is McKean’s inequality.[3]
See also
References
- Lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds.
- A Note on the First Eigenvalue of Spherically Symmetric Manifolds
- On Cheng's eigenvalue comparison theorem, G.P. Bessa, J.F. Montenegro.
- Riemannian foliations and eigenvalue comparison
- S.Y. Cheng, Eigenvalue Comparison Theorems and its Geometric Applications Math. Z. 143, 289{297, (1975).
- McKean, H. P.: An upper bound for the spectrum of △ on a manifold of negative curvature. J. Differ. Geom. 4, 1970, 359-366.
Notes
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