Cheng's eigenvalue comparison theorem

In Riemannian geometry, Cheng's eigenvalue comparison theorem generally states that the larger the domain (or measure), the smaller the first eigenvalue. The 1975 theorem is named after S.Y. Cheng and, using geodesic balls, can be generalized to certain tubular domains.

Theorem

S.Y. Cheng used Barta's theorem to derive the eigenvalue comparison theorem.

Let N be a Riemannian n-manifold and B_N(p, r) be a geodesic ball centered at p with radius r < inj(p), where "inj" is the injectivity radius. Let c be the least upper bound for all sectional curvatures at B_N(p, r) and let N ⁿ(c) be the simply connected n-space form of constant sectional curvature c. Then λ₁(B_N(p, r)) ≥ λ₁(B_N ⁿ(c)(r)) is Cheng's inequality. In particular, when c = −1 and inj(p) = ∞, Cheng’s inequality becomes λ^*(N) ≥ λ^*(H ⁿ(−1)) which is McKean’s inequality.

See also

• Comparison theorem
• Eigenvalue comparison theorem

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
• Bessa, G.P.; Montenegro, J.F. (2008), "On Cheng's eigenvalue comparison theorem", Mathematical proceedings of the Cambridge Philosophical Society, 144 (3): 673–682, ISSN 0305-0041
• Richardson, Ken (1998), "Riemannian foliations and eigenvalue comparison", Ann. Global Anal. Geom., 16: 497–525/
• Cheng, S.Y. (1975), "Eigenvalue Comparison Theorems and its Geometric Applications", Math. Z., 143: 289–297.
• McKean, Henry (1970), "An upper bound for the spectrum of △ on a manifold of negative curvature", J. Differ. Geom., 4: 359–366.