Cheng's eigenvalue comparison theorem

In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that the larger the domain, the smaller the first Dirichlet eigenvalue of the Laplace–Beltrami operator. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its curvature. The theorem is due to Cheng (1975b). Using geodesic balls, it can be generalized to certain tubular domains (Lee 1990).

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. Here λ₁ denotes the first Dirichlet eigenvalue of the Laplacian in a domain.

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

• 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.
• Chavel, Isaac (1984), Eigenvalues in Riemannian geometry, Pure Appl. Math., vol. 115, Academic Press.
• Cheng, Shiu Yuen (1975a), "Eigenfunctions and eigenvalues of Laplacian", Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2, Providence, R.I.: American Mathematical Society, pp. 185–193, MR0378003
• Cheng, Shiu Yuen (1975b), "Eigenvalue Comparison Theorems and its Geometric Applications", Math. Z., 143: 289–297.
• Lee, Jeffrey M. (1990), "Eigenvalue Comparison for Tubular Domains", Proceedings of the American Mathematical Society, 109 (3): 843–848.
• McKean, Henry (1970), "An upper bound for the spectrum of △ on a manifold of negative curvature", J. Differ. Geom., 4: 359–366.
• Richardson, Ken (1998), "Riemannian foliations and eigenvalue comparison", Ann. Global Anal. Geom., 16: 497–525/