Cheng's eigenvalue comparison theorem

Let M be a Riemannian manifold with dimension n, and let B_M(p, r) be a geodesic ball centered at p with radius r less than the injectivity radius of p ∈ M. For each real number k, let N(k) denote the simply connected space form of dimension n and constant sectional curvature k. Cheng's eigenvalue comparison theorem compares the first eigenvalue λ₁(B_M(p, r)) of the Dirichlet problem in B_M(p, r) with the first eigenvalue in B_N(k)(r) for suitable values of k. There are two parts to the theorem:

• Suppose that K_M, the sectional curvature of M, satisfies

${\displaystyle K_{M}\leq k.}$

Then

${\displaystyle \lambda _{1}\left(B_{N(k)}(r)\right)\leq \lambda _{1}\left(B_{M}(p,r)\right).}$

The second part is a comparison theorem for the Ricci curvature of M:

• Suppose that the Ricci curvature of M satisfies, for every vector field X,

${\displaystyle \operatorname {Ric} (X,X)\geq k(n-1)|X|^{2}.}$

Then, with the same notation as above,

${\displaystyle \lambda _{1}\left(B_{N(k)}(r)\right)\geq \lambda _{1}\left(B_{M}(p,r)\right).}$

S.Y. Cheng used Barta's theorem to derive the eigenvalue comparison theorem. As a special case, if k = −1 and inj(p) = ∞, Cheng’s inequality becomes λ^*(N) ≥ λ^*(H ⁿ(−1)) which is McKean’s inequality.[2]

• Comparison theorem
• Eigenvalue comparison theorem