ERRATUM TO EIGENVALUE PINCHING FOR RIEMANNIAN VECTOR BUNDLES

PETER PETERSEN AND CHAD SPROUSE

Lemma 3.1 of [3] is incorrect. The correct statement is the following.

Lemma 0.1. Let u be a function on a compact manifold with Ric ≥ −k^{2}, diam < D such that Δu ≤ αu + β. Then there exist K = K(n, k, D, α, β) and r, with e^{−(n−2) ln(2)/2} < r ≤ 1, such that ∥u∥_{∞} ≤ (K∥u∥_{2})^{r}.

Proof. From [4, eq. (3.8)],

(1)  ∥u∥2p_{\frac{2np}{n−2}} ≤ C_{S} \frac{p^{2}}{2p − 1} (α∥u∥_{2p}^{2p} + β∥u∥_{2p}^{2p−1}) + ∥u∥_{2p}^{2p},

Where C_{S} = C(n, k, D).

Let p_{i} = (\frac{n}{n−2})^{i}. Note that for any A_{i}, B_{i} > 0 we have that (A_{i}+B_{i}) min\{∥u∥_{2p_{i}}^{2p_{i}}, ∥u∥_{2p_{i}}^{2p_{i}−1}\} ≤

A_{i}∥u∥_{2p_{i}}^{2p_{i}} +B_{i}∥u∥_{2p_{i}}^{2p_{i}−1} ≤ (A_{i}+B_{i}) max{∥u∥_{2p_{i}}^{2p_{i}}, ∥u∥_{2p_{i}}^{2p_{i}−1}}. In particular for for each i we can find r_{i} such that 2p_{i} − 1 < r_{i} < 2p_{i} and (2)

(αC_{S} \frac{p_{i}^{2}}{2p_{i} − 1} + 1) ∥u∥_{2p_{i}}^{2p_{i}}+(βC_{S} \frac{p_{i}^{2}}{2p_{i} − 1}) ∥u∥_{2p_{i}}^{2p_{i}−1} = ((α + β)C_{S} \frac{p_{i}^{2}}{2p_{i} − 1} + 1) ∥u∥_{2p_{i}}^{r_{i}}

Therefore,

(3)  ∥u∥_{\frac{2np_{i}}{n−2}} ≤ ((α + β)C_{S} \frac{p_{i}^{2}}{2p_{i} − 1} + 1)^{\frac{1}{2p_{i}}} ∥u∥2p_{i}^{\frac{r_{i}}{2p_{i}}}

Iterating gives

(4)  ∥u∥_{∞} ≤ (∏_{i=1}^{∞} ((α + β)C_{S} \frac{p_{i}^{2}}{2p_{i} − 1} + 1)^{\frac{1}{2p_{i}}} ∥u∥2^{\frac{r_{0}}{2}} )^{∏_{i=1}^{∞}(r_{i}/2p_{i})}

The first product above is well-known to converge to a finite value. Let r = ∏_{i=0}^{∞}(r_{i}/2p_{i}). Since r_{i} ≤ 2p_{i} we clearly have r ≤ 1. To show that r > 0, we use that r_{i} ≥ 2p_{i} −1 which gives r ≥ ∏_{i=0}^{∞}(1− \frac{1}{2p_{i}}) = ∏_{i=0}^{∞}(1− \frac{1}{2}(\frac{n−2}{n})^{i}). Taking a logarithm and estimating by a geometric series shows that this product lies in the interval [e^{−(n−2) ln 2/2}, e^{−(n−2)/4}]. □

This gives the following which is sufficient for the applications in [3], [4]. Let E be a Riemannian vector bundle over M with |R^{E}|, |∇R^{E}| < K, Ric_{M} ≥ −k^{2}, diam(M) < D. Then for L^{2}-orthonormal eigensections S_{1}, ..., S_{m} of E with eigenvalues λ_{1}, ..., λ_{m}, the formulas

∥⟨S_{i}, S_{j}⟩ − δ_{ij}∥_{∞} ≤ τ(λ_{m}|n, k, K, R)
∥∇S_{i}∥_{∞} ≤ τ(λ_{m}|n, k, K, R)

hold, where τ → 0 as λ_{m} → 0.

See [1], [2] for arguments which lead to similar results without requiring a uniform bound on ∇R^{E}. We would like to thank W. Ballman for pointing out this error to us.

References

[1] W. Ballman, J. Bruning, and G. Carron, Eigenvalues and holonomy, preprint.

[2] B. Colbois, P. Ghanaat, and E. Ruh, Curvature and a spectral characterization of nilmanifolds, preprint.

[3] P. Petersen and C. Sprouse, Eigenvalue pinching for Riemannian vector bundles, J. Reine Angew. Math. 511 (1999), 73-86.

[4] P. Petersen and C. Sprouse, Eigenvalue pinching on p-forms, Proceedings of the Fifth Pacific Rim Geometry Conference, Tohoku Math. Pub. 20 (2001), 139-145.