Vafa-witten Estimates for Compact Symmetric Spaces

We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rkG− rkH ≤ 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement. Herzlich gave an optimal upper bound for the lowest eigenvalue of the Dirac operator on spheres with arbitrary Riemannian metrics in [9] using a method developed by Vafa and Witten in [14]. More precisely, he proved that for every metric ḡ on Sn that is pointwise larger than the round metric g, the first eigenvalue λ1(D̄) of the Dirac operator with respect to ḡ is not larger than the first Dirac eigenvalue λ1(D) of the round sphere. Herzlich asked if there are other Riemannian manifolds with optimal Vafa-Witten bounds, in particular if the Fubini-Study metric on CP 2m−1 has this property. In the present note we give positive answers to both questions by generalising Herzlich’s results to symmetric spaces G/H of compact type, where rkG − rkH ≤ 1. In particular, we improve a recent estimate by Davaux and Min-Oo for complex projective spaces in [4], see Example 6.2 below. 1. Theorem. Let M = G/H be a simply connected symmetric space of compact type with rkG − rkH ≤ 1 and assume that M is G-spin. Let g be a symmetric metric, and let D denote the corresponding Dirac operator on M . If ḡ is another metric with ḡ ≥ g on TM and D̄ is the corresponding Dirac operator, then λ1(D̄) ≤ λ1(D) . In the case of equality, we have ḡ = g. For an arbitrary Riemannian metric ḡ such that c2ḡ ≥ g for some suitable positive constant c2, the theorem implies λ1(D̄) ≤ c λ1(D) . We combine the methods of [9] and [4] with related estimates in [7]. In particular, we compare D̄ to an operator D̄1 with nonvanishing kernel acting on the same sections as D̄0 = D̄ ⊗ idRk . We use Parthasarathy’s formula to compute λ1(D), and we exhibit a similar formula to estimate ∥∥D̄1 − D̄∥∥. Both formulas give the same value for g = ḡ. To prove that D̄1 has a kernel, we use the invariance of the Fredholm index if rkH = rkG. If rkH = rkG − 1 we use the invariance of the mod-2-index as in [7]. Unfortunately, both approaches fail if rkG−rkH ≥ 2. Note that in [9], a spectral flow argument was used instead in the case rkH = rkG− 1. In [2], Baum applied the Vafa-Witten approach to Lipschitz maps f of high degree from a closed Riemannian spin manfifold of dimension 2n to S2n. We extend her result to Lipschitz maps of high Â-degree from higher dimensional closed Riemannian spin manifolds to S2n. Recall that if Nn and Mm are closed oriented manifolds, [N ] is the fundamental class of N 2000 Mathematics Subject Classification. 53C27; 53C35; 58J50.