Lower bounds for the eigenvalues of the Dirac operator on quaternionic Kähler spin manifolds
The paper aims to sharpen the well-known lower bound for the eigenvalues of the Dirac operator proved for compact spin manifolds by \textit{Th. Friedrich} [Math. Nachr. 97, 117-146 (1980; Zbl 0462.53027)] for compact manifolds with quaternionic-Kähler spin structures. These structures are locally defined by giving three endomorphisms \(J_\alpha\), \(\alpha=1,2,3\), on the tangent space to \(M\), compatible with the metric and establishing the quaternionic structure. The corresponding Kähler forms \(\Omega_\alpha\) define the (global) fundamental form \(\Omega=\sum_\alpha \Omega_\alpha\wedge \Omega_\alpha\). The achieved estimate for \(4m\)-dimensional manifolds has the form \(\lambda^2\geq C_{m,r}S\), where \(S\) is the scalar curvature (always constant on these structures), while \(r\) is an integer, \(0\leq r\leq m-2\), related to the eigenvalues of the operator \(\Omega-6m\) on the spinors. In fact, \(C_{m,r}\) is decreasing in \(r\) and \(C_{m,0}={m+3\over 4(m+2)}\) which is always bigger than the Friedrich's estimate \({m\over (4m-1)}\) (however very close to if \(m\) big), but the other extreme \(C_{m,m-2}\) is better only for very small \(m\). The authors conjecture that \(C_{m,0}\) should be always a bound, but they prove this for \(m=2\) only. Let us also notice the rigidity of the structures: there are only the quaternionic projective spaces for odd values of \(m\) [cf. \textit{S. Salamon}, Invent. Math. 67, 143-171 (1982; Zbl 0486.53048)]. The proofs are based on sophisticated versions of certain twistor operators \(P^a\) on spinors. Eigenspinors for these operators are studied and an optimal choice of the value of \(a\) yields the result.
scientific article; zbMATH DE number 891047