Bär, C. Metrics with harmonic spinors. (English) Zbl 0867.53037 Geom. Funct. Anal. 6, No. 6, 899-942 (1996). The main result is stated by the author as follows: Theorem A. Let \(M^n\) be a closed spin manifold of dimension \(n\), \(n\equiv 3\text{ mod }4\). Let a spin structure on \(M\) be fixed. Then there exists a Riemannian metric on \(M\) such that the corresponding Dirac operator has a nontrivial kernel, i.e., there are nontrivial harmonic spinors.The proof is complex and original. It involves the computation of the Dirac spectrum of the Berger spheres \(S^{2n+1}\) as well as: Theorem B. Let \(M_1\) and \(M_2\) be two closed Riemannian spin manifolds of odd dimension \(n\). Let \(D_1\) and \(D_2\) be the two Dirac operators. Let \(\varepsilon>0\) and let \(\Lambda>0\) be such that \(-\Lambda\), \(\Lambda\not\in\text{spec}(D_1)\cup\text{spec}(D_2)\). Writei) \((\text{spec}(D_1)\cup \text{spec}(D_2))\cap(-\Lambda, \Lambda)=\{\lambda_1\leq\lambda_2\leq\cdots\leq\lambda_k\}\).Then there exists a Riemannian metric on the connected sum \(M_1\# M_2\) with Dirac operator \(D\) such that the following holds:ii) \(\text{spec}(D)\cap(-\Lambda,\Lambda)= \{\mu_1\leq\mu_2\leq\cdots\leq \mu_k\}\);iii) \(|\lambda_i-\mu_i|< \varepsilon\).To prove Theorem B, the author studies the distribution of the \(L^2\)-norm of the eigenspinors over cylindrical manifolds, in particular over Euclidean annuli. Reviewer: M.Anastasiei (Iaşi) Cited in 3 ReviewsCited in 44 Documents MSC: 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) Keywords:spin structures; Dirac operator; harmonic spinors PDFBibTeX XMLCite \textit{C. Bär}, Geom. Funct. Anal. 6, No. 6, 899--942 (1996; Zbl 0867.53037) Full Text: DOI EuDML References: [1] N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. 36 (1957), 235–249. · Zbl 0084.30402 [2] M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian Geometry I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43–69. · Zbl 0297.58008 [3] M.F. Atiyah, I.M. Singer, The index of elliptic operators: III, Ann. Math. 87 (1968), 546–604. · Zbl 0164.24301 [4] M.F. Atiyah, I.M. Singer, The index of elliptic operators: IV, Ann. Math. 93 (1971), 119–138. · Zbl 0212.28603 [5] M.F. Atiyah, I.M. Singer, The index of elliptic operators: V, Ann. Math. 93 (1971), 139–149. · Zbl 0212.28603 [6] C. Bär, The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces, Arch. Math. 59 (1992), 65–79. · Zbl 0786.53030 [7] C. Bär, Lower eigenvalue estimates for Dirac operators, Math. Ann. 293 (1992), 39–46. · Zbl 0782.58048 [8] C. Bär, The Dirac operator on space forms of positive curvature, J. Math. Soc. Japan, to appear. [9] C. Bär, P. Schmutz, Harmonic spinors on Riemann surfaces, Ann. Glob. Anal. Geom. 10 (1992), 263–273. · Zbl 0763.30017 [10] B. Booss-Bavnbeck, K. Wojciechowski, Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston 1993. [11] S.E. Cappell, R. Lee, E.Y. Miller, Self adjoint elliptic operators and manifold decomposition, Part I: Low eigenmodes and stretching, Preprint. · Zbl 0871.58080 [12] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, 1984. · Zbl 0551.53001 [13] N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55. · Zbl 0284.58016 [14] K.-D. Kirchberg, An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature, Ann. Glob. Anal. Geom. 4 (1986), 291–325. · Zbl 0629.53058 [15] H.B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton University Press, Princeton, 1989. [16] A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris 257 (1963), 7–9. · Zbl 0136.18401 [17] J.W. Milnor, Remarks concerning spin manifolds, in ”Differential and Combinatorial Topology” (S. Cairns, ed.), Princeton, 1965, 55–62. [18] N.R. Wallach, Harmonic analysis on homogeneous spaces, Marcel Dekker, New York, 1973. · Zbl 0265.22022 [19] D.P. Želobenko, Compact Lie Groups and Their Representations, AMS, Providence, 1973. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.