caa a22 4500 463248287 CHVBK 20180405153332.0 cr unu---uuuuu 170326e20070601xx s 000 0 eng 10.1007/s10455-006-9048-2 doi (NATIONALLICENCE)springer-10.1007/s10455-006-9048-2 Manifolds with small Dirac eigenvalues are nilmanifolds [Elektronische Daten] [Bernd Ammann, Chad Sprouse] Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and bounded diameter, and almost non-negative scalar curvature. Let r=1 if n=2,3 and r=2[n/2]-1+1 if n≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost non-negative scalar curvature has one small Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface. Springer Science + Business Media B.V., 2006 Dirac operator nationallicence Nilmanifold nationallicence Small eigenvalues nationallicence Almost positive scalar curvature nationallicence Ammann Bernd Institut Élie Cartan, Université Henri Poincaré, Nancy I, B.P. 239, 54506, Vandoeuvre-Lès-Nancy, Cedex, France aut Sprouse Chad Department of Mathematics, CSU Northridge, 18111 Nordhoff Street, 91330-8313, Northridge, CA, USA aut Annals of Global Analysis and Geometry Kluwer Academic Publishers 31/4(2007-06-01), 409-425 0232-704X 31:4<409 2007 31 10455 https://doi.org/10.1007/s10455-006-9048-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10455-006-9048-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Ammann Bernd Institut Élie Cartan, Université Henri Poincaré, Nancy I, B.P. 239, 54506, Vandoeuvre-Lès-Nancy, Cedex, France aut NATIONALLICENCE 700 1- Sprouse Chad Department of Mathematics, CSU Northridge, 18111 Nordhoff Street, 91330-8313, Northridge, CA, USA aut NATIONALLICENCE 773 0- Annals of Global Analysis and Geometry Kluwer Academic Publishers 31/4(2007-06-01), 409-425 0232-704X 31:4<409 2007 31 10455 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer