Optimal eigenvalues estimate for the Dirac operator on domains with boundary.pdf

a r X i v : m a t h / 0 6 0 3 5 1 2 v 1 [ m a t h .D G ] 2 1 M a r 2 0 0 6 OPTIMAL EIGENVALUES ESTIMATE FOR THE DIRAC OPERATOR ON DOMAINS WITH BOUNDARY SIMON RAULOT Abstract. We give a lower bound for the eigenvalues of the Dirac operator on a compact domain of a Riemannian spin manifold under the MIT bag boundary condition. The limiting case is characterized by the existence of an imaginary Killing spinor. 1. Introduction Let ? be a compact domain in a n-dimensional Riemannian spin manifold (Nn, g) whose boundary is denoted by ??. In [HMR02], the authors studied four elliptic boundary conditions for the Dirac operator D of the domain ?. More precisely, they prove a Friedrich-type inequality [Fri80] which relates the spectrum of the Dirac operator and the scalar curvature of the domain ?. These boundary conditions are the following: the Atiyah-Patodi-Singer (APS) condition based on the spectral resolution of the boundary Dirac operator; a modified version of the APS condition, the mAPS condition; the bound- ary condition CHI associated with a chirality operator; and a Riemannian version of the MIT bag boundary condition. In fact, they show that, if the boundary ?? of ? has non-negative mean curvature, then under the APS, CHI or mAPS boundary conditions, the spectrum of the classical Dirac operator of the domain ? is a sequence of unbounded real numbers {λk : k ∈ Z} satisfying λ2k ≥ n 4(n? 1) R0, (1) where R0 is the infimum of the scalar curvature of the domain ?. Moreover, equality holds only for the CHI and the mAPS conditions and in these cases, ? is respectively isometric to a half-sphere or it carries a non-trivial real Killing spinor and has minimal boundary. In the case of the MIT boundary condition, they show that the spectrum of the Dirac operator on ? is an unbounded discrete set of complex numbers λMIT with positive imaginary part satisfying |λMIT|2 n 4(n? 1) R0, (2) if the mean curvature of the boundary is non-negative. This result leads to the f