Eigenvalues of the Dirac operator on manifolds with boundary. (English) Zbl 0997.58015
Let \(M\) be a compact Riemannian manifold with boundary, equipped with a spin structure. The main subject of the article are eigenvalue estimates for the (classical) Dirac operator on such manifolds. As a first step, the authors investigate different possible boundary conditions. Two kind of boundary conditions will be of importance in the article, Atiyah-Patodi-Singer (APS) boundary conditions and local boundary conditions.
For defining the APS boundary conditions one has to define a “restriction” of the Dirac operator to the boundary, called \(D^{\partial M}\). This is an elliptic, self-adjoint operator, and as such an operator it has discrete spectrum. The sections of the spinor bundle over \(\partial M\) can be decomposed into a sum \(V^+\oplus V^-\), where \(V^\pm\) is the closure of the vector space spanned by eigenspinors to nonnegative resp.negative eigenvalues. APS boundary conditions are defined as \[ \varphi|_{\partial M}\in V^-. \] Local boundary condition are defined by using a chirality operator. This is by definition a parallel section of \(\text{End}(\Sigma \partial M)\), with \(F^2= \text{ Id}\), anticommuting with Cifford multiplication by vectors and acting pointwise unitarily. If such an operator \(F\) exists, the sections of the spinor bundle over \(\partial M\) can be decomposed into a sum \(W^+\oplus W^-\), where \(W^\pm\) is the eigenspace of \(F\) to the eigenvalue \(\pm 1\). Local boundary conditions are defined as \[ \varphi|_{\partial M}\in W^\pm. \] The authors derive lower estimates for the eigenvalues of the square of the Dirac operator on \(M\), both for APS and local boundary conditions. The equality in the estimate for local boundary conditions is attained if the associated eigenspinor is a real Killing spinor and if \(\partial M\) is minimal. In an extra section the authors derive estimates in terms of conformal data. These estimates generalize many previous ones [e.g. the Hijazi inequality, O. Hijazi, Commun. Math. Phys. 104, 151-162 (1986; Zbl 0593.58040)] to manifolds with boundary.
For defining the APS boundary conditions one has to define a “restriction” of the Dirac operator to the boundary, called \(D^{\partial M}\). This is an elliptic, self-adjoint operator, and as such an operator it has discrete spectrum. The sections of the spinor bundle over \(\partial M\) can be decomposed into a sum \(V^+\oplus V^-\), where \(V^\pm\) is the closure of the vector space spanned by eigenspinors to nonnegative resp.negative eigenvalues. APS boundary conditions are defined as \[ \varphi|_{\partial M}\in V^-. \] Local boundary condition are defined by using a chirality operator. This is by definition a parallel section of \(\text{End}(\Sigma \partial M)\), with \(F^2= \text{ Id}\), anticommuting with Cifford multiplication by vectors and acting pointwise unitarily. If such an operator \(F\) exists, the sections of the spinor bundle over \(\partial M\) can be decomposed into a sum \(W^+\oplus W^-\), where \(W^\pm\) is the eigenspace of \(F\) to the eigenvalue \(\pm 1\). Local boundary conditions are defined as \[ \varphi|_{\partial M}\in W^\pm. \] The authors derive lower estimates for the eigenvalues of the square of the Dirac operator on \(M\), both for APS and local boundary conditions. The equality in the estimate for local boundary conditions is attained if the associated eigenspinor is a real Killing spinor and if \(\partial M\) is minimal. In an extra section the authors derive estimates in terms of conformal data. These estimates generalize many previous ones [e.g. the Hijazi inequality, O. Hijazi, Commun. Math. Phys. 104, 151-162 (1986; Zbl 0593.58040)] to manifolds with boundary.
Reviewer: Bernd Ammann (Hamburg)
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 53C27 | Spin and Spin\({}^c\) geometry |
