The Dirac spectrum
This memoir is a survey on the spectral properties of the Dirac operator defined by a spin structure on a Riemannian manifold. I think that it can be used as a valuable guide to get introduced to this subject. The book is self-contained once some basic concepts of differential geometry are known, like vector bundles, Lie groups, principal bundles, connections, curvature, etc. All needed preliminaries on spin geometry are recalled in Chapter~1. The spin group \(\text{Spin}_n\) in dimension \(n\) is the non-trivial \(2\)-fold covering of \(\text{SO}_n\). A spin structure on an oriented Riemannian \(n\)-manifold \(M\) is a \(\text{Spin}_n\)-principal bundle over \(M\) which is a \(2\)-fold covering of the \(\text{SO}_n\)-principal bundle of oriented orthonormal frames of \(TM\) in an equivariant way with respect to the projection \(\text{Spin}_n\to\text{SO}_n\). It is proved that the family of spin structures on \(M\) is in one-to-one correspondence with \(H^1(M,\mathbb{Z}_2)\). It is recalled that \(\text{Spin}_n\) has one fundamental irreducible complex representation if \(n\) is odd, denoted by \(\Sigma_n\), and two if \(n\) is even, denoted by \(\Sigma_n^\pm\). The vector bundle \(\Sigma(M)\to M\) associated to the representation \(\Sigma_n\) (\(n\) odd) or \(\Sigma_n^+\oplus\Sigma_n^-\) (\(n\) even) is called the spinor bundle; in the even dimensional case, each \(\Sigma_n^\pm\) induces a vector bundle \(\Sigma^\pm M\), defining a splitting of \(\Sigma M\). The author also recalls the definitions of the Clifford multiplication \(\mu:TM^*\otimes\Sigma M\to\Sigma M\), the connection \(\nabla^\Sigma\) and the Hermitian inner product on \(\Sigma M\), which satisfy natural relations of compatibility. Then the Dirac operator \(D:\Gamma(\Sigma M)\to\Gamma(\Sigma M)\) is defined as the composition \[ \Gamma(\Sigma M) \overset{\nabla}{\longrightarrow} \Gamma(TM\otimes\Sigma M) \longrightarrow \Gamma(TM^*\otimes\Sigma M) \overset{\mu}{\longrightarrow} \Gamma(\Sigma M)\;, \] where the middle map is the isomorphism defined by the metric. When \(n\) is even, we can write \(D=\begin{pmatrix} 0 & D^- \\ D^+ & 0\end{pmatrix}\) with respect to the splitting of \(\Sigma M\). All basic properties of \(D\) are recalled; in particular, it is elliptic and formally self-adjoint, and it is essentially self-adjoint when \(M\) is complete. As a consequence, when \(M\) is closed, the spectrum of \(D\) consists of a sequence of eigenvalues of finite multiplicity, unbounded from above and from below. Moreover it is symmetric about the origin when \(n\not\equiv3\;(4)\). The Schrödinger-Lichnerowicz formula and the Atiyah-Singer index formula (for \(n\) even) are recalled, respectively stating that \[ D^2=\nabla^*\nabla+\frac{S}{4}\;,\quad\text{ind}(D^+)=\widehat{A}(M)\;, \] where \(S\) is the scalar curvature and \(\widehat{A}(M)\) is the \(\widehat{A}\)-genus of \(M\) (the integral on \(M\) of the \(\widehat{A}\)-class of \(TM\)). A remarkable conformal covariance property of the Dirac operator is also stated. Next, it is explained how a spin structure induces spin structures on hypersurfaces and quotients. The final section of this chapter studies boundary conditions for Dirac operators on manifolds with boundary. Those conditions are required to get the same spectral properties as in the case of closed manifolds. Four types of boundary conditions are specially studied: generalized Atiyah-Patodi-Singer (gAPS) boundary conditions, boundary conditions associated to a chirality operator (CHI), MIT bag boundary condition, and modified generalized Atiyah-Patodi-Singer (mgAPS) boundary conditions. Moreover an appendix is devoted to the so called twistor and Killing spinors, which are used in the book. In Chapter~2, the author presents some of the few examples where the spectrum of the Dirac operator can be explicitly computed: tori, spheres and some other homogeneous spaces; a complete list of the known examples with precise references is also given. Also, a result of J.L.~Milhorat is stated, which allows to compute the first eigenvalue of \(D^2\) for more homogeneous spaces. Chapter~3 deals with lower estimates for the eigenvalues of the Dirac operator \(D\) of a closed spin Riemannian \(n\)-manifold \(M\). The first and most general one is Friedrich's inequality: \[ \lambda^2\geq\frac{n}{4(n-1)}\,\inf_MS \] for all eigenvalue \(\lambda\) of \(D\), where \(S\) is the scalar curvature of \(M\). Moreover this inequality becomes an equality for some eigenvalue if and only if there exists a non-zero Killing spinor on \(M\). This estimate is a consequence of the Schrödinger-Lichnerowicz formula. It is improved in several ways: in the presence of parallel or harmonic forms, parallel distributions, or Kähler structures; by using the conformal equivarince of \(D\), involving the conformal Laplace operator or the Yamabe invariant; with the energy momentum tensor; with the Ricci curvature or the Weil tensor; on surfaces of positive genus, using the spinning systole or the spin-cut diameter; and on boundaries of spin manifolds, involving their mean curvatures. Lower eigenvalue estimates in the case of compact manifolds with boundary is considered in Chapter~4. Versions of the Friedrich's inequality are given for gAPS, CHI, MIT bag and mgAPS boundary conditions. Chapter~5 is devoted to the study of upper bounds of the eigenvalues of the Dirac operator on closed manifolds. Two methods are considered to obtain them. In the first one, due to Vafa-Witten, \(D\) is compared to another Dirac-type operator \(\mathcal{D}\), whose kernel is non-trivial by the index theorem, and then the upper eigenvalue bound follows by estimating the zero order difference \(D-\mathcal{D}\) by geometric quantities. The second method relies on the well known min-max principle, which provides a variational expression of the eigenvalues. An example of the application of the Vafa-Witten's method is the estimate of H.~Baum. Let \(M\) be an even-dimensional closed spin Riemannian manifold with positive sectional curvature \(K\). Then the first eigenvalue \(\lambda_1\) of its Dirac operator \(D\) satisfies \[ |\lambda_1|\leq2^{\frac{n}{2}-1}\,\sqrt{\frac{n}{2}}\,\sqrt{\max_MK}\;. \] An example of application of the min-max principle, combined with the conformal covariance of the Dirac operator, is the following estimate of J.~Lott. Let \(M\) be a spin Riemannian manifold of even dimansion \(\geq2\). Then, for any conformal class \([g]\) of Riemannian metrics on \(M\), there exists some \(b([g])>0\) such that \[ \lambda_1(D_g^2)\geq b([g])\,\sup_M(-S_g) \] for any \(g\in[g]\) with scalar curvature \(S_g<0\), where \(D_g\) is the Dirac operator defined with \(g\), and \(\lambda_1(D_g^2)\) the first eigenvalue of \(D_g^2\). Another upper eigenvalue estimate is shown for the case of surfaces. A much more involved theorem, due to C.~Bär, provides upper estimates for all eigenvalues, not only the first one. Another type of upper eigenvalue estimates is also considered in this chapter, which are extrinsic in the sense that they depend on maps of the given closed spin \(n\)-manifold \(M\) to other manifolds. The first result of this type, due to H.~Baum, holds when \(n\) is even, and assumes the existence of a map \(\iota:M\to\mathbb{S}^n\) of sufficiently high degree: \[ \text{deg}(\iota)\geq1+2^{\frac{n}{2}-1}\sum_{j=1}^{k-1}m_j \] for some \(k\in\mathbb{N}\), where \(m_j\) is the multiplicity of the \(j\)th eigenvalue of \(D^2\). Then the \(k\)th eigenvalue \(\lambda_k\) of \(D\) satisfies \[ |\lambda_k|\leq2^{\frac{n}{2}-1}\,\sqrt{\frac{n}{2}}\,\max_{x\in M}\|T_x\iota\|\;. \] The second result, due to U.~Bunke, also holds for even dimension, and assumes the existence of an isometric immersion \(\iota:M\to\mathbb{R}^N\) fo some \(N\in\mathbb{N}\). Then there is a topologically determined number of eigenvalues \(\lambda\) of \(D\) satisfying \[ \lambda^2\leq2^{\frac{n}{2}}\,\max_{x\in M}\|I\!I_x\|^2\;, \] where \(I\!I_x\) denotes the second fundamental form at \(x\). A more sophisticated result for isometric immersions is also shown in this section, which involves a non-zero twistor-spinor and the mean curvature. The special case of closed curves in \(\mathbb{R}^2\) or \(\mathbb{S}^2\) is also considered. The possibility of prescribing the eigenvalues of the Dirac operator is studied in Section~6. The first section is devoted to Dirac isospectrality (spin Riemannian manifolds whose Dirac operators have the same spectrum). Examples of Dirac isospectral but non-isometric manifolds are given. The first ones are the Milnor \(16\)-tori, and quotients of spheres. Results of Miatello-Podestá and Ammann-Bär are also stated, which exhibit continuous families of non-isometric spin Riemannian manifolds which are Dirac isospectral to each other. The next section of this chapter is devoted to prescribing zero in the Dirac spectrum (existence of non-zero harmonic spinors). For a spin Riemannian \(n\)-manifold \(M\), the dimension of the space of harmonic spinors is denoted by \(d\), which depends on the choice of the spin structure and the conformal class of the metric. In the case of surfaces of genus \(g\), the following properties are satisfied: \(d\leq2[\frac{g+1}{2}]\), there exists a conformal class and a spin structure with \(d=2[\frac{g+1}{2}]\), there exists a conformal class and a spin structure with \(d=0\), if \(g\leq2\) then \(d\) does not depend on the conformal class, if \(g\geq3\) then there exists a spin structure so that \(d\) depends on the conformal class, and if \(g\in\{3,4\}\) then there is also a spin structure so that \(d\) does not depend on the conformal class. In the case of dimension \(n\geq3\), it turns out that the only restriction to prescribe \(d\) is a lower bound given by the index theorem: \[ d\geq \begin{cases} |\widehat{A}(M)| & \text{if \(n\equiv0\;(4)\)}\\ |\alpha(M)| & \text{if \(n\equiv1\;(8)\)}\\ 2\,|\alpha(M)| & \text{if \(n\equiv2\;(8)\)}\\ 0 & \text{otherwise}\;,\\ \end{cases} \] where \(\alpha(M)\) is the \(\alpha\)-genus of \(M\). It was conjectured by C.~Bär and proved with complete generality by Ammann-Dahl-Humbert that this is an equality for a subset of Riemannian metrics that is \(C^1\)-open and \(C^\infty\)-dense. Moreover Bär-Hitchin have shown that, if \(n\equiv0,1,3,7\;(8)\), then \(d\geq1\) for some Riemannian metric. Some ideas of the proof are given. A theorem of M.~Dahl is stated in the last section of this chapter, which asserts that, for dimension \(n\geq3\), it is possible to prescribe the non-zero eigenvalues in any bounded interval centered at zero, with the only restriction of symmetry around zero if \(n\not\equiv3\;(4)\). Chapter~7 deals with non-compact manifolds. In this case, the Dirac spectrum has several components: the point spectrum, the continuous spectrum, the residual spectrum, the essential spectrum and the discrete spectrum. Its explicit computation is made for \(\mathbb{R}^n\), and a list is given with the point and continuous spectra for \(\mathbb{R}H^n\), \(\mathbb{C}H^n\), \(\mathbb{H}H^n\) and \(\mathbb{O}H^n\), indicating the references. A generalization of Goette-Semmelmann states that the Dirac spectrum on a symmetric space of non-compact type is either empty or \(\{0\}\), and in the later case each irreducible factor is of the form \(\mathbb{U}(p+q)/\mathbb{U}(p)\times\mathbb{U}(q)\) with \(p+q\) odd. Another section of this chapter exhibits lower bounds for the spectrum of the Dirac operator \(D\) on a spin Riemannian non-compact manifold of dimension \(n\geq2\). For instance, a theorem of C.~Bär states that, like in the compact case, there is a gap about zero when the scalar curvature is bounded bellow by a positive constant: \[ \min\sigma(D^2)\geq\frac{n}{4(n-1)}\,\inf_MS\;, \] where \(\sigma(D^2)\) denotes the spectrum of \(D^2\). Another estimate of this type is given for connected surfaces of finite volume embedded in \(\mathbb{S}^2\) and carrying the induced spin structure, as well as two more lower bounds involving the conformal Laplace operator due to N.~Große. The next section states several results stating the absence of some component of the Dirac spectrum. For instance, for complete hyperbolic manifolds of finite volume, the discrete or the essential Dirac spectrum is empty depending on the possible triviality the spin structure along some cusp (C.~Bär). More involved results of this kind are also stated. Finally, Chapter~8 is devoted to other topics related with the Dirac spectrum that have been left aside in this overview: other eigenvalue estimates, spectral gap, pinching Dirac eigenvalues, spectrum of other Dirac type operators, conformal spectral invariants, convergence of eigenvalues, eta-invariants and positive mass theorems.
scientific article; zbMATH DE number 5574435