SPECTRAL BOUNDS FOR DIRAC OPERATORS ON OPEN MANIFOLDS CHRISTIAN BAR¨ ABSTRACT. We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich’s estimate for manifolds with positive scalar curvature as well as the au- thor’s estimate on surfaces. 1. INTRODUCTION This paper is concerned with spectral bounds for Dirac operators on noncompact Rie- mannian manifolds. We work with generalized Dirac operators in the sense of Gromov and Lawson as explained in Section 2. Examples are the classical Dirac operatorp acting on spinors, the operator D = d + d acting on forms, and the operator D = 2(¶¯ + ¶¯ ∗) on a Kahler¨ manifold. In Section 3 we show that Friedrich’s lower bound for the eigenvalues of the classical Dirac operator on a compact spin manifold of positive scalar curvature extends to a lower bound for the fundamental tone of the square of any Dirac operator even if the manifold M is incomplete. If M is complete this implies a spectral gap for the Dirac operator itself. We also discuss the equality case. If the spectral bound is an eigen- value of the Dirac operator, then the corresponding eigensection satisfies the Killing spinor equation, an overdetermined elliptic equation of first order. In the case of the classical Dirac operator this implies that M is compact and Einstein. As another spe- cial case we give a simple proof of Lichnerowicz’ classical lower bound for the first positive eigenvalue of the Laplace operator on a compact manifold with positive Ricci curvature. In Section 4 we study the essential spectrum. We show that if the curvature endo- morphism of the Dirac operator is bounded from below at infinity, then a Friedrich- arXiv:0810.5598v1 [math.DG] 31 Oct 2008 Lichnerowicz type estimate holds for the essential spectrum. In particular, this yields a sufficient criterion for a Dirac operator on a complete manifold to be a Fredholm operator. If the curvature endomorphism tends to infinity at infinity, then the spectrum of the square of the Dirac operator is discrete just as in the case of a compact manifold. In the last section we study the classical Dirac operator on surfaces. We show that if M is a connected surface of genus 0 with finite area and if the spin structure is bounding Date: October 29, 2018. 2000 Mathematics Subject Classification. 53C27. Key words and phrases. Dirac operators, point spectrum, continuous spectrum, discrete spectrum, essential spectrum, Killing spinor, Friedrich inequality, Lichnerowicz inequality. 1 2 CHRISTIAN BAR¨ at infinity, then 4p l (D2) ≥ : ∗ area(M) Here M need not be complete and may have infinitely many ends. If one drops the assumption on the spin structure, then the estimate fails. The estimate is proved by reducing it to the case M = S2 which was established by the author almost two decades ago. 2. GENERALIZED DIRAC OPERATORS Let M denote a connected Riemannian manifold. There will be no completeness or compactness assumption on M unless we explicitly say so. We will work with gen- eralized Dirac operators in the sense of Gromov and Lawson [6]. Let E ! M be a Riemannian or Hermitian vector bundle over M equipped with a metric connection ∇E . ¥ For j;y 2 Cc (M;E), the space of smooth, compactly supported sections of E, the L2-product and norms are defined, Z (j;y) = hj;yidV; kjk2 = (j;j): M Here dV denotes the volume element induced by the Riemannian metric of M. The ¥ Hilbert space obtained by completing Cc (M;E) with respect to this scalar product is denoted by L2(M;E). Pointwise scalar products will be denoted by h·;·i and pointwise norms by j · j. We assume that tangent vectors act by Clifford multiplication on E, i. e. there is a vector bundle homomorphism TM ⊗ E ! E; X ⊗ j 7! X · j; satisfying • the Clifford relations X ·Y · j +Y · X · j + 2hX;Yij = 0 for all X;Y 2 TpM, j 2 Ep, p 2 M. • skew symmetry hX · j;yi = −hj;X · yi for all X 2 TpM, j;y 2 Ep, p 2 M. • the product rule E E ∇X (Y · j) = (∇XY) · j +Y · ∇X j for all differentiable vector fields X and Y and for all differentiable sections j in E. Here ∇ is the Levi-Civita connection of M. The Dirac operator is now defined by n D := e · ∇E j ∑ i ei j i=1 SPECTRAL BOUNDS FOR DIRAC OPERATORS ON OPEN MANIFOLDS 3 where e1;:::;en denotes any orthonormal tangent basis, n = dim(M). This definition is independent of the choice of basis. Example 2.1. Let M carry a spin structure. Then one can define the spinor bundle E = SM which has all the required structure. The corresponding generalized Dirac operator is the classical Dirac operator, sometimes also called the Atiyah-Singer operator. See e. g. [11] for the definitions. Example 2.2. Let M be an n-dimensional Riemannian manifold and E = Ln p ∗ p=0 L T M. Using the wedge product and its adjoint one can define a Clifford multi- plication such that the corresponding generalized Dirac operator is given by D = d +d. Here d is the exterior differential and d is its adjoint. See e. g. [16, Ch. 3] for details. Example 2.3. Let M be a Kahler¨ manifold of complex dimension m. Then E = Lm L0;qT ∗M can be equipped with a Clifford multiplication such that the corre- q=0 p sponding generealized Dirac operator is given by D = 2(¶¯ + ¶¯ ∗), compare [16, Ch. 3]. Any generalized Dirac operator is a formally self-adjoint elliptic differential operator of first order. By [6, Thm. 1.17] the Dirac operator is essentially self-adjoint on the ¥ domain Cc (M;E), smooth sections of E with compact support, in the Hilbert space L2(M;E) provided that M is complete. In this case, when we speak about the spectrum of the Dirac operator we will always mean the spectrum of its self-adjoint closure. The spectrum s(D) decomposes into two disjoint parts, the point spectrum sp(D) consisting of all eigenvalues (with square-integrable eigensections), and the continu- ous spectrum sc(D). By elliptic regularity theory eigensections are smooth. If M is compact, then sc(D) is empty and sp(D) is discrete and all eigenvalues have finite multiplicity, compare Corollary 4.4. 2 ¥ Since the operator D is symmetric and nonnegative on the domain Cc (M;E) it has a canonical self-adjoint extension even if M is not complete. This is the Friedrichs 1 extension, the unique self-adjoint extension whose domain is contained in HD(M;E), ¥ the closure of Cc (M;E) with respect to the norm induced by the scalar product (j;y) 1 = (j;y) + (Dj;Dy): HD If M is complete, then this is the only self-adjoint extension of D2 and it coincides with the square of the self-adjoint extension of D. If M is the interior of a compact manifold M with boundary such that the metric extends, then taking the Friedrichs extension corresponds to imposing Dirichlet boundary conditions. When we speak about the spectrum of D2 we always mean the spectrum of the Friedrichs extension. Again, the spectrum decomposes disjointly into the point spectrum and the continuous spectrum, 2 2 2 s(D ) = sp(D ) t sc(D ): 2 2 Since D is nonnegative its spectrum is contained in [0;¥). We call l∗(D ) := mins(D2) the Dirac fundamental tone. One has (D2j;j) (Dj;Dj) (Dj;Dj) l (D2) = inf = inf = inf : ∗ ¥ ¥ j2Cc (M;E) (j;j) j2Cc (M;E) (j;j) j2H1 (M;E) (j;j) 6= 6= D j 0 j 0 j6=0 See [17, Sec. 4.2] for the functional analytic background. 4 CHRISTIAN BAR¨ For a differentiable section j and a differentiable function f : M ! C we have (1) D( f j) = grad f · j + f Dj by the very definition of D. The curvature endomorphism of D is defined by n 1 E K := ∑ ei · e j · R (ei;e j) 2 i; j=1 where RE is the curvature tensor of ∇E . The endomorphism field K is symmetric and we have the Bochner-Lichnerowicz formula [6, Prop. 2.5] (2) D2 = (∇E )∗∇E + K : 1 If D is the classical Dirac operator acting on spinors, then K = 4 scal. If D = d + d, then the restriction of K to 1-forms is Ricci curvature. 3. A FRIEDRICH INEQUALITY Now we show that D has a spectral gap provided K is uniformly positive, a result which is due to Friedrich in the compact case [5, Thm. A]. Since we cannot work with eigensections here, the proof needs modification. Theorem 3.1. Let M be a (possibly incomplete) n-dimensional Riemannian mani- fold. Let D be a generalized Dirac operator on M whose curvature endomorphism is bounded from below by a positive constant k 2 R, i. e. K ≥ k > 0 in the sense of symmetric endomorphisms. Then nk l (D2) ≥ : ∗ n − 1 ¥ Proof. Let j 2 Cc (M;E). By (2) we have (3) kDjk2 = (D2j;j) = (((∇E )∗∇E + K )j;j) ≥ k∇E jk2 + kkjk2: By the Cauchy-Schwarz inequality1 we have n n jD j = j e · ∇E j ≤ j∇E j j ∑ i ei j ∑ ei j i=1 i=1 s s n n p ≤ 12 · j∇E j2 = nj∇E j ∑ ∑ ei j j i=1 i=1 hence 1 j∇E jj2 ≥ jDjj2: n Plugging this into (3) we get 1 kDjk2 ≥ kDjk2 + kkjk2 n 1This trick was shown to me by O.