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The Kernel of the Dirac Operator The kernel of the Dirac operator B. Ammann1 M. Dahl2 E. Humbert3 1Universität Regensburg Germany 2Institutionen för Matematik Kungliga Tekniska Högskolan, Stockholm Sweden 3Laboratoire de Mathématiques et Physique Théorique Université de Tours France SFB-Talk Berlin 2011 Outline Preliminaries The Dirac operator Atiyah-Singer index theorem Comparison to Gauss-Bonnet-Chern operator Is dim ker D a topological invariant? Large kernel conjecture Generic metrics are D-minimal D-minimality theorem History of partial solutions The surgery method Surgery D-minimality and surgery Local D-minimality theorem D-surgery thm implies local D-minimality thm The Dirac operator Let M be a (fixed) compact manifold with spin structure, n = dim M. For any metric g on M one defines I the spinor bundle ΣgM: a vector bundle with a metric, a connection and Clifford multiplication TM ΣgM ΣgM. ⊗ ! Sections M ΣgM are called spinors. ! I the Dirac operator D= g : Γ(ΣgM) Γ(ΣgM): a self-adjoint elliptic differential operator of first! order. = dim ker D= is finite-dimensional. ) g The elements of ker D= g are called harmonic spinors. Why are harmonic spinors interesting? I n = dim M = 2, M a Riemann surface, D= φ = 0 φ is a holomorphic section of pTM pTM. g , ⊕ Such a section corresponds to a conformal immersion of Me into R3 with vanishing mean curvature. (Weierstrass representation 1866) Example I Physics: Stationary state of a fermion with mass 0 or energy 0. I On asymptotically euclidean manifold with scal 0, i.e. a model of a star or black hole: ≥ D= φ = 0 φ is a Witten spinor g , Central ingredient in Witten’s proof of the positive mass theorem. I D= φ = is solvable in φ if ker D= . g ? g The solution is unique up to ker D= g. Atiyah-Singer Index Theorem for n = 4k ! + 0 D= g− Let n = 4k. ΣgM = Σg M Σg−M. D= g = + ⊕ D= g 0 + + + + ind D= = dim ker D= codim im D= = dim ker D= dim ker D= − g g − g g − g Theorem (Atiyah-Singer 1968) Z + ind D= g = Ab(TM) M R Hence: dim ker D= Ab(TM) g ≥ j j Index Theorem for n = 8k + 1 and 8k + 2 n = 8k + 1: α(M) := dim ker D= g mod 2 n = 8k + 2: dim ker D= α(M) := g mod 2 2 α(M) Z=2Z is independent of g. However,2 α(M) depends on the choice of spin structure. Consequence 8 R > Ab(TM) ; if n = 4k; >j j <>1; if n 1 mod 8 and α(M) = 0; dim ker D= g ≡ 6 ≥ >2; if n 2 mod 8 and α(M) = 0; > ≡ 6 :>0; otherwise. Comparison to Gauss-Bonnet-Chern operator The operator d + d ∗ : Γ(Λ•T ∗M) Γ(Λ•T ∗M) is a “twisted” Dirac operator, called Gauss-Bonnet-Chern! operator. Index theorem even odd (d + d ∗)+ : Γ(Λ T ∗M) Γ(Λ T ∗M) ! odd even (d + d ∗) : Γ(Λ T ∗M) Γ(Λ T ∗M) − ! χ(M) = dim ker(d + d ∗)+ dim ker(d + d ∗) − − On the other hand n M ker(d + d ∗) = ker(d + d ∗) i ∗ jΓ(Λ T M) i=0 b (M) := dim ker(d + d ∗) i ∗ i-th Betti number i jΓ(Λ T M) n X dim ker(d + d ∗) = bi (M) i=0 n X dim ker(d + d ∗) = bi (M) i=0 The dimension of the kernel is a topological invariant! The lower bound given by the index theorem n X i dim ker(d + d ∗) χ(M) = ( 1) b (M) ≥ j j j − i j i=0 is attained iff n is even and b1 = b2 = ::: = bn 1 = 0. − I Analogous results for the signature operator. ¯ ¯ I On Kähler manifolds dim ker(@ + @∗) is invariant under deformation of the complex structure. I Kotschick 1996: On some Kähler surfaces the kernel of D= g never attains the Atiyah-Singer bound. Is dim ker D= a topological invariant? No. Definition A metric g on a connected spin manifold is called D= -minimal if dim ker D= g = Bound given by Atiyah-Singer Theorem Generic (=most) metrics are D= -minimal. Conjecture (Large kernel conjecture (Bär-Dahl 2002)) Let dim M 3. For any k N there is a metric gk with g ≥ 2 dim ker D= k k. ≥ Large kernel conjecture Conjecture Let dim M 3. For any k N there is a metric gk with dim ker D ≥k. 2 ≥ This conjecture has been proved by 3 I Hitchin 1974 on M = S for any k N, 2 I Hitchin 1974 in dimensions n 0; 1; 7 mod 8 for k = 1, ≡ I Bär 1996 in dimensions n 3; 7 mod 8 for k = 1, ≡ I Kotschick 1996 n = 4: For any k N there is a Kähler R ^2 surfaces Mk with dim ker D= A + k. 2m ≥ j j I Seeger 2000 on S , m 2, for k = 1, n ≥ I Dahl 2006 on S , n 5, for k = 1. ≥ I Dahl and Große, current research: Systematic approach: Bordism theory. Index on Bordisms. Many open cases! D= -minimality theorem Theorem (D= -minimality theorem, ADH, 2009) Generic metrics on connected compact spin manifolds are D= -minimal. 1 Generic = dense in C1-topology and open in C -topology. The investigations for this result were initiated by Hitchin (1974). The theorem was explicitly conjectured by Bär-Dahl (2002). To prove the D= -minimality theorem it is sufficient to show that there is one D= -minimal metric. History of partial solutions In order to show that generic metrics are D= -minimal, if suffices to show that one D= -minimal metric exists. I Hitchin (1974): dim ker D= g depends on g. I Maier (1996) proved the theorem if n = dim M 4: ≤ I Bär-Dahl (2002) proved the theorem when n 5 and π (M) = e : ≥ 1 f g They used the surgery method (Gromov-Lawson 1980, Stolz 1992). I We (ADH, Adv. Math. 2009) also use the surgery method. It works under no restriction on n or π1. I A new proof (ADH, Math. Res. Lett., to appear) proves a local, stronger version. Surgery k n k Let f : S B − , M be an embedding. We define× ! # k n k k+1 n k 1 M := M f (S B − ) (B S − − )= n × [ × ∼ where = means gluing the boundaries via ∼ k n k 1 M f (x; y) (x; y) S S − − : 3 ∼ 2 × We say that M# is obtained from M by surgery of dimension k. Example: 0-dimensional surgery on a surface. D= -minimality and surgery Theorem (D= -Surgery Theorem, ADH 2009) Let k n 2. If M carries≤ − a D= -minimal metric, then M# carries a D= -minimal metric as well. Bär-Dahl (2002) proved the theorem with other methods for k n 3. ≤ − ADH proved, Math. Res. Lett., to appear Theorem (Local D= -Minimality Theorem) Let M be a compact connected spin manifold with a riemannian metric g. Let U be a non-empty open subset of M. Then there is metric g~ on M which is D= -minimal and which coincides with g on M U. n This theorem implies that generic metrics are D= -minimal. Proof of “D= -surgery Thm = Local D= -minimality Thm” ) We use a theorem from Stolz 1992. The given spin manifold M is spin bordant to N = N0 P, where [ P carries a metric of positive scalar curvature, • 1 N0 is a disjoint union of products of S , a K 3-surface and a • Bott manifold, and carries a D= -minimal metric. N M Assume now n 5. Remove a ball and move M to the other side. ≥ N Sn M − Modify the bordism W such that W is connected and π1(W ) = π2(W ) = 0. n As π1(S ) = 0, the bordism W can be decomposed into pieces corresponding to surgeries of dimension k 0; 1;:::; n 3 2 f − g N Sn M − N N N 0 1 •••• ℓ Invertible Double Proposition Let M be compact, connected and spin. Then there is a metric g on M#( M) with invertible D= . − g See e.g. the book by Booß-Bavnbek and Wojchiekowski. Uses the unique continuation property of D= g. N Sn # M − # M M N N N ′ 0′ 1′ •••• ℓ Copyright: KGB, Darmstadt Back.
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