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Differential Forms, Spinors and Bounded Curvature Collapse

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Differential Forms, Spinors and Bounded Curvature Collapse DIFFERENTIAL FORMS, SPINORS AND BOUNDED CURVATURE COLLAPSE John Lott University of Michigan November, 2000 From preprints \Collapsing and the Differential Form Laplacian" \On the Spectrum of a Finite-Volume Negatively-Curved Manifold" \Collapsing and Dirac-Type Operators" http://www.math.lsa.umich.edu/ lott e 1 2 WHAT I WILL (NOT) TALK ABOUT Fukaya (1987) : Studied the function Laplacian, under a bounded curvature collapse. Cheeger-Colding (preprint) : Studied the function Laplacian, under a collapse with Ricci curvature bounded below. Today : The differential form Laplacian and geometric Dirac-type operators, under a bounded curvature collapse. 3 MOTIVATION : CHEEGER'S INEQUALITY Let M be a connected compact Riemannian manifold. Spectrum of M : 4 0 = λ1(M) < λ2(M) λ3(M) ::: ≤ ≤ Theorem 1. (Cheeger 1969) h2 λ2(M) : ≥ 4 Definition 1. Area(A) h = inf ; A min(vol(M1); vol(M2)) where A ranges over separating hypersurfaces. Question (Cheeger) : Is there a similar inequality for the p-form Lapla- cian? (Problem # 79 on Yau's list.) 4 THE p-FORM LAPLACIAN The p-form Laplacian is M p p = dd∗ + d∗d :Ω (M) Ω (M): 4p ! Spectrum of M : 4p 0 λp;1(M) λp;2(M) λp;3(M) ::: ≤ ≤ ≤ ≤ By Hodge theory, Ker( M ) = Hp(M; R); 4p ∼ so 0 = λp;1(M) = ::: = λp;b (M)(M) < λp;b (M)+1(M) :::: p p ≤ 5 COLLAPSING (Cheeger, Gromov) Notation : RM = Riemann sectional curvatures: diam(M) = sup dM (p; q): p;q M 2 Fix a number K 0. Consider ≥ connected Riem. manifolds M n : RM K and diam(M) 1 : f k k1 ≤ ≤ g There is a number v0(n; K) > 0 such that one has the following di- chotomy : I. Noncollapsing case : vol(M) v0. ≥ Finite number of topological types, C1,α-metric rigidity. In particu- lar, uniform bounds on eigenvalues of p. 4 or II. Collapsing case : vol(M) < v0. Special structure. Need to analyze p in this case. 4 6 BERGER EXAMPLE OF COLLAPSING Hopf fibration π : S3 S2 ! Shrink the circles to radius . Look at the 1-form Laplacian 1. Since 1 3 4 H (S ; R) = 0, the first eigenvalue λ1;1 of 1 is positive. 4 Fact (Colbois-Courtois 1990) lim λ1;1 = 0: 0 ! New phenomenon : (uncontrollably) small eigenvalues. When does this happen? 7 FUKAYA'S WORK ON THE FUNCTION LAPLACIAN Suppose that Mi i1=1 are connected n-dimensional Riemannian man- Mfi g ifolds with R K and diam(Mi) 1. k k1 ≤ ≤ GH Suppose that Mi X. −! Consider the function Laplacian on Mi, with eigenvalues λj(Mi) 1 . f gj=1 Question : Suppose that X is a smooth Riemannian manifold. Is it true that limi λj(Mi) = λj(X)? !1 Answer : In general, no. Need to add a probability measure µ to X. Laplacian on weighted L2-space : < f; X,µf > f 2 dµ 4 = RX jr j : < f; f > f 2 dµ RX Theorem 2. (Fukaya 1987) If dvolMi lim Mi; = (X; µ) i vol(M ) !1 i in the measured Gromov-Hausdorff topology then lim λj(Mi) = λj(X; µ): i !1 8 GOAL Want a \p-form Laplacian" on the limit space X so that after taking a subsequence, λp;j(Mi) λp;j(X): −! Question : What kind of structure do we need on X? 9 SUPERCONNECTIONS (Quillen 1985, Bismut-L. 1995) Input : B a smooth manifold, E = m Ej a Z-graded real vector bundle on B. Lj=0 The (degree-1) superconnections A0 that we need will be formal sums of the form A0 = A[0]0 + A[1]0 + A[2]0 where +1 A[0]0 C1 (B; Hom(E∗;E∗ )), • 2 E A[1]0 is a grading-preserving connection on E and • 2 1 r A0 Ω (B; Hom(E∗;E∗− )). • [2] 2 Then A0 : C1(B; E) Ω(B; E) extends by Leibniz' rule to an operator ! A0 : Ω(B; E) Ω(B; E). ! 2 Flatness condition : (A0) = 0; i.e. 2 2 A0 = A0 = 0, • [0] [2] E E A0 = A0 = 0 and • r [0] r [2] E 2 + A0 A0 + A0 A0 = 0. • r [0] [2] [2] [0] Note : A[0]0 gives a differential complex on each fiber of E. 10 THE NEEDED STRUCTURE ON THE LIMIT SPACE X E A triple (E; A0; h ), where 1. E is a Z-graded real vector bundle on X, 2. A0 is a flat degree-1 superconnection on E and 3. hE is a Euclidean inner product on E. We have A0 : Ω(X; E) Ω(X; E): ! Using gTX and hE, we get (A0)∗ : Ω(X; E) Ω(X; E): ! Put E = A0(A0)∗ + (A0)∗A0; 4 the superconnection Laplacian. Example : If E is the trivial R-bundle on X, A0 is the trivial connec- tion and hE is the standard inner product on E then E is the Hodge Laplacian. 4 11 ANALYTIC COMPACTNESS GH Theorem 3. If Mi X with bounded sectional curvature then after −! E taking a subsequence, there is a certain triple (E; A0; h ) on X such that Mi E lim σ p = σ p : i !1 4 4 Remark 1 : This is a pointwise convergence statement, i.e. for each j, the j-th eigenvalue converges. Remark 2 : Here the limit space X is assumed to be a Riemannian manifold. There is an extension to singular limit spaces (see later). Remark 3 : The relation to Fukaya's work on functions: For functions, only Ω0(X; E0) is relevant. Here E0 is a trivial R- bundle on X with a trivial connection. But its metric hE0 may be nontrivial and corresponds to Fukaya's measure µ. 12 IDEA OF PROOF 1. The individual eigenvalues λp;j are continuous with respect to the metric on M, in the C0-topology (Cheeger-Dodziuk). 2. By Cheeger-Fukaya-Gromov, if M is Gromov-Hausdorff close to X then we can slightly perturb the metric to get a Riemannian affine fiber bundle. That is, affine fiber bundle : M is the total space of a fiber bundle M X with infranil fiber Z, whose holonomy can be reduced from Diff(Z!) to Aff(Z). Riemannian affine fiber bundle : In addition, one has a. A horizontal distribution T H M on M with holonomy in Aff(Z), and b. Fiber metrics gTZ which are fiberwise affine-parallel. So it's enough to just consider Riemannian affine fiber bundles. M E 3. If M is a Riemannian affine fiber bundle then σ( p ) equals σ( p ) 4 2 4 up to a high level, which is on the order of dGH (M; X)− . Here E is the vector bundle on X whose fiber over x X is 2 Ex = affine-parallel forms on Zx : f g Ei 4. Show that the ensuing triples (Ei;Ai0 ; h ) i1=1 have a convergent subsequence (modulo gauge transformation).f g 13 APPLICATION TO SMALL EIGENVALUES Fix M and K 0. Consider ≥ g : RM (g) K and diam(M; g) 1 : f k k1 ≤ ≤ g Question : Among these metrics, are there more than bp(M) small eigenvalues of M ? 4p Suppose so, i.e. that for some j > bp(M), there are metrics gi i1=1 so that f g i λp;j(M; gi) !1 0: −! Step 1. Using Gromov precompactness, take a convergent subsequence of spaces i (M; gi) !1 X: −! Since there are small positive eigenvalues, we must be in the collaps- ing situation. Step 2. Using the analytic compactness theorem, take a further sub- E sequence to get a triple (E; A0; h ) on X. Then E λp;j( ) = 0: 4 In the limit, we've turned the small eigenvalues into extra zero eigen- values. E Recall that has the Hodge form A0(A0)∗ + (A0)∗ A0. Then from Hodge theory,4 p dim(H (A0)) j: ≥ Analysis Topology −! p Fact : There is a spectral sequence to compute H (A0). Analyze the spectral sequence. 14 RESULTS ABOUT SMALL EIGENVALUES Theorem 4. Given M, there are no more than b1(M)+dim(M) small eigenvalues of the 1-form Laplacian. More precisely, if there are j small eigenvalues and j > b1(M) then in terms of the limit space X, j b1(X) + dim(M) dim(X): ≤ − (Sharp in the case of the Berger sphere.) More generally, where do small eigenvalues come from? Theorem 5. Let M be the total space of an affine fiber bundle M X, which collapses to X. Suppose that there are small positive eigenvalues! of p in the collapse. Then there are exactly three possibilities : 4 1. The infranil fiber Z has small eigenvalues of its q-form Laplacian for some 0 q p. That is, bq(Z) < dim affine-parallel q-forms on Z . ≤ ≤ f g OR 2. The \direct image" cohomology bundle Hq on X has a holonomy q representation π1(X) Aut(H (Z; R)) which fails to be semisimple, for some 0 q p. ! ≤ ≤ OR 3. The Leray spectral sequence to compute Hp(M; R) does not de- generate at the E2 term. Each of these cases occurs in examples. 15 UPPER EIGENVALUE BOUNDS Theorem 6. Fix M. If there is not a uniform upper bound on λp;j (among metrics with RM K and diam(M) = 1) then M collapses to a limit spacek X withk1 ≤1 dim(X) p 1. In addition, the generic fiber Z of≤ the fiber≤ bundle− M X is an infranilmanifold which does not admit nonzero affine-parallel! k-forms for p dim(X) k p. − ≤ ≤ Example : Given M, if there is not a uniform upper bound on the j-th eigenvalue of the 2-form Laplacian then M collapses with bounded curvature to a 1-dimensional limit space.
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