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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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