Fibered cusp versus <span class="mathjax-formula">$d$</span>-index theory

Fibered Cusp Versus d-Index Theory.

SERGIUMOROIANU(*)

ABSTRACT- We prove that the indices of fibered-cusp andd-Dirac operators on a spin manifold with fibered boundary coincide if the associated family of Dirac operators on the fibers of the boundary is invertible. This answers a question raised by Piazza. Under this invertibility assumption, our method yields an in- dex formula for the Dirac operator of horn-cone and of fibered horn metrics.

Let X be a compact manifold whose boundary is the total space of a locally trivial fiber bundleW:@X!Yof closed manifolds. Letx:X!R‡

be a defining function for@Xand denote byXthe interior ofX.

Thefibered cusp(orF-) tangent bundle^FTXis a smooth vectorbundle onXdefined in terms of the above data by its global sections:

C¹(X;^FTX):ˆfV 2 C¹(X;TX);V_j@X is tangent to the fibers ofW; hdx;Vi 2x²C¹(X)g:

When restricted to X, theF tangent bundle is canonically isomorphic to the usual tangent bundleTX. By definition, afibered cuspmetricgFis the restriction toXof a Euclidean metric in the bundle^FTX, smooth down to the boundary ofX. Let

g_d:ˆx²gF

be the conformally equivalentd-metric. Such metrics appear naturally in a variety of geometric situations.

EXAMPLE 1. Let (X;g) be a complete hyperbolic manifold of finite volume. Then outside a convex set,Xis isometric to the disjoint union of a (*) Indirizzo dell'A.: Institutul de MatematicaÆ al Academiei RomaÃne, P.O. Box 1-764, RO-014700 Bucharest, Romania.

E-mail: [email protected]

Partially supported by the CERES contract 4-187/2004 and by a CNCSIS contract (2006).

finite numberof ``cusps'', i.e., cylinders [0;1)M with metric dt²‡e ^2th_M;

where hM is flat. Compactify X by setting X:ˆXt(f1g M). This space becomes a smooth manifold with boundary if we impose that e ^t be a boundary-defining function. By the change of variablesxˆe ^t, the metric becomes ad-metric for the trivial boundary fibrationM! fptg.

More generally, a locally symmetric space X with Q-rank 1 cusps is diffeomorphic, outside a compact set, to [R;1)M where M is the total space of a fibrationf:M!Y with a canonical connection; more- over, Xhas a natural Riemannian metric which near infinity takes the form

dt²‡WgY‡e ^2tgZ

whereg_Y;g_Zare a metric on the base, respectively a family of metrics on the fibers.

EXAMPLE 2. The standard metric on XˆRⁿ is an example of a F metric for X the radial compactification, x:ˆ1=r and forthe identity fibration at infinity 1:S^{n 1}!S^{n 1} (F-metrics for the identity fibration are also calledscattering metrics).

More generally, several examples of complete Ricci-flat metrics are of Ftype.

To understand better these metrics, choose a product decomposition of Xnearthe boundary ofXand fix a connection in the fibrationW.

DEFINITION3. AproductF-metric (with respect to the above choices) is aF-metric which near infinity takes the form

dx² x⁴ ‡Wg^Y

x² ‡h

where g^Y is a metric on Y andh a family of metrics on the fibers, both independent ofx.

Thus locally symmetric spaces with Q-rank 1 cusps have product F- metrics.

It is straightforward to decide on the completeness of metrics con- formal to aF-metric.

LEMMA 4. Let g be a F-metric. Then for p2R, the metric x^2pg is complete if and onlyif p1.

PROOF. By compactness of X, there exists a productF-metric g⁰ on

FTXand a constantC>0 such that

C ¹g⁰gCg⁰:

Hence,x^2pgis complete if and only if x^2pg⁰ is. It is evident that a metric which outside a compact set has the form

x^{2p 4}dx²‡h(x)

on (0;e)M(where infinity corresponds toxˆ0), is complete if and only if the length of the segment (0;e) fmgis infinite forallm2M, orin other words ifR^e

0x^{p 2}dxˆ 1, which is equivalent top1. p DEFINITION5. AnexactF-metricgFonXis aF-metric which differs from a productF-metric by a tensor in xC¹(X;S²(^FTX)), i.e., by a sym- metric bilinear form on the bundle ^FTX, smooth down toxˆ0 and van- ishing atxˆ0.

This agrees with the definition from [8]. The name ``exact'' is used by analogy with Melrose's exact b-metrics. In the rest of the paper we will work with an exact F-metric. These metrics were also considered by Leichtnam, Mazzeo and Piazza [2], underthe additional hypothesis that the product decomposition [c;1)M is orthogonal near infinity. Note that what we calld- respectively fibered cusp (orF) metrics is called fi- bered cusp, respectively fibered boundary (or F) metrics in [2]. We keep our terminology for historical reasons.

1. - The main Theorem.

We assume thatX,@Xand the fibers ofWhave fixed (and compatible) spin structures. LetE!Xbe a Hermitian vector bundle with connection (smooth down toxˆ0). Since bothg_Fandg_dare complete by Lemma 1.1, their associated Dirac operators onX (twisted byE) are essentially self- adjoint inL².

TheL² index of the Dirac operatorD^dwas computed by Vaillant [8]

by making extensive use ofFoperators. He assumes that the dimension

of the kernel of the family of Dirac operators on the fibers of Wis con- stant. In this note we observe that in the fully elliptic case (i.e., when the above family of Dirac operators is invertible) the indices of D^d and D^F are the same. The possibility of such a result was conjectured by Paolo Piazza in a private communication. In general, there exists a link be- tween the kernels, which implies that under Vaillant's hypothesis, the index ofD^F is finite.

Our proof works for a more general metricx^2pgF forsomep0. We denote the associated Dirac operator byD^p. The metricg_dis obtained by settingpˆ1, but interesting geometries are also obtained for other values ofp. Using Vaillant's work, we give at the end of the paper an index for- mula fora manifold with various such ends.

THEOREM6. Let gFbe an exact F-metric on X, such that the twisted Dirac operators on the fibers of@X with respect to E, the induced metric on the fibers of the boundaryand the induced spin structure are invertible.

Then the map of multiplication byx^p(n21)defines an isomorphism between the L²-kernels of the operators D^pand D^F. In particular the L²indices (of the chiral operators) coincide.

PROOF. The spinorbundles onXwith respect tox^2pgF, respectivelygF

are the same (i.e., canonically identified) and inherit the same induced metric. We denote this unique spinor bundle byS. It extends naturally to a smooth vectorbundle overX. In the rest of the proof, we suppress the coefficient bundleSEfrom the notation.

LEMMA7. The unbounded operator D^pacting in L²(X;x^2pg_F)with in- itial domain C¹_c (X) is unitarilyequivalent to x ^p=2D^Fx ^p=2 acting in L²(X;gF)with domainC¹_c (X).

PROOF. These are the full (symmetric) Dirac operators. The Dirac operators are linked by the formula of Hitchin (also used by Vaillant [8])

D^pˆx ^p(n‡1)=2D^Fx^{p(n 1)=2}: …1†

The two volume forms onXare related by vol (x^2pg_F)ˆx^npvol (g_F):

Thus the map

C¹_c (X)! C¹_c (X); f7!x ⁿ²f

is an isometry with respect to theL²innerproducts. The conjugation ofD^p via this isometry is illustrated by the diagram (of unbounded operators with domainC¹_c (X)):

L²(X;x^2pg_F) ƒƒƒ!^Dp L²(X;x^2pg_F)

x^pn2

# ^x

pn2

# L²(X;g_F) ƒƒƒƒƒ!^x

pn2D^px^pn2 L²(X;g_F) …2†

Using (1), we see thatx^pn2D^px ^pn2 ˆx ^p2D^Fx ^p2. p Unitarily equivalent operators have isomorphic kernels, hence the in- dex of the chiral partD^p_‡ofD^pequals the index ofx ^p2D^F_‡x ^p2.

The analytic facts that we need from the general theory ofFoperators [3] are the existence of the weightedF-Sobolev spacesx^aH^b_F(X) on which F-operators act, as well as the existence of parametrices inside the F calculus [3, Proposition 8]; that is, for everyA2C^a;b_F elliptic there exists B2C_F^{a; b}with

BA 12C_F^1;0:

By definition,A2C^a;b_F is calledfullyellipticif the normal operator ofx^bAis invertible as a family overYof suspended operators acting on the fibers of W. IfAis fully elliptic, thenBA 1 can even be made to belong toC_F^{1; 1}. LEMMA 8. The operators D^Fand x ^p2D^Fx ^p2are simultaneouslyfully- elliptic.

PROOF. Note that D^F2C^1;0_F (X) and x ^p2D^Fx ^p22C^1;p_F (X;S). In cusp- type calculi, it is a basic fact that commutation by a powerofxdecreases the totalx-order (i.e.,increasesthe powerofx; the filtration is defined by the negative of the powerofxso that it is decreasing, like the symbol filtration).

Namely, forA2C^a;b_F (X), we have

[A;x^p]2C^a_F ^{1;b‡p 1}(X):

This implies that the normal operator satisfies

N(x^px ^p2D^Fx ^p2)ˆ N(D^F): p

REMARK9. This lemma fails forb-operators, see [4].

The proof of the following ``elliptic regularity'' lemma is standard.

LEMMA10. Let A2C^a;b_F (X;E;F)be fullyelliptic. Then the L²solutions of Acˆ0belong to the ideal x¹C¹(X;E).

PROOF. Since A is fully elliptic there exists B2C_F^{a; b}(X;F;E) in- vertingAup toR2x¹C_F¹(X;E), i.e.,

BAˆ1‡R:

Let c2L²(X;E) be a distributional solution of the pseudo-differential equationAcˆ0. It follows

0ˆBAcˆ(1‡R)cˆc‡Rc

socˆ Rc. ButR2x¹C_F¹(X;E) impliesRc2x¹C¹(X;E). p We can now finish the proof of Theorem 1.1. Recall that we assumed the family of Dirac operators D^F on the fibers ofW to be invertible. By [8, Lemma 3.7] this implies that the normal operator ofD^Fis invertible, i.e., thatD^Fis fully elliptic. For the convenience of the reader, we include below a direct proof. Recall that an exactF-metric is assumed to coincide with a productF-metric up to first order terms inx. SincexC¹(X;^FTX) is an ideal in the Lie algebra of fibered cusp vector fields, it follows that the Levi- Civita connection and the Dirac operator also agree, up to first order terms, with the corresponding objects for the productF-metric. For that metric, it is straightforward to see (once the definition of the normal op- erator is recalled) that

N(D^F)²(t)ˆ ktk²_gY‡(D^F)²:

ThereforeN(D^F)²(t) is invertible for allt2TYif and only if the familyD^F is invertible.

In conclusion, the map

ker(x ^p2D^Fx ^p2)!kerD^F f7!x ^p2f

is well-defined by the Lemma 2, has an obviously well-defined inverse f7!x^p2f, and preserves parity, hence it defines a graded isomorphism. We compose this isomorphism with the Hilbert space isometry (2). p

The above proof holds more generally for an arbitrary F-metric whose associated Dirac operatorD^Fis fully elliptic. This condition seems howeverdifficult to check in practice outside the exact case. In the cusp case (i.e., Y is a point), a small improvement was achieved in [6], where one could decide foraclosedcusp metric whether the Dirac operator is fully-elliptic ornot.

Forcompleteness of the exposition, we describe the domains of the operators discussed here. They are certain weightedF-Sobolev spaces.

LEMMA11. The operators D^Fand x ^p2D^Fx ^p2are essentiallyself-adjoint in L²_F. The domain of the adjoint of D^Fis H¹_F. If we assume moreover that D^Fis fullyelliptic, then the domain of the adjoint of x ^p2D^Fx ^p2coincides with x^pH¹_F.

REMARK 12. Since for0p1 the metrics are complete, we know that the Dirac operators have precisely one self-adjoint extension; the extra fact here for suchpis identifying the domain of the extension.

PROOF. The domains of the closures of the operatorsx ^p2D^Fx ^p2andD^F always contains the Sobolev spacesx^pH¹_F, respectivelyH¹_F(see [3]).

Since D^F is elliptic in the F-sense, there exists B2C_F¹ with R:ˆBD^F 12C_F¹. Letf2L²_F be in the domain of the adjoint ofD^F. Then

fˆBD^Ff Rf:

Since by hypothesis D^Ff2L²_F, we get BD^Ff2H¹_F. At the same time, Rf2H¹_F. Thusf2H¹_F. Therefore

Dom(D^F)H¹_FDom(D^F):

If moreover we assume thatD^Fis fully elliptic, there existsB2C_F^{1; p} with R:ˆBx ^p2D^Fx ^p2 12C_F^{1; 1}. Let f2L²_F be in the domain of the adjoint of x ^p2D^Fx ^p2, so fˆBx ^p2D^Fx ^p2f Rf: By hypothesis x ^p2D^Fx ^p2f2L²_F, so Bx ^p2D^Fx ^p2f2x^pH¹_F. Moreover, Rf2x¹H¹_F, so in conclusionf2x^pH¹_F. Thus

Domx ^p2D^Fx ^p2

x^pH¹_FDomx ^p2D^Fx ^p2 :

But for every symmetric operatorDwe have Dom(D)Dom(D). p

Since the embeddingx^aH^b_F,!L²_Fis compact fora;b>0, it follows that fully elliptic F-operators are Fredholm [3]. Hence the kernels discussed here are all finite-dimensional.

In a recent work, Leichtnam, Mazzeo and Piazza [2] computed the index of the Dirac operatorD^Fcorresponding to an exactFmetric for which the decomposition [c;1)M is orthogonal near infinity. They deform the metric to ab-metric using results of Melrose and Rochon [5], apply Mel- rose's b-index formula [4], and use the adiabatic limit formula of Bismut and Cheeger [1]. Together with Theorem 1.1 and a careful analysis of the index density their argument gives a short proof of Vaillant's index for- mula [8] in the exact, orthogonal, fully elliptic case. Although the ortho- gonality hypothesis is used in the proof, the authors informed us in a private communication that the argument goes through under a weaker hypothesis of orthogonality only up to second order at the boundary.

2. - Horn-cone and fibered horn metrics.

The case pˆ1 corresponds to the d-metric, our original motivation, andpˆ0 corresponds to the initialF-metric. Forp>1 the metricx^2pg_Fis incomplete, nevertheless with the invertibility assumption along the fibers, the associated Dirac operator is essentially self-adjoint, by Lemma 2. In this case, if we start with a product F metric, the change of variables yˆx^{p 1} gives (up to a constant) a metric depending on the parameter aˆ p

p 1>1, with singularity atyˆ0:

x^2pg_Fˆdy²‡y²Wg^Y‡y^2ah …3†

whereg_Yis a metric on the base of the boundary fibration, andhis a family of metrics on the fibers. We call (3) a horn-cone metric. Similarly, for 0<p<1, the change of variablesyˆx^{1 p}gives foraˆ p

p 1<0:

x^2pg_Fˆdy² y⁴ ‡Wg^Y

y² ‡y ^2ah …4†

We call this afibered hornmetric. It is singular atyˆ0. The metric (3) is incomplete, while (4) is complete, expands with time in the base di- rections and shrinks in the fibers. In the result below, we could allow some weakerasymptotics forthese metrics, such that afterreverting back to the variablexthe resulting metric is smooth inxand exact as a conformalF-metric.

COROLLARY 13. Let (X; g) be a spin Riemannian manifold with a boundaryfibration W:@X!Y, isometric outside a compact set to a disjoint union of ends of the following types:

horn-cone type(3);

fibered horn(4);

exactd-metric;

exactF-metric.

The conformal weights p maybe different for each end. Let E!X be an auxiliaryvector bundle endowed with a Hermitian connection smooth in x down to xˆ0, where xˆy^p11for ends of type(3),respectivelyxˆy^11p for ends of type(4).If the familyof (twisted) Dirac operators on the fibers on each end with respect to the induced vertical familyof metrics is in- vertible, then the Dirac operator D^g on X is essentiallyself-adjoint and Fredholm. If the dimension of X is even, the index of D^g is given by

index(D^g_‡)ˆ Z

X

A(g)ch(E)^ ‡ Z

Y

A(g^ _Y)^h(D^h):

Moreover, D^ghas compact resolvent (thus, purelydiscrete spectrum) if and onlyif X has noF-type end. In that case a generalized Weyl law holds as in [6].

The right-hand side involves the eta-form defined in [1].

PROOF. We compactifyXusing the variablexˆy^{a 1}foreach end of horn-cone type, and xˆy^{1 a} for ends of fibered horn type, thus our metric becomes of the formx^2pgF, wher epis now a function from the set of ends into [0;1). The metric gF is exact forall the ends. Then by [8, Chapter 2], the Pontrjagin forms ofgare smooth down toxˆ0, so there is no need to regularize the local index integral. The operatorD^gis fully elliptic by the invertibility assumption. By Lemma 2, it follows that it is essentially self-adjoint. The resolvent lives in C ^{1; p} so it is compact if and only if p takes values in (0;1), i.e., if there is no F-type end. By Theorem 1.1, the index is the same as that of the associated conformald- metricg_d, which is computed by Vaillant by the above formula, only that it involves the index density forg_d, not forg. But of course, the Pontrjagin forms are conformal invariants so the formula follows. Alternately, in- stead of Vaillant's result we may use [2] (see the discussion on ortho- gonality at the end of the previous Section).

The Weyl law follows entirely as in [6], based on the fact that the zeta function ofD^g has a meromorphic extension to the complex plane, with a

possibly double first pole. p

3. - The non-fully elliptic case.

Let us mention also a partial result in the non-fully elliptic case. Assume that the hypothesis of Vaillant [8] holds, namely that the kernels of the family of Dirac operators along the fibers of the boundary form a vector bundle. Then Vaillant [8] showed that the dimension of ker(D^d) is finite.

PROPOSITION14. Under the above hypothesis, for every0p1, the kernel of the Dirac operator on X associated to x^2pg_F is finite-dimen- sional.

PROOF. Let D(p) denote the Dirac operator of x^2pgF. Following the proof of Theorem 1.1, we see that the map of multiplication byx^12pinside L²(X;g_d) maps the kernel of an operator inL²(X;g_d) unitarily equivalent to D(p), into the kernel of D^d. This multiplication map is injective but not surjective. Thus kerD(p) injects into ker(D^d), whose dimension if finite by

[7, Chapter3]. p

For pˆ0, underVaillant's hypothesis, the index of D^F is finite, al- though from Melrose's Fredholm criterion we see that in the non-fully elliptic case, D^Fis not Fredholm on any Sobolev space with exponential weighte^axx^bL²(X;gF).

Acknowledgments. I am indebted to Paolo Piazza forproposing this question and forhis valuable suggestions and comments.

REFERENCES

[1] J.-M. BISMUT- J. CHEEGER,h-invariants and their adiabatic limits,J. Amer.

Math. Soc.2(1989), pp. 33-70.

[2] E. LEICHTNAM- R. MAZZEO - P. PIAZZA, The index of Dirac operators on manifolds with fibered boundaries, to appear in Proc. Joint BeNeLuxFra Confer. Math., Ghent, May 20-22, 2005, Bull. Belg. Math. Soc. - Simon Stevin.

[3] R. R. MAZZEO- R. B. MELROSE,Pseudodifferential operators on manifolds with fibered boundaries,Asian J. Math.,2(1998), pp. 833-866.

[4] R. B. MELROSE,The Atiyah-Patodi-Singer index theorem,Research Notes in Mathematics4, A. K. Peters, Wellesley, MA (1993).

[5] R. B. MELROSE- F. ROCHON,Index in K-theoryfor families of fibred cusp operators,preprint math. DG/0507590.

[6] S. MOROIANU,Weyl laws on open manifolds,preprint math. DG/0310075.

[7] W. MUÈLLER,Manifolds with cusps of rank one. Spectral theoryand L²-index theorem, Lecture Notes in Math.1244, Springer-Verlag, Berlin, 1987.

[8] B. VAILLANT, Index- and spectral theoryfor manifolds with generalized fibered cusps,Dissertation, Bonner Math. Schriften 344 (2001), Rheinische Friedrich-Wilhelms-UniversitaÈt Bonn.

Manoscritto pervenuto in redazione il 7 giugno 2006