The Hypoelliptic Laplacian and Ray-Singer Metrics. (AM-167)

Jean-Michel Bismut
Gilles Lebeau
Pages: 376
https://www.jstor.org/stable/j.ctt7sstb

1. Front Matter
(pp. i-iv)
(pp. v-x)
3. Introduction
(pp. 1-10)

The purpose of this book is to develop the analytic theory of the hypoelliptic Laplacian and to establish corresponding results on the associated Ray-Singer analytic torsion. We also introduce the corresponding theory for families of hypoelliptic Laplacians, and we construct the associated analytic torsion forms. The whole setting will be equivariant with respect to the action of a compact Lie group$G$.

Let us put in perspective the various questions which are dealt with in this book. In [B05], one of us introduced a deformation of classical Hodge theory. Let$X$be a compact Riemannian manifold, let$(F,\;{\nabla ^F},\;{g^F})$be a...

4. Chapter One Elliptic Riemann-Roch-Grothendieck and flat vector bundles
(pp. 11-24)

The purpose of this chapter is to recall the results on elliptic analytic torsion forms obtained by Bismut and Lott [BLo95] and later extended by Bismut and Goette [BG01] to the equivariant context.

This chapter is organized as follows. In section 1.1, we state elementary results on Clifford algebras.

In section 1.2, we recall some basic results of standard Hodge theory.

In section 1.3, we give a short account of the construction of the Levi-Civita superconnection in the context of [BLo95].

In section 1.4, we review the relations of this construction to Poincaré duality.

In section 1.5, we introduce a...

5. Chapter Two The hypoelliptic Laplacian on the cotangent bundle
(pp. 25-43)

The purpose of this chapter is to recall the main results in Bismut [B05] on the hypoelliptic Laplacian and also on the families version of this operator.

This chapter is organized as follows. In section 2.1, we recall the construction of the exotic Hodge theory given in [B05]. This construction depends in particular on the choice of a Hamiltonian$\mathcal{H}$.

In section 2.2, we give the Weitzenböck formula for the corresponding hypoelliptic Laplacian.

In section 2.3, we state the results in [B05] according to which this new Hodge theory is a deformation of classical Hodge theory.

In section 2.4, we...

6. Chapter Three Hodge theory, the hypoelliptic Laplacian and its heat kernel
(pp. 44-61)

The purpose of this chapter is to give a short summary of the results on the analysis of the operator$A_{\phi ,\;{\mathcal{H}^c}}^2$, these results being established in detail in chapters 15 and 17. In particular we state in detail convergence results on the resolvent of$\mathfrak{A}'_{{\phi _b},\; \pm \mathcal{H}}^2$as$b\; \to \;0$which are established in chapter 17. Also we derive various results on the spectral theory of$A_{\phi ,\;\mathcal{H}{^c}}^2$. In particular we show that for$b\; > \;0$small enough, the standard consequences of Hodge theory hold, and we prove that the set of$b\; > \;0$where the Hodge theorem does hold...

7. Chapter Four Hypoelliptic Laplacians and odd Chern forms
(pp. 62-97)

In this chapter, given$b\; > \;0$, we construct the odd Chern forms associated to a family of hypoelliptic Laplacians. The idea is to adapt the construction of the forms of [BLo95, BG01] which was explained in chapter 1.

Our Chern forms depend on the parameters$b\; > \;0$,$t\; > \;0$. We will study their asymptotics as$t\; \to \;0$. The asymptotics of the forms rely on local index theoretic techniques which we adapt to the hypoelliptic context.

The proofs of some of the probabilistic results which are needed in the proof of the localization properties of the heat...

8. Chapter Five The limit as $t\; \to \; + \infty$ and $b \to 0$ of the superconnection forms
(pp. 98-112)

The purpose of this chapter is to establish the asymptotics as$t\; \to \; + \infty$or$b \to \;0$of the hypoelliptic superconnection forms which were constructed in chapter 4. We show that for$b > \;0$small enough, convergence as$t\; \to \; + \infty$is uniform, and also that as$b \to \;0$, the hypoelliptic superconnection forms converge to the elliptic superconnection forms, and this occurs uniformly when$t > \;0$stays away from 0.

This chapter is organized as follows. In section 5.1, we define what will eventually be limit superconnection forms, for$t\; = \; + \infty$or for$b = 0$....

9. Chapter Six Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics
(pp. 113-130)

The purpose of this chapter is to define hypoelliptic torsion forms and corresponding hypoelliptic Ray-Singer metrics on the line$\lambda \; = \;\det \;{\mathfrak{H}^ \cdot }\;(X,\;F)$. The main result of this chapter is that these objects verify transgression equations very similar to the corresponding equations we gave in chapter 1 for their elliptic counterparts. It is then natural to try to compare the hypoelliptic objects to the elliptic ones. This will in fact be done in chapters 8 and 9.

The present chapter is organized as follows. In section 6.1, we construct the hypoelliptic torsion forms, and we briefly study their dependence on the parameter$b > \;0$....

10. Chapter Seven The hypoelliptic torsion forms of a vector bundle
(pp. 131-161)

The purpose of this chapter is to calculate the hypoelliptic torsion forms of a real Euclidean vector bundle equipped with a Euclidean connection. These explicit computations will play a key role in establishing the formula which compares the elliptic to the hypoelliptic torsion forms. The fact that our computations are closely related to computations in [BG01, section 4] in the elliptic case can be considered as a microlocal version of our comparison formula.

The present chapter is also related in spirit with similar computations which were done in a holomorphic context in [B90, B94].

This chapter is organized as follows....

11. Chapter Eight Hypoelliptic and elliptic torsions: a comparison formula
(pp. 162-170)

In this chapter, we establish the main result of the book. Namely, we give an explicit formula relating the hypoelliptic torsion forms to the corresponding elliptic torsion forms. The proofs of several intermediate results are deferred to the following chapters.

This chapter is organized as follows. In section 8.1, we construct natural secondary Chern classes attached to two couples of generalized metrics on${\mathfrak{H}^ \cdot }\;(X,\;F)$.

In section 8.2, we state our main result. The next sections are devoted to the proof of this result.

In section 8.3, we introduce a rectangular contour$\Gamma$in${\text{R}}_ + ^{*2}$on which the form a of section 4.2...

12. Chapter Nine A comparison formula for the Ray-Singer metrics
(pp. 171-172)

We make the same assumptions as in chapter 6, and we use the corresponding notation. Also we assume here that$S$is reduced to a point.

Recall that$b\; \in \;{\mathbf{R}}_ + ^*$and that$c\; = \; \pm 1\,/\,{b^2}$. By Theorem 6.7.1, the generalized metric$\parallel \;\parallel _{\lambda ,\;b}^2$does not depend on$b$. Recall that a priori,$\parallel \;\parallel _{\lambda ,\;b}^2$is only a generalized equivariant metric, in the sense that the sign of the$\parallel \;\parallel _{\lambda \;W}^2$,$W\; \in \;\hat G$\is not necessarily positive.

When$n$is even, or when$n$is odd and$c\; > \;0$, we denote by$\parallel \;\parallel _{\lambda ,\;b}^2$the corresponding more classical Ray-Singer metric on$\lambda$...

13. Chapter Ten The harmonic forms for $b\; \to \;0$ and the formal Hodge theorem
(pp. 173-181)

The purpose of this chapter is twofold.

On the one hand, in section 10.1, we prove Theorem 8.4.2, i. e., we compute the asymptotics of the generalized metrics$\mathfrak{h}_b^{{\mathfrak{H}^ \cdot }\,(X,\;T)}$as$b\; \to \;0$.

On the other hand, in section 10.2, we give a direct proof of a formal Hodge theorem as$b\; \to \;0$. Namely, we prove that up to some trivial scaling, the space of formal power series in the variable$b\; > \;0$which lies in$r_b^*\mathbb{H}_b^ \cdot \;(X,\;F)$is in one to one correspondence with${\mathfrak{H}^ \cdot }\;(X,\;F)$. More precisely, given a fixed class in${\mathfrak{H}^ \cdot }\;(X,\;F)$, we compute the formal...

14. Chapter Eleven A proof of equation (8.4.6)
(pp. 182-189)

The purpose of this chapter is to establish equation (8.4.6) in Theorem 8.4.3. We will thus compute explicitly the limit as$t\; \to \;0$of${\upsilon _{\sqrt {tb} ,t}}$. The techniques are closely related to the ones we used in chapter 4. However, a direct application of the results of that chapter would lead to spurious divergences. This forces us to modify our trivializations, very much in the spirit of the local version of the families index theorem of [B86].

This chapter is organized as follows. In section 11.1, we introduce our new trivialization and rescaling of the creation and annihilation variables....

15. Chapter Twelve A proof of equation (8.4.8)
(pp. 190-193)

The purpose of this chapter is to establish equation (8.4.8) in Theorem 8.4.3. We thus establish a uniform bound on$\,{v_{b,\;\varepsilon }}\,|$for$\varepsilon \; \in \;[0,\;1],\;b\; \in \;[\sqrt \varepsilon ,\;{b_0}]$. The proof relies on techniques already used in chapters 4 and 11.

This chapter is organized as follows. In section 12.1, we combine the techniques of chapters 4 and 11 to obtain a uniform expansion of the rescaled operator in the considered range of parameters.

In section 12.2, we prove the required estimate.

In this chapter, we fix${b_0} \geqslant \;1$. We take$t\; \in \;[0,\;1],\;b\; \in \;[\sqrt t ,\;{b_0}]$. We use the notation of section 4.7, while making in the whole chapter$z\; = \;0,\;dc\; = \;0,\;db\; = \;0,\;dt\; = \;0$....

16. Chapter Thirteen A proof of equation (8.4.7)
(pp. 194-213)

The purpose of this chapter is to establish the estimate (8.4.7), which gives an estimate for$\left| {{\upsilon _{\sqrt {tb} ,t}} - {\upsilon _{0,t}}} \right|$which is uniform in$b\; \in \;[0,\;1],\;t\; \in \;[0,\;1]$. The idea is to combine the local index techniques of chapter 11 with the functional analytic machine which is extensively developed in chapter 17 to prove that in the proper sense, as$b\; \to \;0$, the hypoelliptic Laplacian converges to the standard Laplacian. Indeed our estimate is compatible with the convergence result for${\upsilon _{\sqrt {tb} ,t}}$, as$t\; \to \;0$, which is established in chapter 11. This explains why the methods of that chapter play...

17. Chapter Fourteen The integration by parts formula
(pp. 214-223)

The purpose of this chapter is to apply the basic techniques of the Malliavin calculus to the hypoelliptic diffusion which is associated with the hypoelliptic Laplacian.

This chapter should be put in historical context. Malliavin invented his calculus in [M78], as a technique to derive integration by parts formulas with respect to the Brownian measure. He showed in particular that solutions of stochastic differential equations were accessible to his calculus. One main application was the proof by Malliavin of the regularity of the heat kernel associated to a second order hypoelliptic differential operator of the form considered by Hörmander [Hör67]....

18. Chapter Fifteen The hypoelliptic estimates
(pp. 224-246)

The purpose of this chapter is to prove the basic hypoelliptic estimates on the Laplacian$\mathfrak{A}'_{{\phi _b},\; \pm \mathcal{H}}^2.$The proofs extend very easily to the curvature$\mathfrak{C}_{{\phi _b},\; \pm \mathcal{H} - b{\omega ^H}}^{\mathcal{M},\,2}$of the superconnection$\mathfrak{C}_{{\phi _b},\; \pm \mathcal{H} - b{\omega ^H}}^\mathcal{M}$defined in (2.4.15).

We consider$\mathfrak{A}'_{{\phi _b},\; \pm \mathcal{H}}^2$,$\mathfrak{C}_{{\phi _b},\; \pm \mathcal{H} - b{\omega ^H}}^{\mathcal{M},\,2}$instead of$\mathfrak{A}'_{\phi ,\;{\mathcal{H}^c}}^2,\;\mathfrak{C}_{\phi ,\;{\mathcal{H}^c} - {\omega ^H}}^2$because in chapter 17, we will establish estimates which are uniform in$b\; \in \;[0,\;{b_0}],$where${b_0} > 0$is a positive constant, for which the choice of$\mathfrak{A}'_{{\phi _b},\; \pm \mathcal{H}}^2,\;\mathfrak{C}_{{\phi _b},\; \pm \mathcal{H} - b{\omega ^H}}^{\mathcal{M},\,2}$is more natural. Part of the work which is needed in chapter 17 will then have been already done in the present chapter.

This chapter is organized as follows. In section 15.1,...

19. Chapter Sixteen Harmonic oscillator and the ${J_0}$ function
(pp. 247-263)

The purpose of this chapter is to introduce the basic tools which are needed in the proof of the convergence of the resolvent of the operator$2'_{{\phi _b},\; \pm {\Cal H}}^2$to the resolvent of${^X}/2$as$b \to 0$.

Here we essentially consider the case where$X$is a flat torus, or even${{\bf{R}}^n},$and we give an explicit formula for the resolvent for$b\; = \;1.$In chapter 17, this will be used when studying the semiclassical symbol of the resolvent of$2'_{{\phi _b},\; \pm {\Cal H}}^2$and establishing its main properties.

This chapter is organized as follows. In section 16.1, we introduce the formalism...

20. Chapter Seventeen The limit of $\mathfrak{A}'_{{\phi _b},\; \pm \mathcal{H}}^2$ as $b \to 0$
(pp. 264-352)

The purpose of this chapter is to study the asymptotics of the hypoelliptic Laplacian${L_c} = \;2\mathfrak{A}'_{{\phi _b},\; \pm \mathcal{H}}^2$as$b \to 0$. Our main result is that, as anticipated in [B05], it converges in the proper sense to the standard Laplacian${^X}/2.$. As in chapter 15, we only consider the case of one single fiber, the more general case of the hypoelliptic curvature of a family does not introduce any significant new difficulty.

Since this chapter is analytically quite involved, we will try to describe its organization in painstaking detail. The operator${L_c}$is of order 1 in the horizontal directions, while${^X}/2$is of...

21. Bibliography
(pp. 353-358)
22. Subject Index
(pp. 359-360)
23. Index of Notation
(pp. 361-367)