Fundamental properties of manifolds

The goal of this chapter is to obtain information about the heat kernel of a complete Riemannian manifold. There is a tremendous literature on this subject, much of which concerns the asymptotic form of the heat kernel K(t, x, y) as t → 0; we, however, shall be mainly interested in finding uniform upper and lower bounds over the whole range of t, x, y. For manifolds of non-negative Ricci curvature this problem is now largely solved as a result of the efforts of Li and Yau, whose work we shall describe below; see Corollary 5.3.6 and Theorems 5.5.6 and 5.6.3.

If one merely assumes that the Ricci curvature of the manifold is bounded below by a negative constant, the above methods can still be applied but give a much less complete picture. Indeed even for hyperbolic space and its quotients by Kleinian groups the variety of phenomena which can occur is vast and only partly understood. In Section 5.7 we give a brief summary of some recent results, without proofs.

We start with a brief introduction to manifold theory in order to fix notation. Let M be an n-dimensional (connected) manifold with tangent space TM and cotangent space T*M. Smooth sections of TM are called vector fields and smooth sections of T*M are called forms.

This book is a study of linear, self-adjoint, second order, elliptic differential operators. The goal is to investigate spectral properties and obtain pointwise bounds on eigenfunctions by studying the heat kernels.

There is an enormous literature on heat kernels which stretches back half a century, so it is easy to imagine that the subject has already reached its final form. However, we shall make almost no reference to this literature, and shall rely entirely upon results proved within the last five years using quadratic form techniques and logarithmic Sobolev inequalities. These new techniques have led to radically better global bounds on the heat kernels.

We shall be concerned to obtain pointwise upper and lower bounds on various functions in terms of effectively computable constants. In a number of cases the new methods yield constants which are sharp or at least of the correct order of magnitude. This is in sharp distinction to much of the older theory, where various constants appeared to depend upon the magnitudes of the derivatives of the second order coefficients of the differential operator, although in fact they do not.

Because of our approach we are able to deal simply and naturally with operators in divergence form whose second order coefficients are measurable. Earlier treatments of this problem such as that of Gilbarg and Trudinger have relied heavily upon Moser's Harnack inequality. In spite of its fundamental historical and conceptual importance, we make no mention of Moser's approach.

Abstract: Technics recently developped in non commutative geometry through properties of C*Algebras are presented without proofs here to investigate some properties of 2D electrons in a uniform magnetic field. The Peierls substitution, a commonly used approximation for Bloch electrons, is justified and leads to the Rotation Algebra. A new differential calculus, analogous to the Ito calculus for stochastic processes, is introduced to investigate the fine structure of the energy spectrum. We announce the proof of the Wilkinson Rammal formula according to which the derivative of the energy gap boundaries are discontinuous at each rational value of the magnetic flux. We also announce that the derivative is continuous at irrational values of this flux. At last we review and improve the results previously obtained for the Quantum Hall Effect and sketch the proof that in the region of localized states the Hall conductance exhibits plateaux at integer values of the universal constant e2/h.

Introduction:

Several problems in Solid State Physics proved recently to be understandable through using technics developped in non commutative topology and geometry. Let us mention the study of SchrOdinger operators with almost periodic potential [14,86] which may describe the behavior of a Bloch electron in a uniform magnetic field [see 88 for a review and discussion below], the stability of a dynamical system near a quasi periodic orbit [69,86], the periodic or quasi periodic solutions of the KdV equation [35,69, see 86 and references therein], the ground state properties of one dimensional organic conductors [7, 8], the metal-superconductor for superconductors in a network [3, 40, 72, 80], weak localization in normal metal networks [32, 33, 34] or the electronic properties of quasi crystals [36,58,66,83].

INTRODUCTION The purpose of this note is to show how ideas from cyclic homology may be used to study geometrical and analytic properties of loop spaces. As such, the material in this note is not directly related to operator algebras; on the other hand it is not too distant either since cyclic homology is one of several good things to come out of the study of operator algebras in recent years.

Geometry and analysis on loop spaces is a subject of much current interest. One of the ideas which has emerged is the following, rather striking, point of view on the Chern character and the index theorem. Let X be a smooth compact oriented manifold with no boundary and let LX be the space of all smooth loops in X. This loop space is an infinite dimensional manifold, modelled on a Fréchet space. The circle T acts on LX by rotating loops; this action is smooth and the fixed point set is the manifold X regarded as the space of constant loops. We are principally interested in properties of the smooth T-manifold LX. The relation between the Chern character, the index theorem and loop spaces is given by the following facts. Let E be a complex vector bundle on X equipped with a connection ∇.

1. (a) There is a T-invariant, equivariantly closed differential form ω(E, ∇) on LX with the property that the restriction of ω(E, ∇) to X, the space of constant loops, is precisely the usual Chern character form, ch(E, ∇) = Trace(eF) where F is the curvature of ∇. This form ω(E, ∇) is Bismut's equivariant extension of the Chern character.

2. […]

Given a subfactor N of a II, factor M with the same identity, one defines the index of N in M as [M:N] = dimN(L2 (M)) = the Murray von Neumann coupling constant for N on the Hilbert space L2 (M), (= completion of M with respect to the inner product <a,b> = tr(ab*), tr being the unique normalized trace on M). The following result shows the interest of this notion.

Theorem

1. a) If [M:N] < 4 then there is an n ∈ ℤ, n ≥ 3, with [M:N] =4 cos2π/n.

2. b) For any real r ≥ 4 there is a pair N ⊆ M with [M:N] = r.

The “basic construction” of the theory is as follows. If N ⊆ M are finite von Neumann algebras and tr is a faithful normal normalized trace on M, one considers the von Neumann algebra <M,eN> = {M,eN}” on L2 (M,tr) where eN, is the orthogonal projection onto L2 (N,tr) (tr restricted to N). If N and M are factors then [M:N] < ∞ iff <M,eN> is a II1, factor and then [M:N]tr(eN) = 1.

Ocneanu has made great progress on classifying subfactors of the hyperfinite II1 factor with given index, which he will explain in his talk. It would appear that the classification is complete for index < 4.