Kengo Hirachi

Conformally invariant powers of the Laplacian - A complete nonexistence theorem

Conformally invariant operators and the equations they determine play a central role in the study of manifolds with pseudo-Riemannian, Riemannian, conformai and related structures. This observation dates back to at least the very early part of the last century when it was shown that the equations of massless particles on curved space-time exhibit conformai invariance. In this setting a key operator is the con formally invariant wave operator which has leading term a pseudo-Laplacian. The Riemannian signature variant of this operator is a fundamental tool in the Yam abe problem on compact manifolds. Here one seeks to find a metric, from a given conformai class, that has constant scalar curvature. Recently it has become clear that higher order analogues of these operators, viz., conformally invariant operators on weighted functions (i.e., conformai densities) with leading term a power of the Laplacian, have a central role in generating and solving other curvature prescription problems as well as other problems in geometric spectral theory and mathematical physics [2, 5, 15]. In the flat setting, the existence of such operators dates back to [16], where it is shown that, on 4-dimensional Minkowski space, for k G N = {1,2,...}, the kth power of the flat wave operator Ak, acting on densities of the appropriate weight, is invariant under the action of the conformai group. More generally, if ?[w] denotes the space of conformai densities of weight uiGl, then on a flat conformai manifold of dimension n > 3 (and any signature) there exists, for each k E N, a unique conformally invariant operator

Construction of boundary invariants and the logarithmic singularity of the Bergman kernel

in which case the coefficients aj are expressed, by the Weyl invariant theory, in terms of the Riemannian curvature tensor and its covariant derivatives. The Bergman kernel’s counterpart of the time variable t is a defining function r of the domain Ω. By [F1] and [BS], the formal singularity of K at a boundary point p is uniquely determined by the Taylor expansion of r at p. Thus one has hope of expressing φ modulo On+1(r) and ψ modulo O∞(r) in terms of local biholomorphic invariants of the boundary, provided r is appropriately chosen. In [F3], Fefferman proposed to find such expressions by reducing the problem to an algebraic one in invariant theory associated with CR geometry, and indeed expressed φ modulo On−19(r) invariantly by solving the reduced problem partially. The solution in [F3] was then completed in [BEG] to give a full invariant expression of φ modulo On+1(r), but the reduction is still

Q-prime curvature on CR manifolds

Abstract Q -prime curvature, which was introduced by J. Case and P. Yang, is a local invariant of pseudo-hermitian structure on CR manifolds that can be defined only when the Q -curvature vanishes identically. It is considered as a secondary invariant on CR manifolds and, in 3-dimensions, its integral agrees with the Burns–Epstein invariant, a Chern–Simons type invariant in CR geometry. We give an ambient metric construction of the Q -prime curvature and study its basic properties. In particular, we show that, for the boundary of a strictly pseudoconvex domain in a Stein manifold, the integral of the Q -prime curvature is a global CR invariant, which generalizes the Burns–Epstein invariant to higher dimensions.

The second variation of the Bergman kernel of ellipsoids

Introduction. In this paper we shall study Feffermans asymptotic expansion of the Bergman kernel of (real) ellipsoids in C, w^>2. Regarding ellipsoids as perturbations of the ball, we compute the variations of the Bergmen kernel, and give the Taylor expansion of the log term coefficient to the second order in Websters invariants. (The ellipsoids in normal form are parametrized by n real numbers, which we call Websters invariants, and we shall consider the ellipsoids with small parameters.) As a consequence, we show that the vanishing of the log term of the Bergmen kernel characterizes the ball among these ellipsoids. In addition, we derive, from the procedure of computing the variation, a relation among the Bergmen kernels of different dimensional ellipsoids. Let Ω be a smoothly bounded strictly pseudoconvex domain in C, with defining function/>0 in Ω. It has been known since the work of Fefferman [5] that the Bergman kernel of Ω is written in the form

Variation of total Q-prime curvature on CR manifolds☆

Abstract We derive variational formulas for the total Q -prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q -prime curvature agrees with the renormalized volume of such domains with respect to the complete Kahler–Einstein metric.

Inhomogeneous Ambient Metrics

The ambient metric, introduced in [FG1], has proven to be an important object in conformal geometry. To a manifold M of dimension n with a conformai class of metrics [g] of signature (p, q) it associates a diffeomorphism class of formal expansions of metrics \( \tilde g \) of signature (p + 1, q + 1) on a space Open image in new window of dimension n + 2. This generalizes the realization of the conformal sphere Sn as the space of null lines for a quadratic form of signature (n + 1, 1), with associated Minkowski metric \( \tilde g \) on ℝn+2. The ambient space Open image in new window carries a family of dilations with respect to which \( \tilde g \) is homogeneous of degree 2. The other conditions determining \( \tilde g \) are that it be Ricci-flat and satisfy an initial condition specified by the conformal class [g].

Local Sobolev—Bergman Kernels of Strictly Pseudoconvex Domains

This article grew out of an attempt to understand analytic aspects of Fefferman’s invariant theory [F3] of the Bergman kernel on the diagonal of Ω × Ω for strictly pseudoconvex domains Ω in ℂ n with smooth (C ∞ or real analytic) boundary. The framework of his invariant theory applies equally to the Szego kernel if the surface element on ∂Ω is appropriately chosen, while the Szego kernel is regarded as the reproducing kernel of a Hilbert space of holomorphic functions in Ω which belong to the L 2 Sobolev space of order 1/2. This fact is our starting point. For each s ∈ ℝ, we first globally define the Sobolev-Bergman kernel K s of order s/2 to be the reproducing kernel of the Hilbert space H S /2(Ω) of holomorphic functions which belong to the L 2 Sobolev space of order s/2, where the inner product is specified arbitrarily.

Integral Kähler invariants and the Bergman kernel asymptotics for line bundles

Abstract On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q -curvature on a Sasakian manifold is a divergence.

Variation of the Total Q -Prime Curvature in CR Geometry

>Tom Branson introduced the concept of Q-curvature in conformal geometry, in connection with the study of conformal anomaly of determinants of conformally invariant differential operators. The definition can be generalized to CR manifolds via Fefferman’s conformal structure on a circle bundle over CR manifolds, see [2]. Using this correspondence, one can translate the properties of conformal Q-curvature to the CR analogue. However, there has been an important missing piece in this correspondence.

Ambient Metric Construction of CR Invariant Differential Operators

These notes are based on my lectures at IMA, in which I tried to explain basic ideas of the ambient metric construction by studying the Szego kernel of the sphere. The ambient metric was introduced in Fefferman [F] in his program of describing the boundary asymptotic expansion of the Bergman kernel of strictly pseudoconvex domain. This can be seen as an analogy of the description of the heat kernel asymptotic in terms of local Riemannian invariants. The counterpart of the Riemannian invariants for the Bergman kernel is invariants of the CR structure of the boundary. Thus the program consists of two parts: (1) Construct local invariants of CR structures; (2) Prove that (1) gives all invariants by using the invariant theory. In the case of the Szego kernel, (1) is replaced by the construction of local invariants of the Levi form that are invariant under scaling by CR pluriharmonic functions. We formulate the class of invariants in Sections 2 and 3. To simplify the presentation, we confine ourself to the case of the sphere in ℂn. It is the model case of the ambient metric construction and the basic tools already appears in this setting. We construct invariants (formulated as CR invariant differential operators) by using the ambient space in Section 4 and then explain, in Section 5, how to prove that we have got all.