Heat kernel expansions, ambient metrics and conformal invariants

Abstract

The conformal powers of the Laplacian of a Riemannian metric which are known as the GJMS-operators admit a combinatorial description in terms of the Taylor coefficients of a natural second-order one-parameter family H(r; g) of self-adjoint elliptic differential operators. H(r; g) is a non-Laplace-type perturbation of the conformal Laplacian P 2 (g) = H(0; g). It is defined in terms of the metric g and covariant derivatives of the curvature of g. We study the heat kernel coefficients a 2k (r; g) of H(r; g) on closed manifolds. We prove general structural results for the heat kernel coefficients a 2k (r; g) and derive explicit formulas for a 0 (r) and a 2 (r) in terms of renormalized volume coefficients. The Taylor coefficients of a 2k (r; g) (as functions of r) interpolate between the renormalized volume coefficients of a metric g (k = 0) and the heat kernel coefficients of the conformal Laplacian of g (r = 0). Although H(r; g) is not conformally covariant, there is a beautiful formula for the conformal variation of the trace of its heat kernel. Its proof rests on the combinatorial relations between Taylor coefficients of H(r; g) and GJMS-operators. As a consequence, we give a heat equation proof of the conformal transformation law of the integrated renormalized volume coefficients. By refining these arguments, we also give a heat equation proof of the conformal transformation law of the renormalized volume coefficients itself. The Taylor coefficients of a 2 (r) define a sequence of higher-order Riemannian curvature functionals with extremal properties at Einstein metrics which are analogous to those of integrated renormalized volume coefficients. Among the various additional results the reader finds a Polyakov-type formula for the renormalized volume of a Poincaré-Einstein metric in terms of Q-curvature of its conformal infinity and additional holographic terms.The present study of relations between spectral theoretic heat kernel coefficients and geometric quantities like renormalized volume coefficients is stimulated by the holographic perspective of the AdS/CFT-duality (proposing relations between functional determinants and renormalized volumes).

The trace and the mass of subcritical GJMS operators

“…in [Juh13]. However, the GJMS operators have a recursive structure which was investigated by Juhl [Juh16].…”

Asymptotic expansions and conformal covariance of the mass of conformal differential operators

“…The nonexistence of GJMS operators with m > n 2 on general even-dimensional manifolds was shown in [13]. The recursive structure of these operators was investigated by Juhl in [21][22][23], and [24].…”