Heat kernel expansions, ambient metrics and conformal invariants

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 P2(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 a2k(r;g) of H(r;g) on closed manifolds. We prove general structural results for the heat kernel coefficients a2k(r;g) and derive explicit formulas for a0(r;g) and a2(r;g) in terms of renormalized volume coefficients. The Taylor coefficients of a2k(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 a2(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).