Heat kernel expansions, ambient metrics and conformal invariants

  title={Heat kernel expansions, ambient metrics and conformal invariants},
  author={Andreas Juhl},
  journal={arXiv: Differential Geometry},
  • A. Juhl
  • Published 2014
  • Mathematics
  • arXiv: Differential Geometry
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… Expand
4 Citations

Tables from this paper

Asymptotic expansions and conformal covariance of the mass of conformal differential operators
We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of m-Laplace type operators L on compact Riemannian manifolds in terms of RieszExpand
Shift operators, residue families and degenerate Laplacians
We introduce new aspects in conformal geometry of some very natural second-order differential operators. These operators are termed shift operators. In the flat space, they are intertwining operatorsExpand
Volume renormalization for singular Yamabe metrics
This paper carries out a renormalization of the volume of the Loewner-Nirenberg singular Yamabe metric in a given conformal class on a compact manifold-with-boundary. This generalizes the usualExpand
The trace and the mass of subcritical GJMS operators
Let L g L g be the subcritical GJMS operator on an even-dimensional compact manifold (X,g) ( X , g ) and consider the zeta-regularized trace Tr ζ ( L g − 1 ) of its inverse. We show that if ker⁡L gExpand


On conformally covariant powers of the Laplacian
We propose and discuss recursive formulas for conformally covariant powers $P_{2N}$ of the Laplacian (GJMS-operators). For locally conformally flat metrics, these describe the non-constant part ofExpand
Invariants of conformal Laplacians
The conformal Laplacian D = d*d + (n - 2)s/4(n - 1), acting on functions on a Riemannian manifold Mn with scalar curvature s, is a conformally invariant operator. In this paper we will use D toExpand
Leading terms in the heat invariants
Let D be a second-order differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold. The asymptotics of the heat kernel based on D are given by homogeneous,Expand
Laplacian Operators and Q-curvature on Conformally Einstein Manifolds
A new definition of canonical conformal differential operators Pk (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. TheseExpand
Conformal geometry, contact geometry, and the calculus of variations
for metricsg in the conformal class of g0, where we use the metric g to view the tensor as an endomorphism of the tangent bundle and where σk d notes the trace of the induced map on the kth exteriorExpand
Volume and Area Renormalizations for Conformally Compact Einstein Metrics
Let $X$ be the interior of a compact manifold $\overline X$ of dimension $n+1$ with boundary $M=\partial X$, and $g_+$ be a conformally compact metric on $X$, namely $\overline g\equiv r^2g_+$Expand
We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\unicode[STIX]{x2202}M$ has dimension $n$ even. Its definitionExpand
$Q$-Curvature and Poincaré Metrics
This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of the formal solutionExpand
Conformally covariant differential operators: properties and applications
We discuss conformally covariant differential operators, which under local rescalings of the metric, , transform according to for some r if is of order s. It is shown that the flat space restrictionsExpand
Heat kernel expansion: user's manual
Abstract The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action. The aim of this report is to collect usefulExpand