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Andreas Juhl

Publications and source records attributed to Andreas Juhl.

14 recordsLinked to original sources
arXiv2022

Extrinsic Paneitz operators and $Q$-curvatures for hypersurfaces

For any hypersurface $M$ of a Riemannian manifold $X$, recent works introduced the notions of extrinsic conformal Laplacians and extrinsic $Q$-curvatures. Here we derive explicit formulas for the extrinsic version ${\bf P}_4$ of the Paneitz operator and the corresponding extrinsic fourth-order $Q$-curvature ${\bf Q}_4$ in general dimensions. This result involves a series of obvious local conformal invariants of the embedding $M^4 \hookrightarrow X^5$ (defined in terms of the Weyl tensor and the trace-free second fundamental form) and a non-trivial local conformal invariant $\mathcal{C}$. In turn, we identify $\mathcal{C}$ as a linear combination of two local conformal invariants $J_1$ and $J_2$. Moreover, a linear combination of $J_1$ and $J_2$ can be expressed in terms of obvious local conformal invariants of the embedding $M \hookrightarrow X$. This finally reduces the non-trivial part of the structure of ${\bf Q}_4$ to the non-trivial invariant $J_1$. For closed $M^4 \hookrightarrow {\mathbb R}^5$, we relate the integrals of $J_i$ to functionals of Guven and Graham-Reichert. Moreover, we establish a Deser-Schwimmer type decomposition of the Graham-Reichert functional of a hypersurface $M^4 \hookrightarrow X^5$ in general backgrounds. In this context, we find one further local conformal invariant $J_3$. Finally, we derive an explicit formula for the singular Yamabe energy of a closed $M^4 \hookrightarrow X^5$. The resulting explicit formulas show that it is proportional to the total extrinsic fourth-order $Q$-curvature. This observation confirms a special case of a general fact and serves as an additional cross-check of our main result.

math.DG↗
arXiv2022

Residue families, singular Yamabe problems and extrinsic conformal Laplacians

Let $(X,g)$ be a compact manifold with boundary $M^n$ and $σ$ a defining function of $M$. To these data, we associate natural conformally covariant polynomial one-parameter families of differential operators $C^\infty(X) \to C^\infty(M)$. They arise through a residue construction which generalizes an earlier construction in the framework of Poincaré-Einstein metrics. The main ingredient of the definition of residue families are eigenfunctions of the Laplacian of the singular metric $σ^{-2}g$. We prove that if $σ$ is an approximate solution of a singular Yamabe problem, these families can be written as compositions of certain degenerate Laplacians. This result implies that the notions of extrinsic conformal Laplacians and extrinsic $Q$-curvature introduced in recent works by Gover and Waldron can naturally be rephrased in terms of residue families. The new spectral theoretical perspective enables us to relate the extrinsic conformal Laplacians and the critical extrinsic $Q$-curvature to the scattering operator of the asymptotically hyperbolic metric $σ^{-2}g$ extending the work of Graham and Zworski. The latter relation implies that the extrinsic conformal Laplacians are self-adjoint. We describe the asymptotic expansion of the volume of a singular Yamabe metric in terms of Laplace-Robin operators. We also derive new local holographic formulas for all extrinsic $Q$-curvatures in terms of renormalized volume coefficients, the scalar curvature of the background metric, and the asymptotic expansions of eigenfunctions of the Laplacian of the singular metric $σ^{-2}g$. Furthermore, we prove a new formula for the singular Yamabe obstruction $B_n$, and we use the latter formula to derive explicit expressions for the obstructions in low-order cases (confirming earlier results). Finally, we relate the obstruction $B_n$ to the supercritical $Q$-curvature $Q_{n+1}$.

math.DG↗
arXiv2022

Extrinsic Paneitz operators and Q-curvatures for hypersurfaces

For any hypersurface of a Riemannian manifold, recent works introduced the notions of extrinsic conformal Laplacians and extrinsic Q-curvatures. Here we announce explicit formulas for the extrinsic Paneitz operators P_4 and the corresponding extrinsic Q-curvatures for totally umbilic hypersurfaces in any dimension. Moreover, we state explicit formulas for the critical extrinsic P_4 and the total integral of the critical extrinsic Q_4 in the general case. This integral is a global conformal invariant. Finally, we establish an analog of the Alexakis-Deser-Schwimmer decomposition of the critical extrinsic Q_4.

math.DG↗
arXiv2021

On singular Yamabe obstructions

We discuss the singular Yamabe obstruction $\mathcal{B}_3$ of a hypersurface in a four-dimensional general background. We derive various explicit formula for $\mathcal{B}_3$ from the original definition. We relate these formulas to corresponding formulas in the literature. The proofs are elementary.

math.DG↗
arXiv2014

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 $P_2(g) = \H(0;g)$. 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. 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.

math.DG↗
arXiv2012

Explicit formulas for GJMS-operators and $Q$-curvatures

We describe GJMS-operators as linear combinations of compositions of natural second-order differential operators. These are defined in terms of Poincaré-Einstein metrics and renormalized volume coefficients. As special cases, we find explicit formulas for conformally covariant third and fourth powers of the Laplacian. Moreover, we prove related formulas for all Branson's $Q$-curvatures. The results settle and refine conjectural statements in earlier works. The proofs rest on the theory of residue families.

math.DG↗
arXiv2010

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 of any GJMS-operator as the sum of a certain linear combination of compositions of lower order GJMS-operators (primary part) and a second-order operator which is defined by the Schouten tensor (secondary part). We complete the description of GJMS-operators by proposing and discussing recursive formulas for their constant terms, i.e., for Branson's $Q$-curvatures, along similar lines. We confirm the picture in a number of cases. Full proofs are given for spheres of any dimension and arbitrary signature. Moreover, we prove formulas of the respective critical third power $P_6$ in terms of the Yamabe operator $P_2$ and the Paneitz operator $P_4$, and of a fourth power in terms of $P_2$, $P_4$ and $P_6$. For general metrics, the latter involves the first two of Graham's extended obstruction tensors. In full generality, the recursive formulas remain conjectural. We describe their relation to the theory of residue families and the associated $Q$-curvature polynomials.

math.DG↗
arXiv2009

On Branson's $Q$-curvature of order eight

We prove a universal recursive formulas for Branson's $Q$-curvature of order eight in terms of lower-order $Q$-curvatures, lower-order GJMS-operators and holographic coefficients. The results prove a special case of a conjecture in {arXiv:0905.3992}.

math.DG↗
arXiv2009

Universal recursive formulae for Q-curvatures

We formulate and discuss two conjectures concerning recursive formulae for Branson's $Q$-curvatures. The proposed formulae describe all $Q$-curvatures on manifolds of all even dimensions in terms of respective lower order $Q$-curvatures and lower order GJMS-operators. They are universal in the dimension of the underlying space. The recursive formulae are generated by an algorithm which rests on the theory of residue families. We attempt to resolve the algorithm by formulating a conjectural description of the coefficients in the recursive formulae in terms of interpolation polynomials associated to compositions of natural numbers. We prove that the conjectures cover $Q_4$ and $Q_6$ for general metrics, and $Q_8$ for conformally flat metrics. The result for $Q_8$ is proved here for the first time. Moreover, we display explicit (conjectural) formulae for $Q$-curvatures of order up to 16, and test high order cases for round spheres and Einstein metrics.

math.DG↗
arXiv2007

Holographic formula for Q-curvature

This paper derives an explicit formula for Branson's Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincare metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses Q-curvature as a multiple of the Pfaffian and the divergence of a natural one-form. The paper also outlines the relation between holographic formulae for Q-curvature and a new theory of conformally covariant families of differential operators due to the second author.

math.DG↗
arXiv1995

Secondary invariants and the singularity of the Ruelle zeta-function in the central critical point

The Ruelle zeta-function of the geodesic flow on the sphere bundle $S(X)$ of an even-dimensional compact locally symmetric space $X$ of rank $1$ is a meromorphic function in the complex plane that satisfies a functional equation relating its values in $s$ and $-s$. The multiplicity of its singularity in the central critical point $s = 0$ only depends on the hyperbolic structure of the flow and can be calculated by integrating a secondary characteristic class canonically associated to the flow- invariant foliations of $S(X)$ for which a representing differential form is given.

math.DS↗