arXiv research

GJMS operators connect geometry, analysis, and physics.

Researchers develop weighted GJMS operators for smooth metric measure spaces.

Direct proofs are given of Juhl's formulae for GJMS operators and Q-curvatures starting from the original construction of GJMS.

We describe a set of conformally covariant boundary operators associated to the sixth-order GJMS operator on a conformally invariant class of manifolds which includes compactifications of Poincaré--Einstein manifolds. This yields a conformally covariant energy functional for the sixth-order GJMS operator on such manifo…

The paper studies inequalities for fractional GJMS operators on conformal infinity.

The GJMS operators of special Einstein products are factored into simpler operators and applied to solve the Q-Yamabe problem.

Essential self-adjointness and spectrum of CR GJMS operator proved.

Green functions for GJMS operators on spheres derived, linking geometry and rigidity.

We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet-to-Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli--Silvestre extension for $(-Δ)^γ$ when $γ\in(0,1)$, and both…

We give sufficient conditions on a function invariant under the action of an isometry group to be Branson's Q-curvature of a metric in a given conformal class, using the conformal GJMS operators.

Derives GJMS operators and Q-curvatures for submanifolds.

In this paper we consider a singular elliptic equation involving the GJMS (Graham-Jenne-Mason-Sparling) operator of order k on n-dimensional compact Riemannian manifold with 2k<n. Mutiplicity and nonexistence results are established.

Defines a new energy for submanifolds, comparing to Willmore energy.

We investigate the prescribed Q-curvature flow for GJMS operators with non-trivial kernel on compact manifolds of even dimension. When the total Q-curvature is negative, we identify a conformally invariant condition on the nodal domains of functions in the kernel of the GJMS operator, allowing us to prove the global ex…

For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical {\sf GJMS} operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue…

Study of scattering on singular Yamabe spaces using asymptotically hyperbolic manifolds.

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) a…

For the Möbius spheres $S^{q,p}$, we give alternative elementary proofs of the recursive formulas for GJMS-operators and $Q$-curvatures due to the first author [Geom. Funct. Anal. 23, (2013), 1278-1370; arXiv:1108.0273]. These proofs make essential use of the theory of hypergeometric series.

Motivated by AdS/CFT, the extension is made to spin-half of a scalar calculation of the conformal anomalies and functional determinants of GJMS operators. The formal aspects are heuristic but sufficient. A Barnes zeta function representation again proves effective. The determinants are calculated for the two factorisat…

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. …

We argue that the AdS/CFT calculational prescription for double-trace deformations leads to a holographic derivation of the conformal anomaly, and its conformal primitive, associated to the whole family of conformally covariant powers of the Laplacian (GJMS operators) at the conformal boundary. The bulk side involves a…

Under a spectral assumption on the Laplacian of a Poincaré--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order $2γ\in(0,2)$ or $2γ\in(2,4)$ and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption…

The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersufaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in t…

A numerical expression in the form of an integral is given for the determinant of the scalar GJMS operator on an odd--dimensional sphere. Manipulation yields a curious sum formula for the logdet in terms of the logdets of the ordinary conformal Laplacian for other dimensions. A few graphs are drawn.

The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.

A new definition of canonical conformal differential operators $P_k$ ($k=1,2,...)$, with leading term a $k^{\rm th}$ power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction …

Functional determinant for mixed signature sphere products depends on sphere dimensions and parity.

We establish an algorithm which computes formulae for the CR GJMS operators, the $P^\prime$-operator, and the $Q^\prime$-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an explicit factorisation of the CR GJMS operators and the $P^\prime$-operator…

By refining Matsumoto's construction of Einstein ACH metrics, we construct a one parameter family of ACH metrics which solve the Einstein equation to infinite order and have a given three dimensional CR structure at infinity. When the parameter is 0, the metric is self-dual to infinite order. As an application, we give…

We extend the notions of CR GJMS operators and Q-curvature to the case of partially integrable CR structures. The total integral of the CR Q-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR manifolds equipped with an orientation of the bundle of contact forms, which is nontr…

Let $L_g$ be the subcritical GJMS operator on an even-dimensional compact manifold $(X, g)$ and consider the zeta-regularized trace $\mathrm{Tr}_ζ(L_g^{-1})$ of its inverse. We show that if $\ker L_g = 0$, then the supremum of this quantity, taken over all metrics $g$ of fixed volume in the conformal class, is always g…

The paper improves CR Sobolev inequalities and classifies minimizers.

On an even conformal manifold $(M,c)$, such that the critical GJMS operator has non-trivial kernel, we identify and discuss the role of a finite dimensional vector space $N(Q)$ of functions determined by the conformal structure. Using these we describe an infinite dimensional class of functions that cannot be the Q-cur…

In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal…

We prove universal recursive formulas for Branson's $Q$-curvatures in terms of respective lower-order $Q$-curvatures, lower-order GJMS-operators and holographic coefficients.

Proves Juhl formulas for curved Ovsienko--Redou operators, confirming conjectures.

This work generalizes a construction by Habermann and Jost of a canonical metric in a Yamabe-positive conformal class, which uses the Green function of the conformal Laplacian. In dimension $n=2k+1$, $2k+2$, or $2k+3$, if the $k$-th GJMS operator $P_k$ admits a Green function, the constant term of its singularity is sh…

Researchers investigate extremal eigenvalues of GJMS operators in fixed conformal classes.

This paper proves a Liouville type result for a specific higher-order equation on the sphere.

Study describes how conformal metrics behave as Q-curvature changes, forming spherical bubbles.

We study analysis aspects of the sixth order GJMS operator $P_g^6$. Under conformal normal coordinates around a point, the expansions of Green's function of $P_g^6$ with pole at this point are presented. As a starting point of the study of $P_g^6$, we manage to give some existence results of prescribed $Q$-curvature pr…

Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.

We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conform…

We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension $n\geq 3$, there exist many metr…

For a finite reflection subgroup $G\leq O(n+1,1,\mR)$ of the conformal group of the sphere with standard conformal structure $(S^n,[g_0])$, we geometrically derive differential-difference Dunkl version of the series of conformally invariant differential operators with symbols given by powers of Laplace operator. The co…

The conformal anomalies and functional determinants of the Branson--GJMS operators, P_{2k}, on the d-dimensional sphere are evaluated in explicit terms for any d and k such that k < d/2+1 (if d is even). The determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly, which vanishe…

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}.

Method shows existence of conformal metrics with constant $Q$-curvature on manifolds.