Uniqueness and non-existence theorems for conformally invariant equations

By using the equivalent integral form for the Q-curvature equation, we generalize the well-known non-existence results on two-dimensional Gaussian curvature equation to all dimensional Q-curvature equation. Somehow, we introduce a new approach to Q-curvature equation which is higher order and even pseudo-differential equation. As a by-product, we do classify the solutions for Q=1 solutions on Sⁿ as well as on Rⁿ with necessary growth rate assumption.     

Q-curvature flow on 4-manifolds with boundary

Given a compact four-dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero. Using integral methods, we prove global existence and convergence for the Q-curvature flow to a smooth metric conformal to g of prescribed Q-curvature, zero T-curvature and vanishing mean curvature under conformally invariant assumptions.     

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 Poincaré metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses the Q-curvature as a multiple of the Pfaffian and the divergence of a natural 1-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.     

Constant Q-curvature metrics near the hyperbolic metric

Let (M,g)$(M,g)$ be a Poincaré–Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant Q-curvature in the conformal class of an asymptotically hyperbolic metric close enough to g. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant Q-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.     

What is Q-Curvature?

Branson’s Q-curvature is now recognized as a fundamental quantity in conformal geometry. We outline its construction and present its basic properties.     

The explicit construction of Finsler metrics with special curvature properties

The S-curvature is one of most important non-Riemannian quantities in Finsler geometry. It delicately related to Riemannian quantities. This note gives an explicit construction of 3-parameter family of non-locally projectively flat Finsler metrics of non-constant isotropic S-curvature. The necessary and sufficient condition that these Finsler metrics are of constant flag curvature is given.     

Concentration-compactness phenomena in the higher order Liouville's equation

We investigate different concentration-compactness and blow-up phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in R2m, then that of a closed manifold and, finally, the particular case of the sphere S2m. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in R2m, blow-up phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have compactness.     

GJMS operators and Q-curvature for conformal Codazzi structures

The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersurfaces 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 the volume expansion of the Blaschke metric, and derive the first and second variation formulas for a deformation of strictly convex domains.     

The isoperimetric inequality and Q-curvature

A well-known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions. In dimension 2, the constant can be controlled by the integral of the positive part of the Gaussian curvature. In this paper, we showed that on simply connected conformally flat manifolds of higher dimensions, the role of the Gaussian curvature can be replaced by the Branson's Q  -curvature. We achieve this by exploring the relationship between _Ap$A_p$ weights and integrals of the Q-curvature.     

Constant Q-curvature metrics in arbitrary dimension

Working in a given conformal class, we prove existence of constant Q-curvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a nth-order non-linear elliptic differential (or integral) equation with variational structure, where n is the dimension of the manifold. Since the corresponding Euler functional is in general unbounded from above and below, we use critical point theory, jointly with a compactness result for the above equation.     

CR-invariants and the scattering operator for complex manifolds with CR-boundary

Suppose that M is a CR manifold bounding a compact complex manifold X. The manifold X admits an approximate Kähler–Einstein metric g which makes the interior of X a complete Riemannian manifold. We identify certain residues of the scattering operator as CR-covariant differential operators and obtain the CR Q-curvature of M from the scattering operator as well. Our results are an analogue in CR-geometry of Graham and Zworski's result that certain residues of the scattering operator on a conformally compact manifold with a Poincaré–Einstein metric are natural, conformally covariant differential operators, and the Q-curvature of the conformal infinity can be recovered from the scattering operator. To cite this article: P.D. Hislop et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The Paneitz equation in hyperbolic space

The Paneitz operator is a fourth order differential operator which arises in conformal geometry and satisfies a certain covariance property. Associated to it is a fourth order curvature – the Q-curvature.     

Uniqueness theorem for integral equations and its application

This paper is devoted to answering a question asked recently by Y. Li regarding geometrically interesting integral equations. The main result is to give a necessary and sufficient condition on the parameters so that the integral equation with parameters to be discussed in this paper have regular solutions. In the case such condition is satisfied, we will write down the exact solution. As its application of our method, we should show that the non-existence theory of the solutions of prescribed scalar curvature equation on Sn can be generalized to that of prescribed Branson-Paneitz Q-curvature equations on Sn.     

Prescribed Q-curvature problem on closed 4-Riemannian manifolds in the null case

The main objective of this short note is to give a sufficient condition for a non constant function k to be Q curvature candidate for a conformal metric on a closed Riemannian manifold with the null Q-curvature. In contrast to the prescribed scalar curvature on the two-dimensional flat tori, the condition we provided is not necessary as some examples show. The second author would like to thank Department of Mathematics, University of Paris XII for their invitation, hospitality and financial support during his visit. His research is partially supported by NUS research grant: R-146-000-077-112.     

On the non-Riemannian quantity H in Finsler geometry

In this paper, we study a new non-Riemannian quantity H defined by the S-curvature. We find that the non-Riemannian quantity is closely related to S-curvature. We characterize Randers metrics of almost isotropic S-curvature if and only if they have almost vanishing H-curvature. Furthermore, the Randers metrics actually have zero S-curvature if and only if they have vanishing H-curvature.     

Conformal metrics with prescribed Q-curvature on S n

In this paper we prescribe a fourth order conformal invariant on the standard n-sphere, with n????5, and study the related fourth order elliptic equation. We prove new existence results based on a new type of Euler?CHopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal metrics having the same Q-curvature.     

Generic multiplicity for a scalar field equation on compact surfaces

We prove generic multiplicity of solutions for a scalar field equation on compact surfaces via Morse inequalities. In particular our result improves significantly the multiplicity estimate which can be deduced from the degree-counting formula in Chen and Lin (2003) [12]. Related results are derived for the prescribed Q-curvature equation.     

A fourth order uniformization theorem on some four manifolds with large total Q-curvature

Given a four-dimensional manifold $(M,g)$, we study the existence of a conformal metric for which the Q-curvature, associated to a conformally invariant fourth-order operator (the Paneitz operator), is constant. Using a topological argument, we obtain a new result in cases which were still open. To cite this article: Z. Djadli, A. Malchiodi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).