Paneitz Operator Research Articles

Nonnegativity of CR Paneitz Operator and Its Application to the CR Obata’s Theorem

In this paper, we first prove the CR analogue of M. Obata’s theorem on a closed pseudohermitian (2n+1)-manifold with free pseudohermitian torsion. Secondly, we have the CR analogue of Li-Yau’s eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub-Laplacian on a closed pseudohermitian (2n+1)-manifold with a more general curvature condition for n≥2. The key step is a discovery of CR analogue of Bochner formula which involving the CR Paneitz operator and nonnegativity of CR Paneitz operator P 0 for n≥2.

A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)

This is the original manuscript dated March 9th 1983, typeset by the Editors for the Proceedings of the Midwest Geometry Conference 2007 held in memory of Thomas Branson. Stephen Paneitz passed away on September 1st 1983 while attending a conference in Clausthal and the manuscript was never published. For more than 20 years these few pages were circulated informally. In November 2004, as a service to the mathematical community, Tom Branson added a scan of the manuscript to his website. Here we make it available more formally. It is surely one of the most cited unpublished articles. The differential operator defined in this article plays a key role in conformal differential geometry in dimension 4 and is now known as the Paneitz operator.

Conformally invariant differential operators and bilinear functionals in six dimensions

We review how to construct the Paneitz operator in dimension four and the corresponding operator in dimension six, by constructing symmetric bilinear differential functionals that are conformally invariant.

On the estimate of the first eigenvalue of a sublaplacian on a pseudohermitian 3-manifold

In this paper, we study a lower bound estimate of thefirst positive eigenvalue of the sublaplacian on a three-dimensional pseudohermitian manifold. S.-Y. Li and H.-S. Luk derived the lower bound estimate under certain conditions for curvature tensors bounded below by a positive constant. By using the Li‐Yau gradient estimate, we are able to get an effective lower bound estimate under a general curvature condition. The key is the discovery of a new CR version of the Bochner formula which involves the CR Paneitz operator.

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.We prove the existence of a continuum of conformal radially symmetric complete metrics in hyperbolic space Hn, n>4, all having the same constant Q-curvature.Moreover, similar results can be shown also for suitable non-constant prescribed Q-curvature functions.

Pseudo-Einstein and Q-flat metrics with eigenvalue estimates on CR-hypersurfaces

In this paper, we will use the Kohn's ∂ b -theory on CR-hypersurfaces to derive some new results in CR-geometry. Main Theorem. Let M 2n-1 be the smooth boundary of a bounded strongly pseudo-convex domain Ω in a complete Stein manifold V 2n . Then: (1) For n ≥ 3, M 2n-1 admits a pseudo-Einstein metric. (2) For n > 2, M 2n-1 admits a Fefferman metric of zero CR Q-curvature. (3) In addition, for a compact strictly pseudoconvex CR emendable 3-manifold M 3 , its CR Paneitz operator P is a closed operator. There are examples of non-emendable strongly pseudoconvex CR-manifolds M 3 , for which the corresponding ∂ b -operator and Paneitz operators are not closed operators.

A fourth order curvature flow on a CR 3-manifold

Let (M 3 , J, θ 0 ) be a closed pseudohermitian 3-manifbld. Suppose the associated torsion vanishes and the associated Q-curvature has no kernel part with respect to the associated Paneitz operator. On such a background pseudohermitian 3-manifold, we study the change of the contact form according to a certain version of normalized Q-curvature flow. This is a fourth order evolution equation. We prove that the solution exists for all time and converges smoothly to a contact form of zero Q-curvature. We also consider other background conditions and obtain a priori bounds on high-order norms on the solutions.

Partielle Differentialgleichungen

The workshop Partial differential equations , organised by Tom Ilmanen (ETH Zürich), Reiner Schätzle (Universität Tübingen) and Neil Trudinger (Australian National University Canberra) was held July 24-30, 2005. This meeting was well attended by 46 participants, including 6 females, with broad geographic reprensentation. The program consisted of 15 talks and 9 shorter contributions and left sufficient time for discussions. New results combining partial differential equations and geometric problems were presented in the area of minimal surfaces, free boundaries and singular limits, for example the construction of branched minimal surfaces, the regularity of free boundaries in the wake of the monotonicity formula of Weiss and a proof of a conjecture of De Giorgi. A major part of the leading experts of partial differential equations with conformal invariance attended the workshop. Here new results were presented in conformal geometry, for the Yamabe problem, the Paneitz operator and the Willmore functional.

The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold

In this paper, we study a sharp lower bound of the first eigenvalue of the sublaplacian on a 3-dimensional pseudohermitian manifold with the CR Paneitz operator positive. In general cases, S.-Y. Li and H.-S. Luk ({Proc. Am. Math. Soc.} 132(3), 789–798) (2004) proved the lower bound under a condition on a covariant derivative of the torsion as well as the Ricci curvature and the torsion. We show that if the CR Paneitz operator is positive, then the sharp lower bound is obtained under one simpler condition on only the Ricci curvature and the torsion itself; which is similar to the condition given in high-dimensional cases in ({Commun. Partial Differential Equations}, 10(2/3), 191–217) (1985). We also show examples where our theorem applies, but Theorem 1.2 in ({Proc. Am. Math. Soc.} 132(3), 789–798) (2004) does not.

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).

Compactness and global estimates for the geometric Paneitz equation in high dimensions

Given ( M , g ) (M,g) , a smooth compact Riemannian manifold of dimension n ≥ 5 n \ge 5 , we investigate compactness for the fourth order geometric equation P g u = u 2 ♯ − 1 P_gu = u^{2^\sharp -1} , where P g P_g is the Paneitz operator, and 2 ♯ = 2 n / ( n − 4 ) 2^\sharp = 2n/(n-4) is critical from the Sobolev viewpoint. We prove that the equation is compact when the Paneitz operator is of strong positive type.

The Sobolev inequality for Paneitz operator on three manifolds

The Paneitz operator introduced in [P] has attracted some attention in conformal geometry recently. In particular its associated Q-curvature equation has demonstrated its importance in four dimensional conformal geometry (cf. [CGY]). According to Branson ([B]), this operator enjoys similar conformal covariance properties in dimenion three as well. However there are a number of features of this equation that are quite distinct from its analogues in dimensions four and above. In this dimension, the positivity of the Paneitz operator is an important issue. In a previous paper [XY1], the analogue of the Yamabe equation is veri ed for the conformal classes for which both the conformal Laplacian operator and the Paneitz operator are positive. However the conformal classes of interests namely those that are near the standard 3-sphere fail to be in this class. The main purpose of this article is to study the positivity issue for the family of conformal structures provided by the Berger metrics. Let (M; g) be a three dimensional smooth compact Riemannian manifold. The Q curvature is de ned by

A Fully Nonlinear Equation on Four-Manifolds with Positive Scalar Curvature

We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with a metric of positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. A special case of this deformation provides an alternative proof to the main result in Chang, Gursky & Yang, 2002. We also give a new conformally invariant condition for positivity of the Paneitz operator, generalizing the results in Gursky, 1999. From the existence results in Chang & Yang, 1995, this allows us to give many new examples of manifolds admitting metrics with constant Q-curvature.

Conformal energy in four dimension

The conformal energy for 4-manifolds using the Paneitz operator is introduced in this article. The conformal invariance of the energy functional allows us to find a sharp lower bound in terms of the conformal volume. We also demonstrate certain obstruction to existence of minimal immersions to spheres using the fourth order curvature invariance associated to the operator.

On a Fourth Order Equation in 3-D

In this article we study the positivity of the 4-th order Paneitz operator for closed 3-manifolds. We prove that the connected sum of two such 3-manifold retains the same positivity property. We also solve the analogue of the Yamabe equation for such a manifold.

Positivity of Paneitz operators

In this paper, we give two results concerning the positivity property of the Paneitz operator-- a fourth order conformally covariant elliptic operator. We prove that the Paneitz operator is positive for a compact Riemannian manifold without boundary of dimension at least six if it has positve scalar curvature as well as nonnegative $Q-$curvature. We also show that the positivity of the Paneitz operator is preserved in dimensions greater than four in taking a connected sum.

The Principal Eigenvalue¶of a Conformally Invariant Differential Operator,¶with an Application to Semilinear Elliptic PDE

In this paper we give conditions under which the Paneitz operator, a conformally invariant fourth order differential operator, is positive. We also prove a sharp estimate for a related conformal invariant. These results are combined to study the existence of a semilinear PDE with critical Sobolev growth.

The logarithmic Hardy-Littlewood-Sobolev inequality and extremals of zeta functions onS n

The functional estimated sharply by the logarithmic Hardy-Littlewood-Sobolev inequality onSn, for evenn≥2, is linked to the spectrum of the Paneitz operators. From the point of view of conformal geometry such operators are naturaln-dimensional generalizations of the Laplacian onS2.