Compactness for Conformal Metrics with Constant Q Curvature on Locally Conformally Flat Manifolds
نویسندگان
چکیده
In this note we study the conformal metrics of constant Q curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension n ≥ 5 and with Poincarë exponent less than n−4 2 , the set of conformal metrics of positive constant Q and positive scalar curvature is compact in the C∞ topology.
منابع مشابه
Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries
This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect...
متن کاملCompactness of conformal metrics with constant Q-curvature. I
We establish compactness for nonnegative solutions of the fourth order constant Qcurvature equations on smooth compact Riemannian manifolds of dimension ≥ 5. If the Q-curvature equals −1, we prove that all solutions are universally bounded. If the Qcurvature is 1, assuming that Paneitz operator’s kernel is trivial and its Green function is positive, we establish universal energy bounds on manif...
متن کاملConformally Flat Manifolds with Nonnegative Ricci Curvature
We show that complete conformally flat manifolds of dimension n > 3 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to R n or a spherical spaceform Sn/Γ. This extends previous results due to Q.-M. Cheng and B.-L. Chen and X.-P. Zhu. In this note, we study compl...
متن کاملThe Effect of Linear Perturbations on the Yamabe Problem
In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold (M, g) is compact. Established in the locally conformally flat case by Schoen [43, 44] and for n ≤ 24 by Khuri– Marques–Schoen [26], it has revealed to be generally false for n ≥ 25 as shown by Brendle [8] and Brendle–Marques [9]. A stronger version of it...
متن کاملOn Positive Solutions to Semi-linear Conformally Invariant Equations on Locally Conformally Flat Manifolds
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators Pα were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold (Mn, [g]). We prove that, on a closed and locally conformally flat manifold with...
متن کامل