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Đề tài " Existence of conformal metrics with constant Qcurvature "

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and min-max schemes, jointly with the compactness result of [35]. 1. Introduction In recent years, much attention has been devoted to the study of partial differential equations on manifolds, in order to understand some connections between analytic and geometric properties of these objects. . | Annals of Mathematics Existence of conformal metrics with constant Q-curvature By Zindine Djadli and Andrea Malchiodi Annals of Mathematics 168 2008 813 858 Existence of conformal metrics with constant Q-curvature By Zindine Djadli and Andrea Malchiodi Abstract Given a compact four dimensional manifold we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below we employ topological methods and min-max schemes jointly with the compactness result of 35 . 1. Introduction In recent years much attention has been devoted to the study of partial differential equations on manifolds in order to understand some connections between analytic and geometric properties of these objects. A basic example is the Laplace-Beltrami operator on a compact surface s g . Under the conformal change of metric g e2wg we have 1 Ag e-2w Ag -Ag w Kg Kge2w where Ag and Kg resp. Ag and Kg are the Laplace-Beltrami operator and the Gauss curvature of s g resp. of s g . From the above equations one recovers in particular the conformal invariance of fy KgdVg which is related to the topology of s through the Gauss-Bonnet formula 2 ỈKg dVg 2 x E Jy where x s is the Euler characteristic of s. Of particular interest is the classical Uniformization Theorem which asserts that every compact surface carries a conformal metric with constant curvature. On four-dimensional manifolds there exists a conformally covariant operator the Paneitz operator which enjoys analogous properties to the LaplaceBeltrami operator on surfaces and to which is associated a natural concept of curvature. This operator introduced by Paneitz 38 39 and the corresponding Q-curvature introduced in 6 are defined in terms of the .