Existence of conformal metrics with constant Q-curvature

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].

Existence of Conformal Metrics with Prescribed Q-Curvature

“…In [4][5][6][7], existence results for the constant Qcurvature problem in 4-dimensional manifolds are given. On manifolds of dimension greater than 4, existence results were given for Einstein manifolds in [3].…”

Conformal Metrics with Constant Q-Curvature

“…We give here the main ideas for the proof of Theorem 1: throughout this section we assume for simplicity that P g is positive definite (except on constants), referring to [24] for details when negative eigenvalues are present.…”

“…This note concerns some recent progress about problem (3), in particular an extension of the uniformization result of [18], which is given in [24]. Theorem 1.…”

Q-curvature prescription; forbidden functions and the GJMS null space

“…[2,3,12,15,26]), and especially the uniformisation type problem of whether one may find a metric in c with Q g constant; see for example [13,14,16] and the references therein. Throughout we shall work on an even closed (that is, compact without boundary) and connected conformal manifold (M, c) of dimension n ≥ 4.…”