An Upper Bound of the Total Q-curvature and Its Isoperimetric Deficit for Higher-dimensional Conformal Euclidean Metrics

To begin with, let us agree to some basic conventions. We employ the symbols ∆ and ∇ to denote the Laplace operator ∑nk=1 ∂/∂xk and the gradient vector (∂/∂x1, ..., ∂/∂xn) over the Euclidean space R , n ≥ 2. For notational convenience we use X . Y as X ≤ CY for a constant C > 0. We always assume that u is a smooth real-valued function on R, written u ∈ C∞(Rn), and then it generates a conformal metric g = eg0 which is indeed a conformal deformation of the standard Euclidean metric g0 = ∑n k=1 dx 2 k. The volume and surface area elements of the metric g are given by dvg = e dH and dsg = e(n−1)udHn−1 where Hk stands for the k-dimensional Hausdorff measure. So, the volume and surface area of the open ball Br(x) and its boundary ∂Br(x) with radius r > 0 and center x ∈ R have the following values: