Conformal metrics on ℝ<sup>2m</sup> with constant <em>Q</em>-curvature

  • Luca Martinazzi

    Rutgers University, Piscataway, Switzerland


We study the conformal metrics on R2m\R{2m} with constant QQ-curvature QRQ\in\R{} having finite volume, particularly in the case Q0Q\leq 0. We show that when Q<0Q<0 such metrics exist in R2m\R{2m} if and only if m>1m>1. Moreover we study their asymptotic behavior at infinity, in analogy with the case Q>0Q>0, which we treated in a recent paper. When Q=0Q=0, we show that such metrics have the form e2pgR2me^{2p}g_{\R{2m}}, where pp is a polynomial such that 2degp2m22\leq \deg p\leq 2m-2 and supR2mp<+\sup_{\R{2m}}p<+\infty. In dimension 44, such metrics correspond to the polynomials pp of degree 22 with limx+p(x)=\lim_{|x|\to+\infty}p(x)=-\infty.

Cite this article

Luca Martinazzi, Conformal metrics on ℝ<sup>2m</sup> with constant <em>Q</em>-curvature. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19 (2008), no. 4, pp. 279–292

DOI 10.4171/RLM/525