2008/07/05 by Gideon Maschler, Maschler, Gideon
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research
paper · pdf · doi:10.48550/arxiv.0807.0874
A Riemannian metric \whtg with Ricci curvature \wht\ri is called nontrivial quasi-Einstein, in the sense of Case, Shu and Wei, if it satisfies (-a/f)\wht\nab df+\wht\ri=λ\whtg, for a smooth nonconstant function f and constants λ and a>0. If a is a positive integer, by a result of Kim and Kim, such a metric forms a base for certain warped Einstein metrics. On a manifold M of real dimension at least six, let (g,\t) be a pair consisting of a Kähler metric g which is locally Kähler irreducible, and a nonconstant Killing potential \t. Suppose the metric \whtg=g/\t2 is nontrivial \bee on M∖\t-1(0), and the associated function f is locally a function of \t. Then (g,\t) is an \sk pair, a notion defined by Derdzinski and Maschler. This implies that M is biholomorphic to an open set in the total space of a CP1 bundle whose base manifold admits a Kähler-Einstein metric. If M is additionally compact, it is a total space of such a bundle or complex projective space. Also, the function f is affine in \t-1 with nonzero constants. Conversely, in all even dimensions n≥ 4, there exist \sk pairs (g,\t) and corresponding nonzero constants K and L for which g/\t2 is nontrivial quasi-Einstein with f=K\t-1+L. Additionally, a result of Case, Shu and Wei on the Kähler reducibility of nontrivial Kähler \bers is reproduced in dimension at least six in a more explicit form.