A CLASS OF COMPACT POINCARE-EINSTEIN MANIFOLDS: PROPERTIES AND CONSTRUCTION

作者：Gover A Rod^*; Leitner Felipe

来源：Communications in Contemporary Mathematics, 2010, 12(4): 629-659.

DOI：10.1142/S021919971000397X

• Conformal differential geometry
• tractor calculus
• Poincare-Einstein spaces
• almost Einstein spaces
• conformal holonomy
• constant mean curvature hypersurfaces
• asymptotically hyperbolic
• Poincare-Einstein manifolds

We develop a geometric and explicit construction principle that generates classes of Poincare-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalization of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scale singularity set that is a conformal infinity for the Einstein metric. In particular, the construction may be applied to yield families of compact Poincare-Einstein manifolds, as well as classes of almost Einstein manifolds that are compact without boundary. We obtain classification results which show that the construction essentially exhausts a class of almost Einstein (and Poincare-Einstein) manifolds. We develop the general theory of fixed conformal structures admitting multiple compatible almost Einstein structures. We also show that, in a class of cases, these are canonically related to a family of constant mean curvature totally umbillic embedded hypersurfaces.

• 出版日期2010-8