Low regularity Poincaré–Einstein metrics

Abstract

Abstract. We prove the existence of a C 1,1 conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to −1 plus terms of order e −2r where r is the distance from any fixed compact set. This metric has no C 2 conformal compactification.

CR Compactification for Asymptotically Locally Complex Hyperbolic Almost Hermitian Manifolds

“…Notice the similarity between equations (1.1) and (1.3). The real analogue of this result, involving a compactification by a conformal boundary for asymptotically locally real hyperbolic manifolds, has been proven by E. Bahuaud, J. M. Lee, T. Marsh and R. Gicquaud [2][3][4][5]12], pursuing the seminal work of M. T. Anderson and R. Schoen [1]. Notice that, contrary to the real hyperbolic setting, the independence of the compactification with respect to the choice of the essential subset is not established in this article and would deserve further investigations.…”

“…Following [2][3][4][5]12], an essential subset K ⊂ M is a codimension 0 compact submanifold, with smooth boundary ∂K which is convex with respect to its unit outward vector field ν, and such that the normal exponential map…”

“…Lemma 5. 4 Assume that (M, g, J ) satisfies the (ALCH+) and (AK) conditions of order a > 1 2 with sec(M \ K ) < 0. Let π be the orthogonal projection onto {∂ r } ⊥ .…”