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- weighted holder
- smooth
- back metrics ??i
- riemannian metric
- conformally compact
- compact einstein
- equation
- any tensor
- metrics

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BOUNDARY REGULARITY OF CONFORMALLY COMPACT EINSTEIN METRICS

´ PIOTR T. CHRUSCIEL, ERWANN DELAY, JOHN M. LEE, AND DALE N. SKINNER

Abstract.We show thatC2conformally compact Riemannian Einstein metrics have conformal compactiﬁcations that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.

1.Introduction

SupposeMis a smooth, compact manifold with boundary, and letM denote its interior and∂Mits boundary. (By “smooth,” we always mean C∞ Riemannian metric.) AgonMis said to beconformally compact if for some smooth deﬁning functionρfor∂MinM,ρ2gextends by continuity to a Riemannian metric (of class at leastC0) onM. The rescaled metricg=ρ2gis called aconformal compactiﬁcationofg. If for some (hence any) smooth deﬁning functionρ,gis inCk(M) or Ck,λ(M), then we saygis conformally compact of classCkorCk,λ, respectively. Ifgis conformally compact, the restriction ofg=ρ2gto∂Mis a Riemannian metric on∂M, whose conformal class is determined byg, independently of the choice of deﬁning functionρ. This conformal class is called theconformal inﬁnityofg. Several important existence and uniqueness results [1, 2, 5, 9, 13] con-cerning conformally compact Riemannian Einstein metrics have been established recently. For many applications to physics and geometry, it turns out to be of great importance to understand the asymptotic behaviour of the resulting metrics near the boundary. This question has been addressed by Michael Anderson [2], who proved that ifgis a 4-dimensional conformally compact Einstein metric with smooth con-formal inﬁnity, then the conformal compactiﬁcation ofgis smooth up

Date: March 20, 2004. First author partially supported by Polish Research Council grant KBN 2 P03B 073 24; second author partially supported by the ACI program of the French Ministry of Research. 1

BOUNDARY REGULARITY

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to the boundary in suitable coordinates. It has long been conjectured that in higher dimensions, conformally compact Einstein metrics with smooth conformal inﬁnities should have inﬁnite-order asymptotic expan-sions in terms ofρand logρ. The purpose of this paper is to conﬁrm that conjecture. The choice of special coordinates in Anderson’s result cannot be dis-pensed with. Because the Einstein equation is invariant under diﬀeo-morphisms, we cannot expect that the conformal compactiﬁcation of an arbitrary conformally compact Einstein metric will necessarily have optimal regularity for allC∞structures onM example, suppose. Forg is an Einstein metric onMwith a smooth conformal compactiﬁcation, and let Ψ :M→Mbe a homeomorphism that restricts to the iden-tity map of∂Mand to a diﬀeomorphism fromMto itself. Ψ Then∗g will still be Einstein with the same conformal inﬁnity, but its conformal compactiﬁcationρ2Ψ∗g Thusmay no longer be smooth. the best one might hope for is that an arbitrary conformally compact Einstein met-ric can be made smoothly conformally compact after pulling back by an appropriate diﬀeomorphism. Even this is not true in general, because Feﬀerman and Graham showed in [7] that there is an obstruction to smoothness in odd dimensions. Since Einstein metrics are always smooth (in fact, real-analytic) in suitable coordinates in the interior, only regularity at the boundary is at issue. For that reason, instead of assuming thatMis compact, we will assume only that it has a compact boundary componentY, and restrict our attention to a collar neighborhood ofYinM, which we may assume without loss of generality is diﬀeomorphic toY×[01). Throughout this paper, then,Ywill be an arbitrary smooth, connected, compact,n-dimensional manifold without boundary, and we make the following identiﬁcations: M=Y×[01) M=Y×(01) ∂M=Y× {0}. Letρ:M→[01) denote the projection onto the [01) factor; it is a smooth deﬁning function for∂MinM. For 0< R <1, we deﬁne MR=Y×(0 R] MR=Y×[0 R]. In this context, we extend the deﬁnition of conformally compact metrics by saying that a smooth Riemannian metricgonMorMRis confor-mally compact ifρ2gextends to a continuous metric onMorMR, respectively. A continuous map Ψ :MR→M(for someR) that re-stricts to the identity map of∂Mand to a (C∞) diﬀeomorphism from MRits image will be called ato collar diﬀeomorphism Ψ and its. If