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DEHN FILLING AND EINSTEIN METRICS IN HIGHER DIMENSIONS MICHAEL T. ANDERSON
 

Summary: DEHN FILLING AND EINSTEIN METRICS IN HIGHER DIMENSIONS
MICHAEL T. ANDERSON
Abstract. We prove that many features of Thurston's Dehn surgery theory for hyperbolic 3-
manifolds generalize to Einstein metrics in any dimension. In particular, this gives large, infinite
families of new Einstein metrics on compact manifolds.
1. Introduction.
In this paper, we construct a large new class of Einstein metrics of negative scalar curvature on
n-dimensional manifolds M = Mn, for any n 4. Einstein metrics are Riemannian metrics g of
constant Ricci curvature, and we will assume the curvature is normalized as
(1.1) Ricg = -(n - 1)g,
so that the scalar curvature s = -n(n-1). The construction is a direct generalization of Thurston's
theory of Dehn surgery or Dehn filling on hyperbolic 3-manifolds [31] to Einstein metrics in any
dimension; in fact the proof gives a new, analytic approach to Thurston's cusp closing theorem
[31], [32].
To describe the construction, start with any complete, non-compact hyperbolic n-manifold N =
Nn of finite volume, with metric g-1 of constant curvature -1. The manifold N has a finite number
of cusp ends {Ej}, 1 j q, with each end E diffeomorphic to F R+, where F is a compact
flat manifold, with flat metric g0 induced from (N, g-1). For simplicity, assume that each F is an
(n - 1)-torus Tn-1; this can always be achieved by passing to a finite covering space if necessary,
cf. [6].

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics