On the structure of manifolds with positive scalar curvature

Schoen, Richard; Yau, Shing–Tung

doi:10.1007/bf01647970

Cited by 500 publications

(444 citation statements)

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“…However, if there is such a Cauchy surface, then it follows from the constraint equations that the scalar curvature on this maximal surface must be non-negative. This is known to be incompatible with the topology T 3 #T 3 [16]. Thus we have a contradiction, from which it follows that the space-time development of initial data with the reflection properties described above cannot contain a CMC Cauchy surface.…”

Section: Space-times With No Cmc Slicesmentioning

confidence: 88%

Gluing Initial Data Sets for General Relativity

2004

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We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this construction we establish the existence of cosmological, maximal globally hyperbolic, vacuum space-times with no constant mean curvature spacelike Cauchy surfaces.

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Section: Space-times With No Cmc Slicesmentioning

confidence: 88%

Gluing Initial Data Sets for General Relativity

2004

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“…However most of the results are concerned with the connected sum at points (point-wise connected sum), whereas the case of connected sum along a submanifold (generalized connected sum or fiber sum) has received less attention. This kind of construction is clearly less flexible than the first one, nevertheless it has revealed to be a very powerful tool in studying for example the structure of the manifolds with positive scalar curvature (see [6,17]). …”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…We also assume that the normal bundles of K in (M i , g i ) can be diffeomorphically identified (this is necessary to perform the fiber sum, see [17]). Another natural assumption is that the starting manifolds have the same volume and in particular we assume that vol…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Generalized connected sum construction for scalar flat metrics

Mazzieri

2009

manuscripta math.

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Abstract. In this paper we construct constant scalar curvature metrics on the generalizedHere we present two constructions: the first one produces a family of "small" (in general nonzero) constant scalar curvature metrics on the generalized connected sum of M 1 and M 2 . It yields an extension of Joyce's result for point-wise connected sums in the spirit of our previous issues for nonzero constant scalar curvature metrics. When the initial manifolds are not Ricci flat, and in particular they belong to the (1 + ) class in the Kazdan-Warner classification, we refine the first construction in order to produce a family of scalar flat metrics on M. As a consequence we get new solutions to the Einstein constraint equations on the generalized connected sum of two compact time symmetric initial data sets, extending the Isenberg-Mazzeo-Pollack gluing construction.

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“…By [Ker69, p.68] the desired homology n-sphere can be constructed from a connected sum of several copies of S 1 × S n−1 by surgeries on circles and 2-spheres. Thus the surgeries have codimension ≥ 3 and the homology sphere have scal > 0 by [SY79,GL80].…”

Section: Equivariant Connected Sum and Positive Scalar Curvaturementioning

confidence: 99%