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  Generalized connected sum construction for scalar flat metrics

Mazzieri, L. (2009). Generalized connected sum construction for scalar flat metrics. Manuscripta Mathematica, 129(2), 137-168. doi:10.1007/s00229-009-0250-y.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0012-6478-3 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0012-6479-1
Genre: Journal Article

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Mazzieri, Lorenzo1, Author              
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1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24012              

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 Abstract: In this paper we construct constant scalar curvature metrics on the generalized connected sum $${M = M_1 \, \sharp_K \, M_2}$$ of two compact Riemannian scalar flat manifolds (M 1, g 1) and (M 2, g 2) along a common Riemannian submanifold (K, g K ) whose codimension is ≥3. Here 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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 Dates: 2009-06
 Publication Status: Published in print
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 Rev. Method: Peer
 Identifiers: eDoc: 442840
ISI: 000266010200001
ISSN: 0025-2611
DOI: 10.1007/s00229-009-0250-y
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Title: Manuscripta Mathematica
Source Genre: Journal
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Pages: - Volume / Issue: 129 (2) Sequence Number: - Start / End Page: 137 - 168 Identifier: -