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manuscripta mathematica

, Volume 129, Issue 2, pp 137–168 | Cite as

Generalized connected sum construction for scalar flat metrics

  • Lorenzo MazzieriEmail author
Open Access
Article

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.

Mathematics Subject Classification (2000)

53C21 58J60 53A30 57R65 83C05 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Université Paris 12ParisFrance
  3. 3.Max Planck Institut für Gravitationsphysik, Albert Einstein InstitutGolmGermany

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