Critical metrics on connected sums of Einstein four-manifolds

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• 作者：Matthew J. Gursky^a ; ^{mgursky@nd.edu" class="auth_mail" title="E-mail the corresponding author} ; Jeff A. Viaclovsky^b ; ^{jeffv@math.wisc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Einstein metrics ; Quadratic curvature functionals ; Gluing ; Critical metrics
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：9 April 2016
• 年：2016
• 卷：292
• 期：Complete
• 页码：210-315
• 全文大小：1122 K

文摘

We develop a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini–Study metric on CP²${\mathbb{CP}}^2$ and the product metric on S²×S²$S^2\times S^2$. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific Riemannian functional, which depends on the global geometry of the factors. Furthermore, using certain quotients of S²×S²$S^2\times S^2$ as one of the gluing factors, critical metrics on several non-simply-connected manifolds are also obtained.