Generalized connected sum construction for scalar flat metrics

Abstract

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.

Connected Sum Construction for σ k -Yamabe Metrics

“…Some comments are due concerning the non degeneracy condition introduced in Definition 1.1. On one hand this kind of hypothesis is common to all the gluing results based on the implicit function theorem and the perturbative approach (such as the previously mentioned works [14], [21], [22] and [23]) for the reasons explained above. On the other hand it must be pointed out that this condition is not fulfilled by the standard sphere S n , since its linearized operator is given by…”

“…Concerning the solvability of the Yamabe equation (k = 1) on the pointwise connected sum of manifolds with constant scalar curvature, we mention the results of Joyce [14] for the compact case and Mazzeo, Pollack and Uhlenbeck [21] for the non compact case. The generalized connected sum has been treated by the second author in [22] and [23]. Most part of the geometric features of these issues are common to our construction.…”

Generalized gluing for Einstein constraint equations

“…In this situation the starting Cauchy data sets are said to be time symmetric and our problem reduces to constructing a scalar flat metric on the generalized connected sum of two scalar flat Riemannian manifolds. This can be done if both (M 1 , g 1 ) and (M 2 , g 2 ) are non Ricci flat, as shown in [18].…”

Gluing Scalar-Flat Manifolds with Vanishing Mean Curvature on the Boundary

“…Integrating (14) yields the desired estimate of λ from the statement of the lemma. In turn, this estimate on λ, (14), and the expression (13) gives an estimate of the form (11) and ( 12), we have arrived at the desired estimate of |ũ 1 |. Now we chose cut-off functions which will be used to glue together the functions ũ1 , u T , and ũ2 from Claims 1 and 2.…”

“…His work generalizes results of D. Joyce [7] on connected sums of closed manifolds of non-zero constant scalar curvature (see also [9]). In [11] Mazzieri considers the more delicate problem of gluing two closed Yamabe-null manifolds to produce another manifold of vanishing scalar curvature. In general, this process may be obstructed if one of the two original manifolds is Ricci-flat.…”