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...m a small neighbourhood of the bridges, or to create wormholes within a given initial data set, if one could show that the mass of the resulting Schwarzschild metrics can be made positive. 4 (Compare =-=[13,14]-=-, where the conformal method is used for the gluings: this leads a priori to conformal deformations of the metrics which are small away from a neighbourhood of the gluing point, but non-zero throughou...

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..., in the degenerate case, qε ∈ S, but equals 0 otherwise. For simplicity, we also write Yj = ∂Mj. We first recall the connected sum construction for metrics of constant scalar curvature from [24] and =-=[16]-=-. (We also note here that the first proof that the connected sum of scalar positive conformal classes is again scalar positive was achieved earlier by O. Kobayashi [18]). The idea of the construction ...

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...to mention some similar constructions in the literature. An analogous surgery formula holds if we replace the Conformal Laplacian by the Dirac operator, see [4] for details and applications. D. Joyce =-=[20]-=-, followed by L. Mazzieri [30, 31], considered a problem tightly related to our result: their goal is to construct a metric on a manifold obtained via a connected sum along a k-dimensional submanifold...

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...h line of our story is that the above gluing construction can be carried out uniformly in these additional parameters, and, setting k = ℓ + 2, a gluing argument suggested by the work of Dominic Joyce =-=[20]-=- allows us to show the existence of 1-parameter family of ASD metrics on M = kCP2 for which the scalar curvature changes sign in an analogous manner: Theorem B For any integer k ≥ 5, the smooth orient...

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...determining the sign of the Yamabe constant for the anti-self-dual conformal classes constructed in Theorem 4.4. The results obtained in this section are loosely inspired by the work of Dominic Joyce =-=[37]-=- on the Yamabe constants of certain conformal classes on connect sums, although the techniques employed here are completely different from Joyce’s. Lemma 6.1. Let ̟ : Z → U be a 1-parameter standard d...

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...3]. It also allows to rule out Einstein metrics on many spaces [Pet98]. However, in the case σ(M) > 0 it is still open under what conditions one has a monotonicity formula σ(M # ) ≥ σ(M). The article =-=[Joy03]-=- studies 0-dimensional surgery in more details, in particular he shows the non-uniqueness of minimizers of the infimum in (1) in the case σ(M) > 0. Some of these results have been recently generalized...

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...linear problems has revealed to be a very powerful tool in understanding solutions of many geometric problems (minimal and constant mean curvature surfaces [7], [8], constant scalar curvature metrics =-=[4]-=-, [9], [6], and recently even Einstein metrics [1]). However, generalized connected sums along a submanifold have not been addressed so much, probably because these constructions are less flexible. In...