Metrics of positive Ricci curvature on the connected sums of products with arbitrarily many spheres
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Metrics of positive Ricci curvature on the connected sums of products with arbitrarily many spheres Bradley Lewis Burdick1 Received: 8 January 2020 / Accepted: 19 August 2020 © Springer Nature B.V. 2020
Abstract We construct Ricci-positive metrics on the connected sum of products of arbitrarily many spheres provided the dimensions of all but one sphere in each summand are at least 3. There are two new technical theorems required to extend previous results on sums of products of two spheres. The first theorem is a gluing construction for Ricci-positive manifolds with corners that gives a sufficient condition to glue together two Ricci-positive Riemannian manifolds with corners along isometric faces so that the resulting smooth manifold with boundary will be Ricci-positive and have convex boundary. The second theorem claims that one can deform the boundary of a Ricci-positive Riemannian manifold with convex boundary along a Ricci-positive isotopy while preserving Ricci-positivity and boundary convexity. Keywords Positive Ricci curvature · Connected sums · Manifolds with corners Mathematics Subject Classification 53C20
1 Introduction 1.1 Background and main results The existence of a metric of positive scalar curvature was shown to be invariant under surgery in codimension at least 3 by Gromov-Lawson [12] and Schoen-Yau [19]. In particular, the existence of a metric of positive scalar curvature is always invariant under connected sum. Under rigid metric conditions, the existence of a metric of positive Ricci curvature was shown to be invariant in surgery in codimension at least 3 and dimension at least 2 by Sha-Yang [21] and Wraith [28]. The techniques of [12, 19] fundamentally cannot work to preserve positive Ricci curvature (see [13, 27] for discussion) while the Ricci-positive surgery constructions of [21, 28] also cannot be extended to connected sum. Motivated by the case of positive scalar curvature, in this paper we will consider under what conditions is the existence of a Ricci-positive metric preserved under connected sum. * Bradley Lewis Burdick [email protected] 1
University of California, Riverside, 900 University Ave., Riverside, CA 92521, USA
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Vol.:(0123456789)
Annals of Global Analysis and Geometry
Other than the fundamental group, there is no known or expected topological obstruction to the existence of a metric of positive Ricci curvature on the connected sum. As explained, we need to take an entirely different approach from the existing curvature stable surgery results. While there are few general results, there are number of interesting examples of Riccipositive connected sums. The earliest nontrivial example is due to Cheeger [8] on the connected sum of any two of the following manifolds: 𝐂Pn , 𝐇Pn , and 𝐎P2 . Using the surgery result alluded to above, Sha and Yang [21] constructed Ricci-positive metrics on #k (Sn × Sm ) for each k after identifying it with iterated surgery on Sn−1 × Sm+1 . Later in [29], Wraith gave a careful generalization of the construction of [21] that allowed
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