Deforming three-manifolds with positive scalar curvature

Abstract

In this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three-manifold is path-connected. The proof uses the Ricci flow with surgery, the conformal method, and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundamental. As one of the applications we prove the path-connectedness of the space of trace-free asymptotically flat solutions to the vacuum Einstein constraint equations on R 3 .

Path-connectedness of the moduli spaces of metrics with positive isotropic curvature on four-manifolds

“…Similar results on three dimensional manifolds with positive scalar curvature were obtained by Marques in [19].…”

“…In paper [19], Marques also gave the notion of a canonical metric, where he allowed to perform connected sum on the principle sphere S 3 with itself, and the other subcomponents are spherical three-manifolds. In dimension four, we have to face with orbifold singularities, and we do not allow to perform connected sums on the principle sphere S 4 with itself in a canonical metric, and we consider the subcomponents of the form (S 3 × R)/G so that we have a uniform description.…”

“…Then we can define the diffeomorphism ψ and the deformation of metrics g µ as in the proof of Lemma 4.1 of [19].…”

“…The above idea has been successfully used by Marques [19] in dimension 3. The main steps of our proof follows the line of arguments in [19].…”

“…The above idea has been successfully used by Marques [19] in dimension 3. The main steps of our proof follows the line of arguments in [19]. To prove Theorem 1.1, one has to prove the canonical metrics of Theorem 1.2 are unique module the group of diffeomorphisms.…”