# Deforming three-manifolds with positive scalar curvature

```@article{Marques2009DeformingTW,
title={Deforming three-manifolds with positive scalar curvature},
author={Fernando Cod{\'a} Marques},
journal={arXiv: Differential Geometry},
year={2009}
}```
• F. C. Marques
• Published 2009
• Mathematics, Physics
• arXiv: Differential Geometry
In this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact 3-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 \$\mathbb… Expand
62 Citations

#### Paper Mentions

Deformations of positive scalar curvature metrics on 3-manifolds with mean-convex boundary
We give a complete topological characterization of those compact 3-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary. In any such case, we prove that theExpand
Deforming 3-manifolds of bounded geometry and uniformly positive scalar curvature
• Mathematics
• 2017
We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the mainExpand
Metrics with nonnegative Ricci curvature on convex three-manifolds
• Mathematics
• 2016
We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path connected; and, moreover, that the associated moduliExpand
Scalar Curvature, Conformal Geometry, and the Ricci Flow with Surgery
In this note we will review recent results concerning two geometric problems associated to the scalar curvature. In the first part we will review the solution to Schoen’s conjecture about theExpand
Equivariant manifolds with positive scalar curvature
• Mathematics
• 2021
In this paper, for any compact Lie group G, we show that the space of G-invariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible.Expand
Constrained deformations of positive scalar curvature metrics, II
We prove that various spaces of constrained positive scalar curvature metrics on compact 3-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given inExpand
Constrained deformations of positive scalar curvature metrics
• Mathematics
• 2019
We present a series of results concerning the interplay between the scalar curvature of a manifold and the mean curvature of its boundary. In particular, we give a complete topologicalExpand
Ricci flow and contractibility of spaces of metrics
• Mathematics
• 2020
Author(s): Bamler, Richard H; Kleiner, Bruce | Abstract: We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that theExpand
Four-manifolds with positive isotropic curvature
• Mathematics
• 2016
We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compactExpand
The moduli space of two-convex embedded spheres
• Mathematics
• Journal of Differential Geometry
• 2021
We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog toExpand

#### References

SHOWING 1-10 OF 50 REFERENCES
Collapsing irreducible 3-manifolds with nontrivial fundamental group
• Mathematics
• 2009
Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics which volume-collapses and whose sectional curvature is locallyExpand
Weak collapsing and geometrisation of aspherical 3-manifolds
• Mathematics
• 2008
Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thickExpand
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. We show that for any initial riemannian metric on M the solution to the Ricci flow with surgery,Expand
The Ricci flow on the 2-sphere
The classical uniformization theorem, interpreted differential geomet-rically, states that any Riemannian metric on a 2-dimensional surface ispointwise conformal to a constant curvature metric. ThusExpand
Quasiconvex Foliations and Asymptotically Flat Metrics of Non-negative Scalar Curvature
• Mathematics
• 2004
where R is the scalar curvature of the Riemannian metric g induced on a maximal time-slice, and k is the second fundamental form of that slice in the ambient Lorentzian 4-manifold. It is clear fromExpand
Ricci flow with surgery on three-manifolds
This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print: theExpand
Construction of manifolds of positive scalar curvature
We prove that a regular neighborhood of a codimension > 3 subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature. A number of useful factsExpand
Manifolds of Positive Scalar Curvature: A Progress Report
In the special case n = 2, the scalar curvature is just twice the Gaussian curvature. This paper will deal with bounds on the scalar curvature, and especially, with the question of when a givenExpand
Completion of the Proof of the Geometrization Conjecture
• Mathematics
• 2008
This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact,Expand
NONCONNECTED MODULI SPACES OF POSITIVE SECTIONAL CURVATURE METRICS
• Mathematics
• 1993
For a closed manifold M let 9\~(M) (resp. 9\~ic(M)) be the space of Riemannian metrics on M with positive sectional (resp. Ricci) cur- vature and let Diff(M) be the diffeomorphism group of M, whichExpand