Related papers: Deforming three-manifolds with positive scalar cur…

We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M,g) such that: a) (M,g) is non-collapsed, b) the Ricci curvature of (M,g) is bounded from below, c) the geometry of (M,g) at infinity is not too extreme.…

In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is…

Given a 3-manifold that can be written as the double of a compression body, we compute the Chern-Simons critical values for arbitrary compact connected structure groups. We also show that the moduli space of flat connections is connected…

We show that for an orientable non-spin manifold with fundamental group $\mathbb{Z}_2$ and universal cover $S^2\times S^3,$ the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. The…

On a compact three-dimensional Riemannian manifold with boundary, we prove the compactness of the full set of conformal metrics with positive constant scalar curvature and constant mean curvature on the boundary. This involves a blow-up…

A flow-spine of a 3-manifold is a spine admitting a flow that is transverse to the spine, where the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. We say that a contact structure on a closed,…

Cartan-Hadamard manifold is a simply connected Riemannian manifold with non-positive sectional curvature. In this article, we have proved that a Cartan-Hadamard manifold satisfying steady gradient Ricci soliton with the integral condition…

Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface.…

We develop two new methods of constructing sequences of manifolds with positive scalar curvature that converge in the Gromov-Hausdorff and Intrinsic Flat sense to limit spaces with "pulled regions". The examples created rigorously within…

In this paper, we prove Hamilton-Ivey estimates for the Ricci-Bourguignon flow on a compact manifold, with $n=3$ and $\rho<0$. As a consequence, we prove that compact ancient solutions have nonnegative sectional curvature for all negative…

The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also…

We show that, given an immortal solution to the Ricci flow on a closed manifold with uniformly bounded curvature and diameter, the Ricci tensor goes to zero as t goes to infinity. We also show that if there exists an immortal solution on a…

We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the…

In this note, we prove an effective linear volume growth for complete three-manifolds with non-negative Ricci curvature and uniformly positive scalar curvature. This recovers the results obtained by Munteanu-Wang. Our method builds upon…

On a closed manifold, consider the space of all Riemannian metrics for which -Delta + kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature…

We consider the problem of preserving positive Ricci curvature along connected sums. In this context, based on earlier work by Perelman, Burdick introduced the notion of core metrics and showed that the connected sum of manifolds with core…

We prove for $n\in\{3,4,5\}$ that the connected sum of a closed aspherical $n$-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. In particular, a special case of our result…

In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the…

It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient $\kappa$-solution. We prove that the every noncompact ancient $\kappa$-solution in dimension $3$ is…

Let $M^7$ be a smooth manifold equipped with a $G_2$-structure $\phi$, and $Y^3$ be an closed compact $\phi$-associative submanifold. In \cite{McL}, R. McLean proved that the moduli space $\bm_{Y,\phi}$ of the $\phi$-associative…