Related papers: A Note on Ricci-pinched three-manifolds

In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic,…

The study of stable minimal surfaces in Riemannian $3$-manifolds $(M, g)$ with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when $(M, g)$ is asymptotically flat and has horizon…

We prove an L^2-estimate involving Ricci curvature and a harmonic 1-form on a closed oriented Riemannian 3-manifold admitting a solution of any rescaled Seiberg-Witten equations. We also give a necessary condition to be a monopole class on…

We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role…

We prove that for a solution $(M^n,g(t))$, $t\in[0,T)$, where $T<\infty$, to the Ricci flow with bounded curvature on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant $C$ on…

We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the…

In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial…

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry…

Let C be a cone in the space of algebraic curvature tensors. Moreover, let (M,g) be a compact Einstein manifold with the property that the curvature tensor of (M,g) lies in the cone C at each point on M. We show that (M,g) has constant…

Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\to 0$ as $x\to \infty$. Then the Ricci flow with…

In this paper, we prove a rigidity theorem for Poincar\'e-Einstein manifolds whose conformal infinity is a flat Euclidean space. The proof relies on analyzing the propagation of curvature tensors over the level sets of an adapted boundary…

Let $M$ be an $n$-dimensional complete Riemannian manifold with Ricci curvature $\ge n-1$. In \cite{colding1, colding2}, Tobias Colding, by developing some new techniques, proved that the following three condtions: 1) $d_{GH}(M, S^n)\to 0$;…

Let M be a Wintgen ideal submanifold of dimension n in a real space form R^{n+m}(k) of dimension (n+m) and of constant curvature k, n > 3, m = 1 or m > 1. Let g, R, Ricc, g /\ Ricc and C be the metric tensor, the Riemann-Christoffel…

Let $M$ be a 4-dimensional open manifold with nonnegative Ricci curvature. In this paper, we prove that if the universal cover of $M$ has Euclidean volume growth, then the fundamental group $\pi_1(M)$ is finitely generated. This result…

This paper aims to study the $(m,\rho)$-quasi Einstein manifold. This article shows that a complete and connected Riemannian manifold under certain conditions becomes compact. Also, we have determined an upper bound of the diameter for such…

It is well known that in Lorentzian geometry there are no compact spherical space forms; in dimension 3, this means there are no closed Einstein 3-manifolds with positive Einstein constant. We generalize this fact here, by proving that…

In this note we prove the following result: Let $X$ be a complete, connected 4-manifold with uniformly positive isotropic curvature, with bounded geometry and with no essential incompressible space form. Then $X$ is diffeomorphic to…

We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a…

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional…

We study the rigidity of Ricci-flat manifolds with quadratic curvature decay under conditions on the Green function. We show that if the gradient of the Green function is uniformly bounded from below, then the manifold is flat. Furthermore,…