Related papers: Rigidity of four-dimensional Gradient shrinking Ri…

In [Cheeger-Tian 2005], Cheeger-Tian proved an $\epsilon$-regularity theorem for $4$-dimensional Einstein manifolds without volume assumption. They conjectured that similar results should hold for critical metrics with constant scalar…

We provide conditions for a compact gradient hyperbolic Ricci and a compact gradient hyperbolic Yamabe soliton to be trivial, hence, the manifold to be an Einstein manifold in the first case, and a manifold of constant scalar curvature, in…

We find new examples of steady gradient Ricci solitons with positive curvature operator in dimensions four and above. Utilising a procedure first introduced by Lai, we construct examples with $O(p) \times O(q)$ symmetry in dimension $p+q$…

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…

In this article we study any 4-dimensional Riemannian manifold $(M,g)$ with harmonic curvature which admits a smooth nonzero solution $f$ to the following equation \begin{eqnarray} \label{0002bx} \nabla df = f(Rc -\frac{R}{n-1} g) + x Rc+…

We show that the combination of non-negative sectional curvature (or $2$-intermediate Ricci curvature) and strict positivity of scalar curvature forces rigidity of complete (non-compact) two-sided stable minimal hypersurfaces in a…

In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $\Sigma$ of a Riemannian 5-manifold $M$…

We show that any $4$-dimensional complete self-shrinker in $\mathbb{R}^{5}$ with constant squared norm $S$ of the second fundamental form, $f_{3}=0$ and constant $f_{4}$ is isometric to $\mathbb{R}^{4}$, where $h_{ij}$ are components of the…

For a shrinking Ricci soliton with Ricci curvature convergent to zero at infinity, it is proved that it must be asymptotically conical.

Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c\neq 0$. Suppose that the structure vector field $\xi$ of $M$ is an eigen vector field of the Ricci tensor $S$, $S\xi=\beta\xi$, $\beta$ being a function. We study on $M$, a…

In this paper we study 4d gradient steady Ricci solitons, which are weak $\kappa$-solutions, and admit O(3)-symmetry. Under a weak curvature decay condition, we find precise geometric asymptotics of such solitons, which are similar to those…

What restrictions are there on a spacetime for which the Ricci curvature is such as to produce convergence of geodesics (such as the preconditions for the Singularity Theorems) but for which there are no singularities? We answer this…

We prove structure results for homogeneous spaces that support a non-constant solution to two general classes of equations involving the Hessian of a function and an invariant 2-tensor. We also consider trace-free versions of these systems.…

In a series of papers, Bamler [Bam20a,Bam20b,Bam20c] further developed the high-dimensional theory of Hamilton's Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity…

In this paper we show that an expanding or steady gradient Ricci soliton warped product $B^n\times_f F^m$, $m>1$, whose warping function $f$ reaches both maximum and minimum must be a Riemannian product. Moreover, we present a necessary and…

We consider solutions (M,g(t)), 0 <= t <T, to Ricci flow on compact, four dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case that the scalar curvature is bounded…

We will give a simple proof that the metric of any compact Yamabe gradient soliton (M,g) is a metric of constant scalar curvature when the dimension of the manifold n>2.

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,…

In this paper, we establish the rigidity of the generalized cylinder $N^n \times \mathbb R^{m-n}$, or a quotient thereof, in the space of Ricci shrinkers equipped with the pointed-Gromov-Hausdorff topology. Here, $N$ is a stable Einstein…

In this announcement, we exhibit the second variation of Perelman's $\lambda$ and $\nu$ functionals for the Ricci flow, and investigate the linear stability of examples. We also define the "central density" of a shrinking Ricci soliton and…