Related papers: A gap theorem on complete shrinking gradient Ricci…

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…

In this paper, we study volume growth, Liouville theorem and the local gradient estimate for $f$-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral…

We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian…

In this article we have showed that a gradient $\rho$-Einstein soliton with a vector field of bounded norm and satisfying some other conditions is isometric to the Euclidean sphere. Later, we have proved that a non-trivial complete gradient…

In this paper, we consider a left-invariant Riemannian metric $g$ on the Lie group $F^4$. We classify Ricci solitons on $(F^4,g)$ and show that all such solitons are expanding and non-gradient. Moreover, we study the existence of harmonic…

We bring new insights into the long-standing Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are…

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-\'{E}mery Ricci tensor has a positive lower bound, and either of the…

In this paper, we investigate curvature pinching phenomena in complete non-compact asymptotically conical gradient expanding Ricci solitons and establish several Hamilton-Ivey type curvature pinching estimates. These results are parallel to…

Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $\mathcal{G}\subset…

We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb{R}^n$. In addtion, we partially give analogous rigidity results of the…

Our main aim in this paper is to investigate the rigidity of complete noncompact gradient steady Ricci solitons with harmonic Weyl tensor. More precisely, we prove that an $n$-dimensional ($n\geq 5$) complete noncompact gradient steady…

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 classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat…

We prove the existence of a unique complete shrinking gradient K\"ahler-Ricci soliton with bounded scalar curvature on the blowup of $\mathbb{C}\times\mathbb{P}^{1}$ at one point. This completes the classification of such solitons in two…

In this paper we prove that the Sobolev inequality, the logarithmic Sobolev inequality, the Schr\"odinger heat kernel upper bound, the Faber-Krahn inequality, the Nash inequality and the Rozenblum-Cwikel-Lieb inequality all equivalently…

A gradient Ricci soliton is a triple $(M,g,f)$ satisfying $R_{ij}+\nabla_i\nabla_j f=\lambda g_{ij}$ for some real number $\lambda$. In this paper, we will show that the completeness of the metric $g$ implies that of the vector field…

In this article, we provide some volume growth estimates for complete noncompact gradient Ricci solitons and quasi-Einstein manifolds similar to the classical results by Bishop, Calabi and Yau for complete Riemannian manifolds with…

This paper studies a complete gradient Ricci soliton with an isoparametric potential function. Our first theorem asserts that, for the steady case, there is a critical level set of codimension greater than one. This is consistent with…

In this work, we derive the complete and explicit solution for the conformal Ricci-Bourguignon soliton on Vaidya spacetime. We provide the closed-form expression for the vector field and establish the necessary conditions for the existence…

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…