相关论文: A note on the uniformization of gradient K\"ahler-…

In this survey paper, we analyse and compare the recent curvature estimates for three types of $4$-dimensional gradient Ricci solitons, especially between Ricci shrinkers [58] and expanders [17]. In addition, we provide some new curvature…

We mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. In section 2, we estimate the area growth of level sets and the volume growth of…

We show that expanding K\"ahler-Ricci solitons which have positive holomorphic bisectional curvature and are asymptotic to K\"ahler cones at infinity must be the U(n)-rotationally symmetric expanding solitons constructed by Cao.

We provide necessary and sufficient conditions for some particular couples $(g,\nabla)$ of pseudo-Riemannian metrics and affine connections to be statistical structures if we have gradient almost Einstein, almost Ricci, almost Yamabe…

We derive a sharp lower bound for the scalar curvature of non-flat and non-compact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. We also give an upper bound for the…

Let $B$ be a K\"ahler-Einstein Fano manifold, and $L \to B$ be a suitable root of the canonical bundle. We give a construction of complete Calabi-Yau metrics and gradient shrinking, steady, and expanding K\"ahler-Ricci solitons on the total…

In this paper, we study the uniformly strong convergence of K\"ahler-Ricci flow on a Fano manifold with varied initial metrics and smooth deformation complex structures. As an application, we prove the uniqueness of K\"ahler-Ricci solitons…

In $N(k)$-contact metric manifolds and/or $(k,\mu)$-manifolds, gradient Ricci solitons, compact Ricci solitons and Ricci solitons with $V$ pointwise collinear with the structure vector field $\xi $ are studied.

Let $(g, X)$ be a K\"ahler-Ricci soliton on a complex manifold $M$. We prove that if the K\"ahler manifold $(M, g)$ can be K\"ahler immersed into a definite or indefinite complex space form of constant holomorphic sectional curvature $2c$,…

Haslhofer and M\"uller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly…

The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In…

Let $(M, g, \omega, f, \lambda)$ be a K\"{a}hler gradient Ricci soliton in real dimension four. One first observes that it is an integrable Hamiltonian system in a classical sense. Indeed, all known complete examples are toric and the…

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…

The investigation of Ricci solitons is the focus of this work. We have proved triviality results for compact gradient Ricci soliton under certain restriction. Later, a rigidity result is derived for a compact gradient shrinking Ricci…

In this article, we study the higher-order regularity of the K\"ahler-Ricci flow on compact K\"ahler manifolds with semi-ample canonical line bundle. We proved, using a parabolic analogue of Hein-Tosatti's work on collapsing Calabi-Yau…

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…

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature.

B List has recently studied a geometric flow whose fixed points correspond to static Ricci flat spacetimes. It is now known that this flow is in fact Ricci flow modulo pullback by a certain diffeomorphism. We use this observation to…

Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on $M$, with an appropriate control on the Ricci curvature makes $M$…

Motivated by the long-time behavior of Ricci flows that collapse with bounded curvature, we study expanding Ricci solitons with nilpotent symmetry on vector bundles over a closed manifold. We prove that, under mild assumptions that are…