Related papers: Gap theorems for K\"ahler-Ricci solitons

For compact K\"ahlerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry…

In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or…

We obtain a formal obstruction, i.e. a necessary condition for the existence of polarised complex deformations of K\"ahler-Ricci solitons. This obstruction is expressed in terms of the harmonic part of the variation of the complex…

In this paper, we deduce a Bochner-type identity for compact gradient Einstein-type manifolds with boundary. As consequence, we are able to show a rigidity result for Einstein-type manifolds assuming the parallel Ricci curvature condition.…

In this paper we prove a gap theorem for K\"ahler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author. We also prove a Liouville theorem for…

We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases…

For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.

We use the momentum construction for $S^1$-invariant K\"ahler metrics as developed by Hwang-Singer to construct new examples of steady K\"ahler-Ricci solitons. We also prove that these solitons are unique in their K\"ahler class, provided…

In $1996$, H.-D. Cao constructed a $U(n)$-invariant steady gradient K\"ahler-Ricci soliton on $\mathbb{C}^{n}$ and asked whether every steady gradient K\"ahler-Ricci soliton of positive curvature on $\mathbb{C}^{n}$ is necessarily…

This undergraduate thesis is focused on introducing the reader to concepts related to the search for topological obstructions to the existence of compact gradient shrinking Ricci soliton metrics in dimension four. It contains a discussion…

We prove that any noncompact $\kappa$-noncollapsed steady Ricci soliton with nonnegative curvature operator must be rotationally symmetric if it has a linear curvature decay.

We show that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union…

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…

We prove that there is no nontrivial $L^2$-integrable harmonic 1-form on noncompact complete gradient steady Ricci solitons or noncompact complete gradient shrinking K\"{a}hler-Ricci solitons. As an application, it can be used to…

We study the modified Ricci solitons as a new class of Einstein type metrics that contains both Ricci solitons and $n$-quasi-Einstein metrics. This class is closely related to the construction of the Ricci solitons that are realised as…

In this paper, we give various curvature pinching conditions such that shrinkers are compact. On one hand, we prove that shrinkers with positive Ricci curvature are compact when they have bounded curvature and certain curvature pinching…

We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that strict linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the strict linear…

This paper is concerned with the study of generalized gradient Ricci-Yamabe solitons. We characterize the compact generalized gradient Ricci-Yamabe soliton and find certain conditions under which the scalar curvature becomes constant. The…

A classical theorem of Bochner asserts that the isometry group of a compact Riemannian manifold with negative Ricci curvature is finite. In this paper we give several extensions of Bochner's theorem by allowing "small" positive Ricci…

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…