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

We prove that all ends of a gradient shrinking $\rho$-Einstein soliton are $\varphi$-non-parabolic, provided $\rho$ is nonnegative and the soliton has bounded and nonnegative scalar curvature, where the weight $\varphi$ is a negative…

In this paper, we study the rigidity of eigenvalues of shring Ricci solitons. It is known that the drifted Laplacian on shrinking Ricci solitons has discrete spectrum, its eigenvalues have a lower bound and a rigidity result holds. Firstly,…

We use the theory of isoparametric functions to investigate gradient Ricci solitons with constant scalar curvature. We show rigidity of gradient Ricci solitons with constant scalar curvature under some conditions on the Ricci tensor, which…

We study the properties of Ricci curvature of ${\mathfrak{g}}$-manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between Ricci curvature of the manifold and the Ricci curvature of the…

In this paper, we prove that any compact 2-sided smooth stable minimal hypersurface in gradient Ricci soliton $(M^{n},g,f)$ with scalar curvature $R\geq(n-1)\lambda$ must have vanished second fundamental form and vanished normal Ricci…

In this paper, we prove that any non-flat ancient solution to K\"ahler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also prove that any gradient shrinking solitons with positive bisectional…

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 show that smooth polarized Fano fibrations have no nontrivial finite covers. Using results by Sun-Zhang and Wylie, it follows that shrinking K\"ahler-Ricci solitons are simply-connected.

Recently, Sun-Zhang have developed an algebraic theory for K\"ahler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration $(\pi: X \to Y, \xi)$. In particular, they conjecture that existence of a K\"ahler-Ricci…

We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher…

Among other results, a compact almost K\"ahler manifold is proved to be K\"ahler if the Ricci tensor is semi-negative and its length coincides with that of the star Ricci tensor or if the Ricci tensor is semi-positive and its first order…

We prove a precompactness theorem for invariant metrics on compact homogeneous spaces without injectivity radius bounds, assuming uniform bounds on the diameter and on all derivatives of the curvature tensor. As a consequence, we prove that…

In this paper, we study the geometry and topology of complete gradient shrinking Sasaki-Ricci solitons. We first prove that they must be connected at infinity. This is a Sasaki analogue of gradient shrinking K\"ahler-Ricci solitons.…

We consider the K\"ahler-Ricci flow $\frac{\partial}{\partial t}g_{i\bar{j}} = g_{i\bar{j}} - R_{i\bar{j}}$ on a compact K\"ahler manifold $M$ with $c_1(M) > 0$, of complex dimension $k$. We prove the $\epsilon$-regularity lemma for the…

Suppose $(M^n, g, f)$ is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in \cite{Petersen-Wylie,Guan-Lu-Xu}. Using maximum principle, we prove that…

In this paper, we study conformal Ricci solitons and conformal gradient Ricci solitons on generalized ($\kappa,\mu$)-space forms. The conditions for the solitons to be shrinking, steady, and expanding are derived in terms of conformal…

We introduce the concept {\it $h$-almost Ricci soliton} which extends naturally the {\it almost Ricci soliton} by Pigola-Rigoli-Rimoldi-Setti and show that a compact nontrivial $h$-almost Ricci soliton of dimension no less than three with…

In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete $3$-dimensional…

In this note, we show that any $n$-dimensional $\kappa$-noncollapsed steady K\"ahler-Ricci soliton with nonnegative bisectional curvature must be flat. The result is an improvement to our former work in \cite{DZ2}.

We obtain a compactness result for Fano manifolds and K\"ahler Ricci flows. Comparing to the more general Riemannian versions by Anderson and Hamilton, in this Fano case, the curvature assumption is much weaker and is preserved by the…