Fano manifolds with weak almost K\"ahler-Ricci solitons

In this paper, we prove that a sequence of weak almost K\"ahler-Ricci solitons under further suitable conditions converge to a K\"ahler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded below, there exists a sequence of weak almost K\"ahler-Ricci solitons which converge to a K\"ahler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology

Structure of spaces with Bakry-\'Emery Ricci curvature bounded below

In this paper, we explore the limit structure of a sequence of Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded below in the Gromov-Hausdorff topology. By extending the techniques established by Cheeger-Cloding for Riemannian manifolds with Ricci curvature bounded below, we prove that each tangent space at a point of the limit space is a metric cone. We also analyze the singular structure of the limit space analogous to a work of Cheeger-Colding-Tian. Our results will be applied to study the limit space of a sequence of K\"ahler metrics arising from solutions of certain complex Monge-Amp\`ere equations for the existence of K\"ahler-Ricci solitons on a Fano manifold via the continuity method.Comment: The proof of Theorem 5.4 is modified. Some typos are correcte

Higher dimensional steady Ricci solitons with linear curvature decay

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.Comment: We improve the main results in the previous version of paper without the assumption of positive Ricci curvatur

Steady Ricci solitons with horizontally ϵ\epsilon-pinched Ricci curvature

In this paper, we prove that any $\kappa$-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $\epsilon$-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $\kappa$-noncollapsed gradient steady Ricci soliton $(M^n, g,f)$ with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature $R(x)$ satisfies $\lim_{r(x)\rightarrow\infty}R(x)f(x)=C_0\sup_{x\in M}R(x)$ with $C_0>\frac{n-2}{2}$.Comment: Corollary 3.11 is adde

A note on the KK-stability on toric manifolds

In this note, we prove that on polarized toric manifolds the relative $K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and also we show that the modified $K$-energy is proper in the space of $G_0$-invariant K\"ahler metrics for the case of toric surfaces which admit the extremal metrics.Comment: 8 page

Properness of log FF-functionals

In this paper, we apply the method developed in [Ti97] and [TZ00] to proving the properness of log $F$-functional on any conic K\"ahler-Einstein manifolds. As an application, we give an alternative proof for the openness of the continuity method through conic K\"ahler-Einstein metrics.Comment: 21 page

Asymptotic behavior of positively curved steady Ricci Solitons

In this paper, we analyze the asymptotic behavior of $\kappa$-noncollapsed and positively curved steady Ricci solitons and prove that any $n$-dimensional $\kappa$-noncollapsed steady K\"ahler-Ricci soliton with non-negative sectional curvature must be flat.Comment: This is a final version. We added some details in the proofs of Proposition 4.3 and Corollary 4.

Convergence of K\"ahler-Ricci flow on Fano manifolds, II

In this paper, we give an alternative proof for the convergence of K\"ahler-Ricci flow on a Fano mnaifold $(M,J)$. This proof differs from that in [TZ3]. Moreover, we generalize the main theorem of [TZ3] to the case that $(M,J)$ may not admit any K\"ahler-Einstein metrics.Comment: 22 page

GG-Sasaki manifolds and K-energy

In this paper, we introduce a class of Sasaki manifolds with a reductive $G$-group action, called $G$-Sasaki manifolds. By reducing K-energy to a functional defined on a class of convex functions on a moment polytope, we give a criterion for the properness of K-energy. In particular, we deduce a sufficient and necessary condition related to the polytope for the existence of transverse $G$-Sasaki Einstein metrics. A similar result is also obtained for transverse $G$-Sasaki Ricci solitons. As an application, we construct several examples of $G$-Sasaki Ricci solitons by an established openness theorem for transverse $G$-Sasaki Ricci solitons.Comment: some typos were corrected and some references were adde

Relative KK-stability and modified KK-energy on toric manifolds

In this paper, we discuss the relative $K$-stability and the modified $K$-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative $K$-stability and the properness of modified $K$-energy. In particular, our results hold for toric Fano manifolds with vanishing Futaki-invariant. We also verify our results on the toric Fano surfaces