Related papers: Numerical invariants for weighted cscK metrics

Mabuchi solitons generalize K\"{a}hler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with K\"{a}hler-Ricci solitons, there is a distinct necessary…

Recent interest among geometers in $f$-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric…

We prove linear semi-stability for a large class of Einstein metrics of non-positive scalar curvature. More precisely, we show that any Einstein $n$-manifold with non-positive scalar curvature carrying a parallel twisted pure spin$^r$…

In this paper, we prove that on a smooth K\"ahler manifold, the $\mathbb{G}$-coercivity of the weighted Mabuchi functional implies the existence of the (v, w)-weighted-cscK (extremal) metric with v log-concave (firstly studied in…

We will study the $1$-weighted Ricci curvature in view of the extrinsic geometric analysis. We derive several geometric consequences concerning stable weighted minimal hypersurfaces in weighted manifolds under a lower $1$-weighted Ricci…

Suppose $(X,J,\omega)$ is a Fano manifold and $t \to r_t$ is a diverging K\"ahler-Ricci trajectory. We construct a bounded geodesic ray $t \to u_t$ weakly asymptotic to $t \to r_t$, along which Ding's $\mathcal F$-functional decreases,…

We obtain a necessary and sufficient condition of existence of a K{\"a}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is…

We prove that a four-dimensional gradient shrinking Ricci soliton with $\delta W^{\pm}=0$ is either Einstein, or a finite quotient of $S^3\times\mathbb{R}$, $S^2\times\mathbb{R}^2$ or $\mathbb{R}^4$. We also prove that a four-dimensional…

For certain compact complex Fano manifolds $M$ with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of $M$ consisting of K\"ahler classes whose Bando-Calabi-Futaki…

In this paper, we establish several sufficient and necessary conditions for the convergence of a K\"ahler-Ricci flow, on a K\"ahler manifold with positive first Chern class, to a K\"ahler-Einstein metric (or a shrinking K\"ahler-Ricci…

We construct a family of non-collapsed, non-K\"ahler, non-Einstein steady Ricci solitons in even dimensions greater or equal to four. These solitons exist on complex line bundles over K\"ahler-Einstein manifolds of positive scalar…

We examine various examples of horosymmetric manifolds which exhibit interesting properties with respect to canonical metrics. In particular, we determine when the blow-up of a quadric along a linear subquadric admits K\"ahler-Einstein…

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…

We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume…

We study the convergence of the K\"ahler-Ricci flow on a compact K\"ahler manifold $(M,J)$ with positive first Chern class $c_1(M;J)$ and vanished Futaki invariant on $\pi c_1(M;J)$. As the application we establish a criterion for the…

We generalize K\"ahler-Ricci solitons to the almost-K\"ahler setting as the zeros of Inoue's moment map \cite{MR4017922}, and show that their existence is an obstruction to the existence of first-Chern-Einstein almost-K\"ahler metrics on…

We show that if a Fano manifold $M$ is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then $M$ admits a K\"ahler-Einstein metric. This is a strengthening of the solution of the…

We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential $K$-theory, or…

In this paper, we extend Tosatti's method to study the lower boundedness of modified $K$-energy on a Fano manifold and apply this result to study the relative $K$-stability of the deformation space of a K\"ahler Ricci soliton.

We introduce new probabilistic and variational constructions of (twisted) K\"ahler-Einstein metrics on complex projective algebraic varieties, drawing inspiration from Onsager's statistical mechanical model of turbulence in two-dimensional…