Related papers: A Bando-Mabuchi Uniqueness Theorem

We interpret the coupled Ding semistability and the reduced coupled uniform Ding stability of log Fano pairs in the notion of coupled stability thresholds and reduced coupled stability thresholds. As a corollary, we solve a modified version…

We prove the boundedness theorem for Fano threefolds with log-terminal singularities of any fixed index. This is an improvement of our earlier result, where we required additionally that the variety is Q-factorial, with Picard number 1. The…

We prove existence of twisted K\"ahler-Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when $-K_X$ is big, we obtain a uniform Yau-Tian-Donaldson existence theorem for K\"ahler-Einstein…

We prove the convergence of the Bergman kernels and the $L^2$-Hodge numbers on a tower of Galois coverings $\{X_j\}$ of a compact K\"ahler manifold $X$ converging to an infinite Galois (not necessarily universal) covering $\widetilde{X}$.…

We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kahler-Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat…

Let $X$ be a Fano manifold of dimension at least $2$ and $D$ be a smooth divisor in a multiple of the anticanonical class, $\frac1\alpha(-K_X)$ with $\alpha>1$. It is well-known that K\"ahler-Einstein metrics on $X$ with conic singularities…

In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampere equations with conical singularities along simple normal crossing divisors. In particular, any two points in the…

We prove a general result about the geometry of holomorphic line bundles over Kahler manifolds.

In this note we adapt the work of Hall to find quasi-Einstein metrics on sphere bundles over products of Fano Kaehler-Einstein manifolds, as well as bundles where only one end is blown down.

We prove that the stability condition for Fano manifolds defined by Saito-Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the…

We establish a uniform Sobolev inequality for K\"ahler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular K\"ahler metrics on K\"ahler spaces with…

We give a general inequality for Bergman kernels of Bergman spaces defined by convex weights in $\C^n$. We also discuss how this can be used in Nazarov's proof of the Bourgain-Milman theorem, as a substitute for H\"ormander's estimates for…

In this essay, we study the sufficient and necessary conditions for a Randers metrc to be of constant Ricci curvature without the restriction of strong convexity (regularity). The classification result for the case $\|\beta\|_{\alpha}>1$ is…

Let $X$ be a Fano manifold which is the zero scheme of a general global section $s$ in an irreducible homogenous vector bundle over a Grassmannian. We prove that the restriction of the Pl\"ucker embedding embeds $X$ projectively normal, and…

In this note, we generalized Berndtsson's result about the Nakano positivity of direct image sheaves to some special singular cases.

We consider the problem of existence of constant scalar curvature Kaehler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight…

The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally K\"ahler…

We prove some Liouville-type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary, thereby confirming some cases of Wang's conjecture (J. Geom. Anal. 31,…

The Klein-Grifone approach to global Finsler geometry is adopted. A global existence and uniqueness theorem for Chern connection is formulated and proved. The torsion and curvature tensors of Chern connection are derived. Some properties…

In this article we focus on multiplier Hermitian-Einstein metrics introduced by Mabuchi which include K\"ahler-Einstein metrics, K\"ahler-Ricci solitons and Mabuchi solitons as special cases. We also focus on KSM-manifolds, which are…