Related papers: Remarks on Singular K\"ahler-Einstein Metrics

We prove the existence of Kahler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kahler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kahler-Ricci flow…

This is the first paper in a series to develop a linear and nonlinear theory for elliptic and parabolic equations on K\"ahler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex…

We extend Tsuji's iterative construction of complete K\"ahler--Einstein metrics with negative scalar curvature to noncompact K\"ahler manifolds with bounded geometry, using Berndtsson's method from the compact setting. Consequently, given a…

We consider an extension of the results of S. Bando, R. Kobyashi, G. Tian, and S. T. Yau on the existence of Ricci-flat K\"{a}hler metrics on quasi-projective varieties Y=X\D with \alpha[D]=c_1(X), \alpha >1. The requirement that D admit a…

In this paper we study a special case of the completion of cusp K\"{a}hler-Einstein metric on the regular part of varieties by taking the continuity method proposed by La Nave and Tian. The differential geometric and algebro-geometric…

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of…

We survey recent results on the existence of K\"ahler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli spaces of K-polystable Fano varieties.…

We explain a characterization of Einstein-Fano manifolds in terms of the lower bound of the density of the volume of the K\"ahler-Ricci Flow. This is a direct consequence of Perelman's uniform estimate for the K\"ahler-Ricci Flow and a…

In this short note, we prove that the space of all admissible piecewise linear metrics parameterized by length square on a triangulated manifolds is a convex cone. We further study Regge's Einstein-Hilbert action and give a much more…

We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our…

Quasi-Einstein manifolds are well-studied generalizations of Einstein manifolds. This includes gradient Ricci solitons and has a natural correspondence with the warped product Einstein manifolds. A quasi-Einstein metric is said to be rigid…

We consider families of conical K\"ahler-Einstein metrics on rank one horosymmetric Fano manifolds, with decreasing cone angles along a codimension one orbit. At the limit angle, which is positive, we show that the metrics, restricted to…

We introduce mu-scalar curvature for a K"ahler metric with a moment map mu and start up a study on constant mu-scalar curvature K"ahler metric as a generalization of both cscK metric and K"ahler-Ricci soliton and as a continuity path to…

We establish the existence of K\"ahler-Ricci flow on pseudoconvex domains with general initial metric without curvature bounds. Moreover we prove that this flow is simultaneously complete, and its normalized version converge to the complete…

Smooth metric measure spaces have been studied from the two different perspectives of Bakry-\'Emery and Chang-Gursky-Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization…

We show the existence of complete negative K\"ahler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature. We prove that any K\"ahler metrics on such manifolds can be deformed to the complete negative…

We consider the K\"ahler-Ricci flow $(X, \omega(t))_{t \in [0,T)}$ on a compact manifold where the time of singularity, $T$, is finite. We assume the existence of a holomorphic map from the K\"ahler manifold $X$ to some analytic variety $Y$…

We produce complete bounded curvature solutions to K\"ahler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth \K metric uniformly equivalent to another complete bounded curvature \K metric. We obtain…

We consider the K\"ahler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the K\"ahler-Ricci flow on total…

We show that integrability of an almost complex structure in complex dimension $m$ is equivalent, in the presence of an almost hermitian metric, to $m(m-1)$ equations involving what we call shear operators. Inspired by this, we give an…