arXiv research

We prove that square integrable holomorphic functions (with respect to a plurisubharmonic weight) can be extended in a square integrable manner from certain singular hypersurfaces (which include uniformly flat, normal crossing divisors) to entire functions in affine space. This provides evidence for a conjecture regard…

The paper studies quaternionic Monge-Ampère equations in weighted energy classes.

Let $(M,ω)$ be a Kahler manifold. An integrable function on M is called $ω^q$-plurisubharmonic if it is subharmonic on all q-dimensional complex subvarieties. We prove that a smooth $ω^q$-plurisubharmonic function is q-convex. A continuous $ω^q$-plurisubharmonic function admits a local approximation by smooth, $ω^q$-pl…

The paper studies Kähler metrics from finite Monge-Ampère mass exhaustion functions.

In this paper, we show that the extremal length functions on Teichmüller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined b…

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds c…

Uniformises Kähler surfaces with positive curvature to complex plane.

Introduces trace operator for quasi-plurisubharmonic functions on Kähler manifolds.

The paper proves a special case of Yau's conjecture for Kähler surfaces.

Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.

In this article, we solve the strong openness conjecture on the multiplier ideal sheaves for the plurisubharmonic functions posed by Demailly. We prove two conjectures about the growth of the volumes of the sublevel sets of plurisubharmonic functions related to the complex singularity exponents and quasi-plurisubharmon…

This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic functi…

Let $X$ be a compact Kähler manifold and $θ$ a smooth closed $(1,1)$-real form representing a big cohomology class $α\in H^{1,1}(X,\R)$. The purpose of this note is to show, using pluripotential and viscosity techniques, that any $θ$-plurisubharmonic function $\f$ can be approximated from above by a decreasing sequence…

Study distances between special functions on Kähler manifolds.

New curvature assumptions prove Nakano positivity for complex vector bundles.

Smoothly bounded domains have special functions that are plurisubharmonic.

This research extends quasiplurisubharmonic functions on compact Kähler manifolds.

Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.

Sharp estimates for Bergman metrics derived from Kähler quantization.

Paper extends Ohsawa-Takegoshi theorem to more general domains, proving removable singularities for plurisubharmonic functions.

In this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy many of their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are gen…

Energy functional on Teichmüller space is plurisubharmonic but not strictly so.

We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.

We prove a Liouville theorem for the plurisubharmonic functions on complete Kaelher manifolds. As the applications, we prove a splitting theorem for complete Kaehler manifolds with nonnegative biscetional curvature in terms of the linear growth harmonic functions and a optomal gap theorem for such manifolds.

New stability thresholds detect K-stability in Fano manifolds.

We prove a linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow. We then use this sharp differential inequality to study the Liouville properties of the plurisubharmonic functions on complete Kaehler manifolds with nonnegative bisectional curvature.

Solves a specific Dirichlet problem on Hermitian manifolds.

Strict plurisubharmonicity proven for Teichmüller energy on Hitchin representations.

Recently the authors showed that there is a robust potential theory attached to any calibrated manifold (X,φ). In particular, on X there exist φ-plurisubharmonic functions, φ-convex domains, φ-convex boundaries, etc., all inter-related and having a number of good properties. In this paper we show that, in a strong sens…

We introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A…

A hypercomplex manifold is a manifold equipped with a triple of complex structures $I, J, K$ satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metri…

We establish plurisubharmonicity of the envelope of Poisson and Lelong functionals on almost complex manifolds. That is, we generalize the corresponding results for complex manifolds and almost complex manifolds of complex dimension two. We also provide some applications to the regularization of J-plurisubharmonic func…

Let $Ω$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $Ω$. We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local Kähler-Ei…

Study on residual Monge-Ampère mass of complex functions with directional Lipschitz continuity.

This is an essay on potential theory for geometric plurisubharmonic functions. It begins with a given closed subset G of the Grassmann bundle $G(p,TX)$ of tangent $p$-planes to a riemannian manifold $X$. This determines a nonlinear partial differential equation which is convex but never uniformly elliptic (p < dim X). …

Sharp inequalities for weighted log canonical thresholds derived.

Let $(X,ω)$ be a compact Kähler manifold. We introduce and study the largest set $DMA(X,ω)$ of $ω$-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set $PSH(X,\om)$ of all…

New method for complex Monge-Ampère equations on Kähler manifolds.

Study on finite entropy and energy in Kähler geometry.

Constructs a function to prove meromorphic differential strata don't have complete subvarieties.

We introduce a wide subclass ${\cal F}(X,ω)$ of quasi-plurisubharmonic functions in a compact Kähler manifold, on which the complex Monge-Ampère operator is well-defined and the convergence theorem is valid. We also prove that ${\cal F}(X,ω)$ is a convex cone and includes all quasi-plurisubharmonic functions which are …

GC Stein manifolds characterized with embeddings and functions.

We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.

Recently the authors have explored new concepts of plurisubharmonicity and pseudoconvexity, with much of the attendant analysis, in the context of calibrated manifolds. Here a much broader extension is made. This development covers a wide variety of geometric situations, including, for example, Lagrangian plurisubhamon…

Study confirms geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.

Researchers solve Dirichlet problem for complex Monge-Ampère equation on Hermitian manifolds.

Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are define…

The aim of this paper is to study the Lelong number, the integrability index and the Monge-Ampère mass at the origin of an $S^1$-invariant plurisubharmonic function on a balanced domain in $\mathbb{C}^n$ under the Schwarz symmetrization. We prove that $n$ times the integrability index is exactly the Lelong number of th…