Related papers: Potential Theory on Almost Complex Manifolds

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…

We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Am\`ere equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and H\"older…

We develop potential theory for $m$-subharmonic functions with respect to a Hermitian metric on a Hermitian manifold. First, we show that the complex Hessian operator is well-defined for bounded functions in this class. This allows to…

We make a systematic study of (quasi-)plurisubharmonic envelopes on compact K\"ahler manifolds, as well as on domains of $\mathbb{C}^n$, by using and extending an approximation process due to Berman [Ber13]. We show that the quasi-psh…

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…

We establish the plurisubharmonicity of the envelope of the Poisson functional on almost complex manifolds. That is, we generalize the corresponding result for complex manifolds and almost complex manifolds of complex dimension two.

Let $X$ be a compact K\"ahler manifold and $\theta$ a smooth closed $(1,1)$-real form representing a big cohomology class $\alpha \in H^{1,1}(X,\R)$. The purpose of this note is to show, using pluripotential and viscosity techniques, that…

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.…

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…

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…

We study boundary properties of plurisubharmonic functions near real submanifolds of almost complex manifolds.

We present some results dealing with the local geometry of almost complex manifolds. We establish mainly the complete hyperbolicity of strictly pseudoconvex domains, the extension of plurisubharmonic functions through generic submanifolds…

We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem…

This paper studies first the differential inequalities that make it possible to build a global theory of pseudo-holomorphic functions in the case of one or several complex variables. In the case of one complex dimension, we prove that the…

Let $\Omega\subseteq M$ be a bounded domain with a smooth boundary $\partial\Omega$, where $(M,J,g)$ is a compact, almost Hermitian manifold. The main result of this paper is to consider the Dirichlet problem for a complex Monge-Amp\`{e}re…

First we extend the theory of subharmonic functions on smooth strictly $k$-analytic curves from Thuillier's thesis to the case of possibly singular analytic curves over a non-archimedean field. Classically psh functions are then defined as…

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…

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel…

The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a…

In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain of a compact K\"ahler manifold. We prove that a precise bound on the complex Monge-Amp\`ere…