Related papers: High Energy plurisubharmonic classes
We study finite energy classes of quasiplurisubharmonic (qpsh) functions in the setting of toric compact K{\"a}hler manifolds. We characterize toric qpsh functions and give necessary and sufficient conditions for them to have finite…
Let $(X,\omega)$ be a compact K\"ahler manifold. We introduce and study the largest set $DMA(X,\omega)$ of $\omega$-plurisubharmonic (psh) functions on which the complex Monge-Amp\`ere operator is well defined. It is much larger than the…
Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Amp\`ere type arising in K\"ahler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric…
In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Amp\`ere equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal…
We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct…
Let $X$ be a compact complex manifold which admits a hermitian metric satisfying a curvature condition introduced by Guan-Li. Given a semipositive form $\theta$ with positive volume, we define the Monge-Amp\`ere operator for unbounded…
The purpose of this article is to study the (residual) Monge-Amp\`{e}re mass of a plurisubharmonic function with an isolated unbounded locus. A general decomposition formula is obtained under the Sasakian structure of the unit sphere. In…
In this paper, we use Takeuchi's Theorem to show that for every Lipschitz pseudoconvex domain $\Omega$ in $\mathbb{CP}^n$ there exists a Lipschitz defining function $\rho$ and an exponent $0<\eta<1$ such that $-(-\rho)^\eta$ is strictly…
We undertake a preliminary step towards studying non-Archimedean pluripotential theory on polarized affine cones over a trivially valued field. We study plurisubharmonic functions and the Monge--Amp\`ere operator defined on the finite…
We study the complex Monge-Amp\` ere operator on compact K\"ahler manifolds. We give a complete description of its range on the set of $\omega-$plurisubharmonic functions with $L^2$ gradient and finite self energy, generalizing to this…
The goal of this short note is to relate the integrability property of the exponential $e^{-2\phi}$ of a plurisubharmonic function $\phi$ with isolated or compactly supported singularities, to a priori bounds for the Monge-Amp\`ere mass of…
We study the Parabolic complex Monge-Amp\'ere equation in a bounded strictly pseudoconvex domain in \mathbb{C}^n, with the boundary condition u=\varphi and the initial condition u=u_0. In this paper, we consider the case where \varphi is…
We investigate the finite $p$-energy classes $E_p$ of quaternionic plurisubharmonic functions of Cegrell type. We also construct an example to show that the optimal constant in the energy estimate is strictly bigger than $1$ for $p>0$,…
On bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function $\phi$ such that $\phi^*=\phi_*$ on the closure of the domain. For $\phi$ satisfying certain additional criteria limiting its…
Let $\mu$ be a non-negative measure defined on bounded $\mathcal F$-hyperconvex domain $\Omega$. We are interested in giving sufficient conditions on $\mu$ such that we can find a plurifinely plurisubharmonic function satisfying $NP (dd^c…
We prove several results showing that plurisubharmonic functions with various bounds on their Monge-Ampere masses on a bounded hyperconvex domain always admit global plurisubharmonic subextension with logarithmic growth at infinity.
We characterize the class of probability measures on a compact Kahler manifold such that the associated Monge-Amp\`ere equation has a solution of finite pluricomplex energy. Our results are also valid in the big cohomology class setting.
For suitable bounded hyperconvex sets $\Omega$ in $\mathbb{C}^N$, in particular the ball or the polydisk, we give estimates for the approximation numbers of composition operators $C_\phi \colon H^2 (\Omega) \to H^2 (\Omega)$ when $\phi…
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…
We study the Dirichlet problem for the complex Monge-Amp\`ere operator on a B-regular domain $\Omega$, allowing boundary data that is singular or unbounded. We introduce the concept of pluri-quasibounded functions on $\Omega$ and $\partial…