Continuous Plurisubharmonic Function Research Articles

On smoothing of plurisubharmonic functions on unbounded domains

We prove that for every n ≥ 2 n \ge 2 , there exists a pseudoconvex domain Ω ⊂ C n \Omega \subset \mathbb {C}^n such that c 0 ( Ω ) ⊊ c 1 ( Ω ) \mathfrak {c}^0(\Omega ) \subsetneq \mathfrak {c}^1(\Omega ) , where c k ( Ω ) \mathfrak {c}^k(\Omega ) denotes the core of Ω \Omega with respect to C k \mathcal {C}^k -smooth plurisubharmonic functions on Ω \Omega . Moreover, we show that there exists a bounded continuous plurisubharmonic function on Ω \Omega that is not the pointwise limit of a sequence of C 1 \mathcal {C}^1 -smooth bounded plurisubharmonic functions on Ω \Omega .

On the Hölder continuous subsolution problem for the complex Monge–Ampère equation, II

We solve the Dirichlet problem for the complex Monge-Amp\`ere equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Amp\`ere measure of a H\"older continuous plurisubharmonic function. If the boundary data is continuous, then the solution is continuous. If the boundary is H\"older continuous, then the solution is also H\"older continuous. In particular, the answer to a question of A. Zeriahi is always affirmative.

Plurisubharmonically separable complex manifolds

Let $M$ be a complex manifold and $PSH^{cb}(M)$ be the space of bounded continuous plurisubharmonic functions on $M$. In this paper we study when functions from $PSH^{cb}(M)$ separate points. Our main results show that this property is equivalent to each of the following properties of $M$: (1) the core of $M$ is empty. (2) for every $w_0\in M$ there is a continuous plurisubharmonic function $u$ with the logarithmic singularity at $w_0$. Moreover, the core of $M$ is the disjoint union of 1-pseudoconcave in the sense of Rothstein sets $E_j$ with the following Liouville property: every function from $PSH^{cb}(M)$ is constant on each of $E_j$.

A cup product lemma for continuous plurisubharmonic functions

A version of Gromov’s cup product lemma in which one factor is the (1, 0)-part of the differential of a continuous plurisubharmonic function is obtained. As an application, it is shown that a connected noncompact complete Kähler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a [Formula: see text]-dimensional complex analytic set at some point has the Bochner–Hartogs property; that is, the first compactly supported cohomology with values in the structure sheaf vanishes.

Hölder continuous solutions of the Monge–Ampère equation on compact Hermitian manifolds

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

A uniform boundedness principle in pluripotential theory

For families of continuous plurisubharmonic functions we show that, in a local sense, separately bounded above implies bounded above.

Levi’s Problem for Pseudoconvex Homogeneous Manifolds

AbstractSuppose G is a connected complex Lie group and H is a closed complex subgroup. Then there exists a closed complex subgroup J of G containing H such that the fibration π:G/H ⟶ G/J is the holomorphic reduction of G/H; i.e., G/J is holomorphically separable and O(G/H)≅ π*O(G/J). In this paper we prove that if G/H is pseudoconvex, i.e., if G/H admits a continuous plurisubharmonic exhaustion function, then G/J is Stein and J/H has no non-constant holomorphic functions.

Plurisubharmonic approximation and boundary values of plurisubharmonic functions

We study the problem of approximating plurisubharmonic functions on a bounded domain Ω by continuous plurisubharmonic functions defined on neighborhoods of Ω¯. It turns out that this problem can be linked to the problem of solving a Dirichlet type problem for functions plurisubharmonic on the compact set Ω¯ in the sense of Poletsky. A stronger notion of hyperconvexity is introduced to fully utilize this connection, and we show that for this class of domains the duality between the two problems is perfect. In this setting, we give a characterization of plurisubharmonic boundary values, and prove some theorems regarding the approximation of plurisubharmonic functions.

Viscosity solutions to degenerate complex monge-ampère equations

Degenerate complex Monge-Ampère equations on compact Kähler manifolds have recently been studied intensively using tools from pluripotential theory. We develop an alternative approach based on the concept of viscosity solutions and systematically compare viscosity concepts with pluripotential theoretic ones. This approach works only for a rather restricted type of degenerate complex Monge-Ampère equations. Nevertheless, we prove that the local potentials of the singular Kähler-Einstein metrics previously constructed by the authors are continuous plurisubharmonic functions. They were previously known to be locally bounded. Another application is a lower-order construction with a C0-estimate of the solution to the Calabi conjecture that does not use Yau's celebrated theorem. © 2011 Wiley Periodicals, Inc.

Plurisubharmonic domination in Banach spaces

We show that if X is a Banach space with a Schauder basis, Ω ⊂ X is a pseudoconvex open subset, and u : Ω → ( − ∞ , ∞ ) is a locally bounded function, then there is a continuous plurisubharmonic function w : Ω → ( − ∞ , ∞ ) with u ( x ) ⩽ w ( x ) for all x ∈ Ω . This has many applications to analytic cohomology of complex Banach manifolds.

On Lelong-Bremermann Lemma

The main theorem of this note is the following refinement of the well-known Lelong-Bremermann Lemma: Let u be a continuous plurisubharmonic function on a Stein manifold. of dimension n. Then there exists an integer m C-m, s = 1, 2,..., such that the sequence of functions u(s) (z) = 1/p(s) max (ln vertical bar g(j)((s)) (z)vertical bar : j = 1,..., m converges to u uniformly on each compact subset of Omega. In the case when Omega is a domain in the complex plane, it is shown that one can take m = 2 in the theorem above (Section 3); on the other hand, for n-circular plurisubharmonic functions in C-n the statement of this theorem is true with m = n + 1 (Section 4). The last section contains some remarks and open questions.

On the behavior of strictly plurisubharmonic functions near real hypersurfaces

We describe the behavior of certain strictly plurisubharmonic functions near some real hypersurfaces in ℂn, n≥3. Given a hypersurface we study continuous plurisubharmonic functions which are zero on the hypersurface and have Monge–Ampere mass greater than one in a one-sided neighborhood of the hypersurface. If we can find complex curves which have sufficiently high contact order with the hypersurface then the plurisubharmonic functions we study cannot be globally Lipschitz in the one-sided neighborhood.

Plurisubharmonic Functions and the Structure of Complete Kähler Manifolds with Nonnegative Curvature

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kahler 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 can be approximated by smooth plurisubharmonic functions through the heat flow deformation. Optimal Liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established. The results are then applied to prove several structure theorems on complete noncompact Kahler manifolds with nonnegative bisectional or sectional curvature.

Partial Harmonicity of Continuous Maximal Plurisubharmonic Functions

If u is a sufficiently smooth maximal plurisubharmonic function such that the complex Hessian of u has constant rank, it is known that there exists a foliation by complex manifolds, such that u is harmonic along the leaves of the foliation. In this paper, we show a partial analogue of this result for maximal plurisubharmonic functions that are merely continuous, without the assumption on the complex Hessian. In this setting, we cannot expect a foliation by complex manifolds, but we prove the existence of positive currents of bidimension (1, 1) such that the function is harmonic along the currents.

A Bounded Plurisubharmonic Function Which is not an Infinite Sum of Continuous Plurisubharmonic Functions

In this note we show an example of a bounded plurisubharmonic function in a bidisc which cannot be represented, in a smaller bidisc, as a sum of a pointwise convergent sequence of continuous plurisubharmonic functions. The problem has been posed by Cegrell and Throbiornson in [2] and is related to Cegrell’s study of the Monge–Ampere operator in [1]. Analogous constructions are possible in higher dimensional spaces. Let us recall that any bounded subharmonic function in a disc is a sum of continuous subharmonic functions (see e.g. [4]).

Bergman kernel and metric on non-smooth pseudoconvex domains

A stability theorem of the Bergman kernel and completeness of the Bergman metric have been proved on a type of non-smooth pseudoconvex domains defined in the following way:D = {z∈U|r(z)} <whereU is a neighbourhood of\(\bar D\) andr is a continuous plurisubharmonic function onU. A continuity principle of the Bergman Kernel for pseudoconvex domains with Lipschitz boundary is also given, which answers a problem of Boas.

Sums of continuous plurisubharmonic functions and the complex Monge-Amp�re operator in ? n

Sums of continuous plurisubharmonic functions and the complex Monge-Amp�re operator in ? n