Plurisubharmonic Exhaustion Function Research Articles

Minimal kernels of weakly complete spaces

Let X be a weakly complete space i.e. X a complex space endowed with a C k -smooth, k⩾0, plurisubharmonic exhaustion function. We give the notion of minimal kernel Σ 1= Σ 1( X) of X by the following property: x∈ Σ 1 if no continuous plurisubharmonic exhaustion function is strictly plurisubharmonic near x. The study of the geometric properties of the minimal kernels is the aim of present paper. After stating that the minimal kernel Σ 1 of a weakly complete space can be defined by a single plurisubharmonic exhaustion function ϕ, called minimal, using the characterization in terms of Bremermann envelopes, we prove the following, crucial, result: if X is a weakly complete manifold and ϕ a minimal function for X, the nonempty level sets Σ c 1= Σ 1∩{ ϕ= c} have the local maximum property. In the last section we discuss the special case of weakly complete surfaces. We prove that if dim c X=2 and c is a regular value of a minimal function ϕ then the nonempty level sets Σ c 1= Σ 1∩{ ϕ= c} are compact spaces foliated by holomorphic curves.

The Bergman metric on complete K�hler manifolds

We use the existence of a bounded uniformly Holder continuous plurisubharmonic exhaustion function to characterize the Bergman completeness of a complete Kahler manifold. As an application, we proved that any simply-connected complete Kahler manifold with sectional curvature bounded above by a negative constant is Bergman complete.

Symplectic geometry and the uniqueness of Grauert tubes

A compact real analytic Riemannian manifold M admits a canonical complexification with plurisubharmonic exhaustion function satisfying the homogeneous complex Monge-Ampere equation, called a Grauert tube. From the point of view of complex analysis, several authors have considered whether a given complex manifold can arise in more than one way from this construction. We show that given a compact M and a finite exhaustion, the underlying Riemannian structure is unique. The proof uses the technique of holomorphic disks spanning two exact Lagrangian submanifolds of the cotangent bundle of M, and Schwarz reflection.

Generically strongly $q$-convex complex manifolds

Suppose ϕ is a real analytic plurisubharmonic exhaustion function on a connected noncompact complex manifold X. The main result is that if the real analytic set of points at which ϕ is not strongly q-convex is of dimension at most 2q+1, then almost every sufficiently large sublevel of ϕ is strongly q-convex as a complex manifold. For X of dimension 2, this is a special case of a theorem of Diederich and Ohsawa. A version for ϕ real analytic with corners is also obtained.

THE PLURICOMPLEX GREEN FUNCTION ON PSEUDOCONVEX DOMAINS WITH A SMOOTH BOUNDARY

In this article we deal with the behavior of the pluricomplex Green function GD(·;w), of a pseudoconvex domain D in [Formula: see text], when the pole tends to a boundary point. In [7], it was shown that, given a boundary point w0 of a hyperconvex domain D, then there is a pluripolar set E⊂D, such that lim sup w→w0 GD(z;w)=0 for z∈D\E. Under an additional assumption on D, that can be viewed as natural, one can avoid the pluripolar exceptional set. Our main result is that on a bounded domain [Formula: see text] that admits a Hoelder continuous plurisubharmonic exhaustion function ρ:D→[-1,0), the pluricomplex Green function GD(·,w) tends to zero uniformly on compact subsets of D, if the pole w tends to a boundary point w0 of D.

On ${\mathbf{R}}^{+}$ and ${\mathbf{C}}$ complete holomorphic vector fields

We show that, on holomorphic manifolds that have a plurisubharmonic exhaustion function and that do not carry nonconstant bounded plurisubharmonic functions (e.g. Ca), holomorphic vector fields that are complete in positive time are complete in complex time.

Quantitative estimates for the Green function and an application to the Bergman metric

Let D⊂ℂ n be a bounded pseudoconvex domain that admits a Hölder continuous plurisubharmonic exhaustion function. Let its pluricomplex Green function be denoted by G D (.,.). In this article we give for a compact subset K⊂D a quantitative upper bound for the supremum sup z∈K |G D (z,w)| in terms of the boundary distance of K and w. This enables us to prove that, on a smooth bounded regular domain D (in the sense of Diederich-Fornaess), the Bergman differential metric B D (w;X) tends to infinity, for X∈ℂ n /{O}, when w∈D tends to a boundary point. Furthermore, we prove that the order of growth of B D (W;.) under nontangential approach of w∈D to a point z 0 ∈∂D of finite type, can be estimated from below by 1∖N, where N denotes the order of pseudoconvex extendability of ∂D at z 0 .

Some remarks on complex Lie groups

First we show that any complex Lie group is complete Kähler. Moreover we obtain a plurisubharmonic exhaustion function on a complex Lie group as follows. Let the real Lie algebra of a maximal compact real Lie subgroup K of a complex Lie group G. Put q := dimC Then we obtain that there exists a plurisubharmonic, strongly (q + 1)-pseudoconvex in the sense of Andreotti-Grauert and K-invariant exhaustion function on G.

A decomposition problem on weakly pseudoconvex domains

Let Ω ⊂ Cn be a bounded pseudoconvex domain with defining function ρ of class CK , 2 ≤ K ≤ ∞. Let ν ≥ 1 be a real number such that ρ = −(−ρ)1/ν is a strictly plurisubharmonic exhaustion function on Ω. For a bounded open set Ω0 ⊃ Ω we obtain for S = Ω \ Ω the following theorem. Theorem 1. Let k be a nonnegative integer and tk = [2νmax(4+3k, n−1 5 ) +1]. Then there exists a Ck map W k = (W k 1 , · · · , W k n ) : Ω × S −→ Cn with the following properties: (i) W k(·, ζ) is holomorphic for all ζ ∈ S and for (z, ζ) ∈ Ω × S one has

Bounded p.s.h. functions and pseudoconvexity in Kähler manifold

Abstract.It is proved that the C2-smoothly bounded pseudoconvex domains in Pn admit bounded plurisubharmonic exhaustion functions. Further arguments are given concerning the question of existence of strictly plurisubharmonic functions on neighbourhoods of real hypersurfaces in Pn.

Immersions and embeddings in domains of holomorphy

Let D 1 {D_1} be a bounded smooth strongly pseudoconvex domain in C N {\mathbb {C}^N} and let D 2 {D_2} be a domain of holomorphy in C M ( 2 ⩽ N , 5 ⩽ M , 2 N ⩽ M ) {\mathbb {C}^M}(2 \leqslant N,5 \leqslant M,2N \leqslant M) . There exists then a proper holomorphic immersion from D 1 {D_1} to D 2 {D_2} . Furthermore if P I ( D 1 , D 2 ) {\mathbf {PI}}({D_1},{D_2}) is the set of proper holomorphic immersions from D 1 {D_1} to D 2 {D_2} and A ( D 1 , D 2 ) A({D_1},{D_2}) is the set of holomorphic maps from D 1 {D_1} to D 2 {D_2} that are continuous on the boundary, then the closure of P I ( D 1 , D 2 ) {\mathbf {PI}}({D_1},{D_2}) in the topology of uniform convergence on compacta contains A ( D 1 , D 2 ) A({D_1},{D_2}) . The approximating proper maps can be made tangent to any finite order of contact at a given point. The same result was obtained for proper holomorphic maps, in one codimension, when the target domain has a plurisubharmonic exhaustion function with no saddle critical points. This includes the case where the target domain is convex. Density in a weaker sense was derived in one codimension when the critical points are contained in a compact subset of the target domain. This occurs (for example) when the target domain is bounded weakly pseudoconvex with C 2 {C^2} -smooth boundary. If the target domain is strongly pseudoconvex then the approximating proper holomorphic maps can also be made continuous on the boundary. A lesser degree of pseudoconvexity is required from the target domain when the codimension is larger than the minimal. A domain in C L {\mathbb {C}^L} is called " M M dimensional-pseudoconvex" (where L ⩾ M L \geqslant M ) if it has a smooth exhaustion function r r such that every point w w in this domain has some M M -dimensional complex affine subspace going through this point for which r r , restricted to this subspace, is strictly plurisubharmonic in w w . In the result mentioned above the assumption that the target domain is pseudoconvex in C M ( M ⩾ 2 N , 5 ) {\mathbb {C}^M}(M \geqslant 2N,5) can be substituted for the assumption that the domain is " M M -dimensional-pseudoconvex". Similarly, the assumption that the target domain D 2 {D_2} is " ( N + 1 ) (N + 1) -dimensional-pseudoconvex" and all the critical points of some appropriate exhaustion function are " ( N + 1 ) (N + 1) -dimensional-convex" (defined in a similar manner) yields that the closure of the set of proper holomorphic maps from D 1 {D_1} to D 2 {D_2} contains A ( D 1 , D 2 ) A({D_1},{D_2}) . All the results are obtained with embeddings when the Euclidean dimensions are such that dim C ( D 2 ) ⩾ 2 dim C ( D 1 ) + 1 {\dim _\mathbb {C}}({D_2}) \geqslant 2{\dim _\mathbb {C}}({D_1}) + 1 . Thus, in this case, when one of the assumptions mentioned above is fulfilled, then the closure of the set of embeddings from D 1 {D_1} to D 2 {D_2} contains A ( D 1 , D 2 ) A({D_1},{D_2}) .

A Remark on Bounded Strictly Plurisubharmonic Exhaustion Functions

A more general version of the theorem of Diederich and Fornaess on the existence of bounded strictly plurisubharmonic exhaustion functions is proved by completely elementary methods.

A remark on bounded strictly plurisubharmonic exhaustion functions

A more general version of the theorem of Diederich and Fornaess on the existence of bounded strictly plurisubharmonic exhaustion functions is proved by completely elementary methods.

Domains of existence for plurisubharmonic functions

Theorem. Let 12 C ~" be a domain with C 2 boundary, and let the Levi form of OQ be nondeyenerate on an open dense subset of Og2. Then there is a plurisubharmonic function on £2 that cannot be extended to be plurisubharmonic on a set f2: CO" with f21c~0:# 0. We note that the hypotheses of the theorem are "generic" and are satisfied, for instance, if £2 is bounded and has real analytic boundary. The methods here are local, but the situation where a large portion of ~£2 is Levi flat seems to pose a global problem. Let P(f2) denote the plurisubharmonic functions on O. It is well known that if g2 is pseudo-convex, then there is a plurisubharmonic exhaustion function for f2. However, if u~ Lipl(O)c~P(f2) and if f2 is strongly pseudoconvex with C 2 boundary, then u extends to a plurisubharmonic function in a neighborhood of O. On the other hand, if

Exhaustion functions and Stein neighborhoods for smooth pseudoconvex domains

A strictly plurisubharmonic exhaustion function with negative values is constructed for arbitrary relatively compact pseudoconvex domains with smooth boundary in a Stein manifold. It is applied to verify the Serre conjecture in a special case. A sufficient condition is given that guarantees the existence of a neighborhood-basis of Stein domains for certain bounded pseudoconvex domains on a Stein manifold.