Evolution by Levi form in $\mathbb{P}^2$

2021 ◽  
Vol 32 (3) ◽  
pp. 549-564
Author(s):  
Giuseppe Tomassini
Keyword(s):  
Levi Form

On the Bergman metric on bounded pseudoconvex domains an approach without the Neumann operator

2014 ◽  
Vol 25 (03) ◽  
pp. 1450025 ◽  
Author(s):  
Gregor Herbort
Keyword(s):  
Distance Function ◽  
Pseudoconvex Domain ◽  
Levi Form ◽  
Pseudoconvex Domains ◽  
Bergman Metric ◽  
Plurisubharmonic Functions ◽  
Euclidean Metric ◽  
Bounded Complex ◽  
Boundary Distance ◽  
Complex Gradient

Let 0 < ε ≤ ½ be fixed. We prove that on a bounded pseudoconvex domain D ⋐ ℂn the Bergman metric grows at least like [Formula: see text] times the euclidean metric, provided that on D there exists a family (φδ)δ of smooth plurisubharmonic functions with a self-bounded complex gradient (uniformly in δ), such that for any δ the Levi form of φδ has eigenvalues ≥ δ-2ε on the set {z ∈ D | δD(z) < δ}. Here, δD denotes the boundary-distance function on D.

Aspects of the Levi form

2019 ◽  
Vol 13 (1) ◽  
pp. 71-89
Author(s):  
Judith Brinkschulte ◽  
C. Denson Hill ◽  
Jürgen Leiterer ◽  
Mauro Nacinovich
Keyword(s):  
Levi Form

Remarks on k-Leviflat Complex Manifolds

1979 ◽  
Vol 31 (4) ◽  
pp. 881-889 ◽  
Author(s):  
B. Gilligan ◽  
A. Huckleberry
Keyword(s):  
Plurisubharmonic Function ◽  
Stein Manifold ◽  
Levi Form ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Complex Manifolds ◽  
Generic Fiber ◽  
Convex Case ◽  
Compact Complex Manifolds

In the theory of functions of several complex variables one is naturally led to study non-compact complex manifolds which have certain types of exhaustions. For example, on a Stein manifold X there is a strictly plurisubharmonic function ϕ: X → R+ so that the pseudoballs Bc = {φ < c } exhaust X. Conversely, a manifold which has such an exhaustion is Stein. The purpose of this note is to study a class of manifolds which have exhaustions along the lines of those on holomorphically convex manifolds, namely the k-Leviflat complex manifolds. Unlike the Stein case, the Levi form may have positive dimensional 0-eigenspaces. In the holomorphically convex case these are tangent to the generic fiber of the Remmert reduction.

scholarly journals Non-degenerate real hypersurfaces in complex manifolds admitting large groups of pseudo-conformal transformations. I

1976 ◽  
Vol 62 ◽  
pp. 55-96 ◽  
Author(s):  
Keizo Yamaguchi
Keyword(s):  
Complex Manifold ◽  
Real Hypersurface ◽  
Levi Form ◽  
Real Hypersurfaces ◽  
Conformal Transformations ◽  
Complex Manifolds ◽  
Real Analytic ◽  
Large Groups

Let S (resp. S′) be a (real) hypersurface (i.e. a real analytic sub-manifold of codimension 1) of an n-dimensional complex manifold M (resp. M′). A homeomorphism f of S onto S′ is called a pseudo-conformal homeomorphism if it can be extended to a holomorphic homeomorphism of a neighborhood of S in M onto a neighborhood of S′ in M. In case such an f exists, we say that S and S′ are pseudo-conformally equivalent. A hypersurface S is called non-degenerate (index r) if its Levi-form is non-degenerate (and its index is equal to r) at each point of S.

scholarly journals An extension theorem for holomorphic mappings - Corrigenda

1983 ◽  
Vol 94 (1) ◽  
pp. 189-189
Keyword(s):  
Tangent Space ◽  
Extension Theorem ◽  
Levi Form ◽  
Holomorphic Mappings ◽  
Additional Hypothesis ◽  
Complex Tangent

J. C. Wood. ‘An extension theorem for holomorphic mappings.’As pointed out by Y.-T. Siu [MR82d:32021] the theorems and corollary require the additional hypothesis that the boundary ∂X, is hyper-(m–1)-convex where m = dim X, i.e. for all p ∈ ∂X, the sum of the eigenvalues of the restriction of the Levi form of ∂X to the complex tangent space Tp(∂X) ∩ JTp(∂X) is non-negative.

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