scholarly journals AN EXPLICIT FORMULA FOR THE LEVI FORM OF THE DISTANCE FUNCTION TO REAL HYPERSURFACES IN Cn

2012 ◽  
Vol 66 (2) ◽  
pp. 375-381 ◽  
Author(s):  
Kazuko MATSUMOTO
Keyword(s):  
Explicit Formula ◽  
Distance Function ◽  
Levi Form ◽  
Real Hypersurfaces

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.

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.

LEVI-UMBILICAL REAL HYPERSURFACES IN A COMPLEX SPACE FORM

2016 ◽  
Vol 229 ◽  
pp. 99-112 ◽  
Author(s):  
JONG TAEK CHO ◽  
MAKOTO KIMURA
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Levi Form ◽  
Complex Space Form ◽  
Real Hypersurfaces ◽  
Complex Projective Plane ◽  
Nonzero Constant ◽  
Local Construction ◽  
Complex Projective

We give a classification of Levi-umbilical real hypersurfaces in a complex space form $\widetilde{M}_{n}(c)$, $n\geqslant 3$, whose Levi form is proportional to the induced metric by a nonzero constant. In a complex projective plane $\mathbb{C}\mathbb{P}^{2}$, we give a local construction of such hypersurfaces and moreover, we give new examples of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$.

scholarly journals Flow of real hypersurfaces by the trace of the Levi form

1999 ◽  
Vol 6 (6) ◽  
pp. 645-661 ◽  
Author(s):  
Gerhard Huisken ◽  
Wilhelm Klingenberg
Keyword(s):  
Levi Form ◽  
Real Hypersurfaces

Parallel Residues

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Distance Function ◽  
Length Function ◽  
Coxeter System ◽  
Vertex Set ◽  
Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

On Asymmetric Distances

2013 ◽  
Vol 1 ◽  
pp. 200-231 ◽  
Author(s):  
Andrea C.G. Mennucci
Keyword(s):  
Total Variation ◽  
Calculus Of Variations ◽  
Distance Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Segment ◽  
Intrinsic Metric ◽  
Typical Application ◽  
Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

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