Let F be a Riemann surface foliation on M∖E, where M is a complex manifold and E⊂M is a closed set. Assume that F is hyperbolic, i.e. all the leaves of the foliation F are hyperbolic Riemann ...surfaces. Fix a Hermitian metric g on M. We consider the Verjovsky modulus of uniformization map η, which measures the largest possible derivative in the class of holomorphic maps from the unit disc into the leaves of F. Various results are known to ensure the continuity of the map η along the transverse directions, with suitable conditions on M, F and E. For a domain U⊂M, let FU be the holomorphic foliation given by the restriction of F to the domain U, i.e. F|U. We consider the modulus of uniformization map ηU corresponding to the foliation FU and study its variation when the corresponding domain U varies in the Caratheodory kernel sense, motivated by the work of Lins Neto and Canille Martins.
A novel score function based on the Poincaré metric is proposed and applied to a decision-making problem. Decision-making on Fuzzy Sets (FSs) has been considered due to the flexibility of the data, ...and it is applied to the decision-making. However, decisions with FSs are sometimes nondecisive even for different membership degrees. Hence, Intuitionistic Fuzzy Sets (IFSs) data is applied to design a score function for the decision-making with the Poincaré metric. This function is supported by the profound information of IFSs; IFSs include hesitation degree together with membership and non-membership degree. Hence, IFS membership and non-membership degree are expressed as two-dimensional vectors satisfying the Poincaré metric for simplification. At the same time, the proposed approach addresses the hesitation information in the IFS data. Next, a score function is proposed, constructed and provided. The proposed score function has a strict monotonic property and addresses the preference without resorting to the accuracy function. The strict monotonic property guarantees the preference of all attributes. Additionally, the existing problem of score function design in IFSs is addressed: they return zero scores even with different meanings for the same membership and non-membership degree. The advantages of the proposed score function over existing ones are demonstrated through illustrative examples. From the calculation results, the proposed decision score function discriminates between all candidates. Hence, the proposed research provides a solid foundation for the hesitation analysis on the decision-making problem.
We study the holonomy cocycle
H
of a holomorphic foliation
F
by Riemann surfaces defined on a compact complex projective surface
X
satisfying the following two conditions:
its singularities
E
are all ...hyperbolic;
there is no holomorphic non-constant map
C
→
X
such that out of
E
the image of
C
is locally contained in a leaf.
Let
T
be a harmonic current tangent to
F
which does not give mass to any invariant analytic curve. Using the leafwise Poincaré metric, we show that
H
is integrable with respect to
T
. Consequently, we infer the existence of the Lyapunov exponent function of
T
.
Let
be a smooth Riemann surface foliation on
, where M is a complex manifold and
is a closed set. Fix a hermitian metric g on
and assume that all leaves of
are hyperbolic. For each leaf
, the ratio ...of
, the restriction of g to L, and the Poincaré metric
on L defines a positive function η that is known to be continuous on
under suitable conditions on M, E. For a domain
, we consider
, the restriction of
to U and the corresponding positive function
by considering the ratio of g and the Poincaré metric on the leaves of
. First, we study the variation of
as U varies in the Hausdorff sense motivated by the work of Lins Neto-Martins. Secondly, Minda had shown the existence of a domain Bloch constant for a hyperbolic Riemann surface S, which in other words shows that every holomorphic map from the unit disc into S, whose distortion at the origin is bounded below, must be locally injective in some hyperbolic ball of uniform radius. We show how to deduce a version of this Bloch constant for
.
A note on convex conformal mappings Chuaqui, Martin; Osgood, Brad
Proceedings of the American Mathematical Society,
06/2019, Letnik:
147, Številka:
6
Journal Article
Recenzirano
We establish a new characterization for a conformal mapping of the unit disk \mathbb{D} to be convex, and identify the mappings onto a half-plane or a parallel strip as extremals. We also show that, ...with these exceptions, the level sets of \lambda of the Poincaré metric \lambda \vert dw\vert of a convex domain are strictly convex.
Let
F
be a smooth Riemann surface foliation on
M
\
E
, where
M
is a complex manifold and the singular set
E
⊂
M
is an analytic set of codimension at least two. Fix a hermitian metric on
M
and assume ...that all leaves of
F
are hyperbolic. Verjovsky’s modulus of uniformization
η
is a positive real function defined on
M
\
E
defined in terms of the family of holomorphic maps from the unit disc
D
into the leaves of
F
and is a measure of the largest possible derivative in the class of such maps. Various conditions are known that guarantee the continuity of
η
on
M
\
E
. The main question that is addressed here is its continuity at points of
E
. To do this, we adapt Whitney’s
C
4
-tangent cone construction for analytic sets to the setting of foliations and use it to define the tangent cone of
F
at points of
E
. This leads to the definition of a foliation that is of
transversal type
at points of
E
. It is shown that the map
η
associated to such foliations is continuous at
E
provided that it is continuous on
M
\
E
and
F
is of transversal type. We also present observations on the locus of discontinuity of
η
. Finally, for a domain
U
⊂
M
, we consider
F
U
, the restriction of
F
to
U
and the corresponding positive function
η
U
. Using the transversality hypothesis leads to strengthened versions of the results of Lins Neto–Martins on the variation
U
↦
η
U
.
Consider a Brody hyperbolic foliation with non-degenerate singularities on a compact complex manifold. We show that the leafwise heat diffusions and the abstract heat diffusions coincide. In ...particular, this will imply that the abstract heat diffusions are unique.
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