aa r X i v : . [ m a t h . C V ] D ec ON A LOWER BOUND OF THE KOBAYASHI METRIC
NIKOLAI NIKOLOV
Abstract.
It is shown that a lower bound of the Kobayashi met-ric of convex domains in C n does not hold for non-convex domains. Let D be a domain in C n , z ∈ D and X ∈ C n . Denote by γ D and κ D the Carath´eodory and Kobayashi metrics of D : γ D ( z ; X ) = sup {| f ′ ( z ) X | : f ∈ O ( D, D ) } ,κ D ( z ; X ) = inf {| α | : ∃ ϕ ∈ O ( D , D ) with ϕ (0) = z, αϕ ′ (0) = X } , where D is the unit disc. Then γ D ≤ κ D . By the Lempert theorem [5], γ D = κ D if D is convex.Set d D ( z ; X ) to be the distance from z to ∂D in the X -direction, i.e. d D ( z ; X ) = sup { r > z + λX ∈ D if | λ | < r } (possibly d D ( z ; X ) = ∞ ). It follows by the definition of κ D that κ D ( z ; X ) ≤ d D ( z ; X ) . On the other hand, if D is convex, then γ D ( z ; X ) ≥ d D ( z ; X )due to S. Frankel [2, Theorem 2.2] and I. Graham [3, Theorem 5]. Thefollowing short proof of this estimate can be found in [1, Theorem 4.1]: γ D ( z ; X ) ≥ γ Π ( z ; X ) = 12 d Π ( z ; X ) = 12 d D ( z ; X ) , where Π is a supporting half-space of D at a boundary point of theform z + λX, | λ | = d D ( z, X ) . It turns out that the converse to the Frankel-Graham result holds.
Proposition 1.
For a domain D in C n the following conditions areequivalent:(a) D is convex;(b) γ D ( z ; X ) ≥ d D ( z ; X ) ; Mathematics Subject Classification.
Key words and phrases.
Carath´eodory and Kobayashi metrics, convex domain. (c) lim inf z → a κ D ( z ; z − a ) − || z − a || ≥ for any a ∈ ∂D. Proof.
We have only to show that (c) implies (a). Assume the contrary.According to [4, Theorem 2.1.27], one may find a point a ∈ ∂D anda real-valued quadratic polynomial q such that q ( a ) = 0 , ∇ q ( a ) = 0 , the set G = { z ∈ C n : q ( z ) < } is contained in D near a and ∂G hasnormal curvature χ < X ∈ T R a ∂G. We may assumeby continuity that X T C a ∂G. Then the planar set F = G ∩ ( a + C X )has smooth boundary near a with curvature χ at a. Let E be theconnected component of F for which a ∈ ∂E. Denote by n the innernormal to ∂E at a. Using [6, Proposition 1], we get the contradiction0 ≤ lim sup n ∋ z → a κ D ( z ; z − a ) − || z − a || ≤ lim n ∋ z → a κ E ( z ; z − a ) − || z − a || = χ < . References [1] E. Bedford, S. I. Pinchuk,
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Sharp constants for the Koebe theorem and for estimates of intrinsicmetrics on convex domains , Proc. Symp. Pure Math. 52(2) (1991), 233-238.[4] L. H¨ormander,
Notions of convexity , Birkh¨auser, Boston, 1994.[5] L. Lempert,
Holomorphic retracts and intrinsic metrics in convex domains ,Anal. Math. 8 (1982), 257-264.[6] N. Nikolov, M. Trybu la, L. Andreev,
Boundary behavior of invariant functionson planar domains , arXiv:1511.03117.
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