On the Hessian of the Carathéodory metric
On the Hessian of the Carathéodory metric
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关于卡拉西奥多里度量的 Hessian 矩阵
DOI：
10.1216/rmj-1978-8-3-555
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发表时间：
1978
影响因子：
0.8
通讯作者：
J. Burbea
中科院分区：
文献类型：
--
作者：
J. Burbea
引入了上半连续函数在C中的点z附近的广义下Hessian(对于n=1，见Heins，Nagoya Math.J.2L(1962)，1-60)。由此，我们引入了“截曲率”，并证明了Carathéodory-ReifFen度量的截面曲率总是^-4。这推广了Suita(Kodai Math.扫描电子显微镜。Rep.25(1973)，215-218)。球的截面曲率和多面体的截面曲率都是-4。本文还给出了上述度规的海森的一些其他性质。1.预赛。对于点z=(zi9··-，zn)GC，我们设||z||=(2 7 1 L%L)，对FLEO，r>；0，B(a，r)={zGC：\\Z-a，r；r}表示以a为中心，半径为r的开球.切向量a与切向量v之间的自然对记为(a，v).特别地，若f是点z附近的C函数，且v=(v19···，！？“)GB O，则<；df(Z)，v)=t-jt-vj。设D是C“中的有界域，U是C中的单位圆，H(D：U)表示从D到U的全纯函数族。对于D中的固定GB，我们记H{(D：U)={/G H(D：U)：/(GB)=0}。对于每个{EUROND D，Cd(Gb；)是由Cd(C；v)=sup{|(Df(Av)\：FGH(D：U))定义在D在Gb的复切空间上的函数。Reiffen[3])。显然，Cd(C；v)=sup{|<；d/(GB)，t>；)|：fgh((D：i/)}.CD称为D的Carathéodory度量。由于H(D：U)是一个正规族，CD(GB；v)的定义中的上界是由某个F G Ht(D：C7)获得的。这里F(Z)=F(z；GB，v)。根据正规族的推论，CD(GB：t>；)在(GB，u)中是连续的。有关上述属性，请参阅[3]。此外，Cd(GB；v)是1976年7月19日由编辑收到的局部Lipschitz函数[！]·，并于1976年11月8日修订后的形式。AMS(MOS)1970年主题分类。32h15、32a30。
The generalized lower Hessian of an upper semi-continuous function / near a point z in C is introduced (for n = 1 see Heins, Nagoya Math. J. 2L (1962), 1-60). With this we introduce a "sectional curvature" and we prove that the sectional curvature of the Carathéodory-ReifFen metric is always ^ — 4. This generalizes a result of Suita (Kodai Math. Sem. Rep. 25 (1973), 215-218) in the one dimensional case. The sectional curvatures of the ball and polydisk are always —4. A few other properties of the Hessian of the above metric are shown. 1. Preliminaries. For a point z = (zi9 • • -, zn) G C , we set ||z|| = ( 2 7 1 l%l) and for flEO, r > 0, B(a, r) = {z G C : \\z — a\\ < r} denotes the open ball centered at a and with radius r. The natural pairing between a contangent vector a and a tangent vector v is denoted by (a, v). Especially, iff is a C function near the point z, and v = (vl9 • • • , !?„)£ O , then <df(z),v)= t-jt-vj. Let D be a bounded domain in C" and let U be the unit disk in C. H(D : U) designates the family of holomorphic functions from D into U. For fixed £ in D we write H{ (D:U)= { / G H(D : U) : /(£) = 0}. For each { € D , CD(£; ) is the function defined on the complex tangent space of D at £ by CD(C; v) = sup{|(df(a v)\:fGH(D: U)} (cf. Reiffen [3] ). Evidently, CD(C;v) = sup{|<d/(£),t>)| :fGH((D : I/)}. CD is called the Carathéodory metric for D. Since H(D : U) is a normal family, the supremum in the definition of CD(£; v) is attained by some F G Ht (D : C7). Here F(z) = F(z; £, v). By a normal family argument CD(£ : t>) is continuous in (£, Ü). For the above mentioned properties see [3]. Moreover, CD(£; v) is a locally Lipschitz function [!]• Received by the editors on July 19, 1976, and in revised form on November 8, 1976. AMS (MOS) 1970 Subject Classifications. 32H15, 32A30.