On the Squeezing Function and Fridman Invariants

Abstract

For a domain D ⊂ C n , the relationship between the squeezing function and the Fridman invariants is clarified. Furthermore, localization properties of these functions are obtained. As applications, some known results concerning their boundary behavior are extended.

On the generalized squeezing functions and Fridman invariants of special domains

“…Recently, Nikolov-Verma [20] have proved that s D (z) ≤ ẽD (z) ≤ e D (z). This still holds for the generalized squeezing function (cf.…”

“…There have also been some studies on ẽD and e D (see e.g. [6,10,18,20]). For more details on various recent results, we refer the readers to the survey papers [7,25].…”

On Fridman invariants and generalized squeezing functions

“…Since Fridman invariants and generalized squeezing functions are similar in spirit to the Kobayashi-Eisenman volume form K D and the Carathéodory volume form C D , respectively, it is natural to study the comparison of them. For this purpose, we will always assume that D is a bounded domain in C n and Ω is a bounded, balanced, convex and homogeneous domain in In [9], Nikolov and Verma have shown that m D (z) is always less than or equal to one. The next result shows that the same is true for m Ω D (z).…”

“…In section 3, we study basic properties of generalized squeezing functions, in particular extending various properties of the squeezing function given in [1,2] to the more general setting. In section 4, we study the comparison of Fridman invariants and generalized squeezing functions, in particular generalizing previous results from [9,10].…”

Fridman’s Invariant, Squeezing Functions, and Exhausting Domains

“…By the decreasing property of the Kobayashi metric, it is obvious that s D ≤ e D for all bounded domains (see [28]). So as a corollary of Theorem 1.1, we get the main result of Diederich-Fornaess-Wold in [10] which says that, for a bounded strictly pseudocovnex domain D with C 2 boundary which is not biholomorphic to the unit ball, s D (z) ≤ Cδ(z) for some constant C.…”