A comparison of two biholomorphic invariants

Abstract

The Fridman invariant, which is a biholomorphic invariant on Kobayashi hyperbolic manifolds, can be seen as the dual of the much studied squeezing function. We compare this pair of invariants by showing that they are both equally capable of determining the boundary geometry of a bounded domain if their boundary behaviour is apriori known.

A note on the boundary behaviour of the squeezing function and Fridman invariant

“…The study of biholomorphic invariants has been attracted much attention in the complex differential geometry to enhance the comprehension and application of biholomorphic classification of complex domains. The squeezing function, the Fridman invariant, and the quotient invariant by using the Carathéodory and Kobayashi-Eisenman volume elements, have received increasing interest as biholomorphic invariants in recent years (see [BK19], [MV19], [NV18], [NN19] and the references therein). We particularly consider both the squeezing function and the Fridman invariant associated to a certain class of pseudoconvex domains in C n in this paper.…”

“…Suppose that Ω is pseudoconvex of D'Angelo finite type near ξ 0 and lim j→∞ s Ω (η j ) = 1 or lim j→∞ h Ω (η j ) = 0. In [JK18] and [MV19], they proved that if the sequence {η j } ⊂ Ω converges to ξ 0 along the inner normal line to ∂Ω at ξ 0 , then ξ 0 must be strongly pseudoconvex (for details, see [JK18] for n = 2 and [MV19] for general case). Moreover, this result was obtained in [Ni18] for the case that {η j } ⊂ Ω converges nontangentially to ξ 0 and in [NN19] for the case that {η j } ⊂ Ω converges 1 m1 , .…”

On the generalized squeezing functions and Fridman invariants of special domains

“…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].…”

Further study of Squeezing function Corresponding to Polydisk

“…For other important properties of squeezing functions corresponding to unit ball one can refer the following papers: [12], [11], [20], [26], [18], [19]. Recently [9] N. Gupta, S. K. Pant introduced squeezing function corresponding to polydisk.…”