An analogue of the squeezing function for projective maps

TL;DR

This paper introduces a projective analogue of the complex squeezing function, demonstrating its boundedness on convex domains and its boundary behavior on strongly convex domains, with applications to projective geometry theorems.

Contribution

It defines a new projective invariant inspired by the complex squeezing function and proves key properties for convex and strongly convex domains.

Findings

The projective squeezing function is uniformly bounded below on convex domains.

It tends to 1 at the boundary of strongly convex domains.

A new proof of a projective analogue of the Wong-Rosay theorem is provided.

Abstract

In the spirit of Kobayashi's applications of methods of invariant metrics to questions of projective geometry, we introduce a projective analogue of the complex squeezing function. Using Frankel's work, we prove that for convex domains it stays uniformly bounded from below. In the case of strongly convex domains, we show that it tends to 1 at the boundary. This is applied to get a new proof of a projective analogue of the Wong-Rosay theorem.