On holomorphic isometric embeddings of the unit disk into polydisks

On Holomorphic Isometric Embeddings of the Unit Disk into Polydisks

We study the classification of holomorphic isometric embeddings of the unit disk into polydisks. As a corollary of our results, we can give a complete classification when the target is the 2-disk and the 3-disk. We also prove that the holomorphic isometric embeddings between polydisks are induced by those of the unit disk into polydisks.

Embeddings of polydisks

Symplectic embeddings of polydisks

If P is a polydisk with radii R1 ≤ ... ≤ Rn and P ′ is a polydisk with radii R 1 ≤ ... ≤ R n , then we prove that P symplectically embeds in P ′ provided that C(n)R1 ≤ R′1 and C(n)R1...Rn ≤ R ′ 1 ...R n . Up to a constant factor, these conditions are optimal.

Geometric Quasi - Isometric Embeddings Into

We show that F has an infinite family of quasi-isometrically embedded subgroups of the form F m × Zn, for integral m, n ≥ 0. These subgroups have simple geometric but more complicated algebraic descriptions We present them to illustrate the intricate geometry of Thompson’s group F as well as the interplay between its standard finite and infinite presentations.

Ribbon R-trees and holomorphic dynamics on the unit disk