Analysis Seminar
Thursday, 09/03/2023, 11.30 am
Venue: Ramanujan Hall
Host: Prachi Mahajan
Speaker: Ratna Pal
Affiliation: IISER Mohali
Title: Rigidity properties of Henon maps in $\mathbb{C}^2$ and Short $\mathbb{C}^2$.
Abstract: The broad research area of my talk is Complex Dynamics in Several Variables. Classically complex dynamics was studied for rational endomorphisms of the Riemann sphere. In the past three decades, this field of research has flourished to a great extent and the holomorphic dynamics in higher dimensions has attracted a lot of attention. In particular, the dynamics of the polynomial automorphisms in higher dimensions mushroomed as one of the central themes of study. In $\mathbb{C}^2$, the most important polynomial automorphisms are the Henon maps and in this talk they will play the role of the protagonist. In the first part of the talk, we shall see a couple of rigidity properties of Henon maps. Loosely speaking, by rigidity properties we mean those properties of Henon maps which determine the underlying Henon maps almost uniquely. In the latter part of the talk, we shall survey a few recent results obtained for Short $\mathbb{C}^2$'s. A Short $\mathbb{C}^2$ is a proper domain of $\mathbb{C}^2$ that can be expressed as an increasing union of unit balls (up to biholomorphism) such that the Kobayashi metric vanishes identically, but allows a bounded above pluri-subharmonic function. The sub-level sets of the Green's functions of Henon maps are classical examples of Short $\mathbb{C}^2$'s. Note that the Green's function of a Henon map $H$ is the global pluri-subharmonic functions on $\mathbb{C}^2$ which is obtained by measuring the normalized logarithmic growth rate of the orbits of points in $\mathbb{C}^2$ under the iterations of the Henon map $H$. In this part of the talk, we shall first see a few interesting natural properties of Short $\mathbb{C}^2$'s. Then we give an effective description of the automorphism groups of the sublevel sets of Green's functions of Henon maps (recall that the sublevel sets of the Green's functions of Henon maps are Short $\mathbb{C}^2$'s). It turns out that the automorphism groups of this class of Short $\mathbb{C}^2$'s are not very large. Thus it shows that, unlike in a bounded set-up, although the Euclidean balls have large automorphism groups, the automorphism group of an increasing union of balls (up to biholomorphism) might flatten out when the final union is unbounded. A part of the results which will be presented in this talk is obtained in several joint works with Sayani Bera, John Erik Fornaess and Kaushal Verma.