Tropical limit of hyperbolic amoebas of complex analytic surfaces

TL;DR

This paper investigates the behavior of complex analytic surfaces in the special linear group under a scaling limit, revealing convergence properties of their images in the associated symmetric space.

Contribution

It establishes a general convergence result for families of analytic surfaces in SL_2(C) under scaling, connecting complex geometry with hyperbolic amoebas.

Findings

Convergence of surface images in the hyperbolic space under scaling.

New insights into the tropical limit of hyperbolic amoebas.

Bridging complex analytic surfaces with hyperbolic geometry.

Abstract

In this letter, we establish a general fact about the convergence of images of families of closed analytic surfaces in the special linear group $\operatorname{SL}_2(\mathbb{C})$ under the quotient by its maximal compact subgroup $\operatorname{SU}(2)$ subject to a contracting scaling sequence.