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Tropical limit of hyperbolic amoebas of complex analytic surfaces

Peter Petrov, Mikhail Shkolnikov

TL;DR

The article studies tropical-type limits of hyperbolic amoebas arising from closed analytic surfaces in $\mathrm{SL}_2(\mathbb{C})$ under a contracting scaling in the hyperbolic quotient $\mathbb{H}^3$. It proves that, for a sequence of scales $s_n\to\infty$, the rescaled images $R_{s_n}\circ\varkappa(V_n)$ either escape to infinity or have Hausdorff-convergent subsequences whose limits are complements of open balls centered at $O$ in $\mathbb{H}^3$. The authors provide a complete, rigorous proof built around a technical lemma on horospheres and leverage two models of hyperbolic geometry to control intersections, establishing a robust link between complex-analytic geometry and hyperbolic tropical limits. The work extends tropical-limit phenomena to analytic surfaces (not only algebraic ones), demonstrates that every ball-complement arises as a tropical limit, and discusses potential generalizations to other reductive groups and phase-structure descriptions of limit configurations.

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.

Tropical limit of hyperbolic amoebas of complex analytic surfaces

TL;DR

The article studies tropical-type limits of hyperbolic amoebas arising from closed analytic surfaces in under a contracting scaling in the hyperbolic quotient . It proves that, for a sequence of scales , the rescaled images either escape to infinity or have Hausdorff-convergent subsequences whose limits are complements of open balls centered at in . The authors provide a complete, rigorous proof built around a technical lemma on horospheres and leverage two models of hyperbolic geometry to control intersections, establishing a robust link between complex-analytic geometry and hyperbolic tropical limits. The work extends tropical-limit phenomena to analytic surfaces (not only algebraic ones), demonstrates that every ball-complement arises as a tropical limit, and discusses potential generalizations to other reductive groups and phase-structure descriptions of limit configurations.

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 under the quotient by its maximal compact subgroup subject to a contracting scaling sequence.

Paper Structure

This paper contains 5 sections, 2 theorems, 2 equations, 2 figures.

Table of Contents

  1. The result
  2. The context
  3. A plausible argument
  4. An actual proof
  5. Remarks

Key Result

Theorem 1

Let $\{s_n\}_{n=1}^{\infty}$ be a sequence of positive real numbers such that $s_n\rightarrow+\infty,$ and let $\{V_n\}_{n=1}^{\infty}$ be a sequence of closed analytic complex surfaces inside $\operatorname{SL}_2(\mathbb{C})$. Then, either the sequence $R_{s_n}\circ\varkappa(V_n)\subset\mathbb{H}^3

Figures (2)

  • Figure 1: An illustration of the Lemma. A part of rescaled horoshpere together with a part of the boundary of the upper ball to surround the ball in the middle.
  • Figure 2: An illustration of the trick used in the proof of Theorem shown in the upper-half space model of $\mathbb{H}^3$ after an inverse of the rescaling $R_{s_{n_K}}$. The region in the middle is where there are some points of a hyperbolic amoeba $\varkappa(V_{n_K}),$ one is marked by X, and the horse-shoe shape around is the region where there are none. Parallel lines represent horospheres moving upwards, and the winding path (stating at X and ending at ?!) corresponds to the hypothetical image of disappearing intersection.

Theorems & Definitions (4)

  • Theorem
  • Lemma
  • proof : Proof of the Theorem
  • proof : Proof of the Lemma