Arrangements of small circles for Morse-Bott functions

TL;DR

This work investigates arrangements of small circles in the plane, termed MB circle-centered arrangements (MBCC), and their connection to Morse-Bott theory. It shows that regions bounded by such arrangements arise as images of real algebraic maps and that composing these images with natural projections yields Morse-Bott functions, linking circle geometry to singularity theory. The core contribution is a systematic, local-operation framework that characterizes how the Poincaré-Reeb graph $W_{D_{\mathcal{S}},i}$ changes when a new small circle is added, encapsulated in a sequence of explicit theorems (Theorems 1–5). This provides a constructive bridge between circle arrangements and real algebraic maps, enabling explicit design of Reeb-type graphs and Morse-Bott structures on manifolds, with potential applications to reconstructing algebraic maps from prescribed graphs and to broader geometric modeling.

Abstract

As a topic of mathematics, "arrangements", systems of hyperplanes, circles, and general (regular) submanifolds, attract us strongly. We present a natural elementary study of arrangements of circles. It is also a kind of new studies. Our study is closely related to geometry and singularity theory of Morse(-Bott) functions. Regions surrounded by circles are regarded as images of real algebraic maps and composing them with projections gives Morse-Bott functions: this observation is natural, and surprisingly, recently presented first, by the author. We present a systematic way of constructing such arrangements by choosing small circles centered at existing circles inductively. We are interested in graphs the regions surrounded by the circles naturally collapse. We have studied local changes of the graphs in adding these circles. These graphs are essentially so-called {\it Reeb graphs} of the previous Morse-Bott functions: they are spaces of all components of preimages of single points for the functions.

Figures (9)

• Figure 1: The case of Example \ref{['ex:1']}. Blue dots show vertical poles and red dots show horizontal poles. Graphs are Poinaré-Reeb graphs of $D_{\mathcal{S}}$.
• Figure 2: Local observations for $p=x_{j^{\prime}} \in S_{x_{j^{\prime}},r_{j^{\prime}}}$. Blue colored lines show vertical lines and horizontal lines. Dots are for vertical poles, horizontal poles, and the centers of circles. Dotted lines are for straight lines tangent to circles at points contained in exactly two circles $S_{x_{j},r_{j}} \in \mathcal{S}$ and $S_{x_{j^{\prime}},r_{j^{\prime}}} \in \mathcal{S^{\prime}}$. Furthermore, each of the first four is for a case where $p$ is not a vertical pole or a horizontal pole. In addition, the remaining four is for a case where $p$ is a vertical pole or a horizontal pole: we omit horizontal lines and vertical lines here.
• Figure 3: A case for the case (\ref{['thm:2.1.2']}). The blue circle shows the circle $S_{x_{j^{\prime}},r_{j^{\prime}}}$ and the change of our graph is depicted in blue. Hereafter, we use colored objects in this way. Vertices and edges are shown by the notation with dotted segments.
• Figure 4: The case (\ref{['thm:2.2.1']}): vertices and edges are shown by the notation.
• Figure 5: The case (\ref{['thm:2.2.2']}): vertices and edges are shown by the notation.

Theorems & Definitions (10)

• Proposition 1
• proof
• Proposition 2
• Theorem 1
• Theorem 2
• Theorem 3
• proof : A proof of Theorems \ref{['thm:1']}--\ref{['thm:3']}
• Theorem 4
• proof
• Theorem 5