Bounded domains in the 3-dimensional space

TL;DR

This paper explores the shapes of bounded domains in 3D space using Morse functions and Reeb graphs, establishing conditions under which these domains are isotopic to handlebodies and analyzing their boundary visibility properties.

Contribution

It introduces weighted Reeb graphs to study domain shapes and proves that low-weight conditions imply the domain is an embedded handlebody, linking shape analysis with isotopy and visibility.

Findings

Bounded domains with small weighted Reeb graphs are isotopic to handlebodies.

Under the minNCP hypothesis, domains with boundary visible from infinity are handlebodies.

The study connects Morse theory, Reeb graphs, and geometric visibility in 3D topology.

Abstract

We study the shapes of compact connected 3-manifolds with connected smooth boundary in the 3-dimensional Euclidean space $\boldsymbol{R}^3$. We call them bounded domains. Since compact connected surfaces in $\boldsymbol{R}^3$ bound unique bounded domains, the objects are the same as compact connected surfaces in $\boldsymbol{R}^3$. To understand their shapes, we use the Morse height functions $F: M\to \boldsymbol{R}$ which are the orthogonal projections from the bounded domains $M$ to lines, and their Reeb graphs $\mathcal{R}_F$ and $\mathcal{R}_{F|\partial M}$ which are obtained by identifying connected components of level sets of maps to points. We introduce the weighted Reeb graphs $\mathcal{R}_F^w$ and the weighted indexed Reeb graphs $\mathcal{R}_F^{wi}$. We investigate whether a bounded domain admits a Morse height function $F$ with the weighted Reeb graphs $\mathcal{R}_F^w$ with small weight. We show that if the weights are less than 2. $M$ can be deformed by isotopy to an embedded handlebody. The original question which lead us to investigate bounded domains is the following question: "Can the domain $M$ be isotoped so that, for every point of the boundary $\partial M$, there is a ray from the point which intersects the domain $M$ only at the end point?" In other words, "Can $M$ be isotoped to $\iota(M)$ so that every point of $\partial \iota (M)$ is visible from the infinity?" Under the minNCP hypothesis, we show that if a bounded domain $M$ can be isotoped to $\iota(M)$ so that every point of the boundary is visible from the infinity, then $M$ is an embedded handlebody. Here the minNCP hypothesis asserts that, if $M$ is isotopic to a visible $\iota(M)$, $\iota(M)$ can be taken so that $z:\iota(M)\to \boldsymbol{R}$ is a Morse height function with minimum number of critical points in the isotopy class of the embedding $M\subset \boldsymbol{R}^3$.