Bounded domains in the 3-dimensional space

TL;DR

This work develops a Morse-theoretic framework for bounded domains in ${\boldsymbol{R}}^3$ by pairing height projections $F$ with (weighted) Reeb graphs to capture domain topology. It proves that if the weighted Reeb graph weights satisfy $<2$, then the domain is a handlebody (a regular neighborhood of a spatial graph), and analyzes obstructions arising at higher weights, including knotted exteriors. The analysis of knot exteriors via Reeb graphs clarifies when a domain is a tubular neighborhood or a knot exterior and shows how weight-2 configurations can be eliminated by Morse perturbations. By linking visibility to these graph invariants through a minNCP hypothesis, the paper connects isotopy, Morse data, and 3-manifold topology to classify when a bounded domain is visible as a simple handlebody versus a more intricate knotted object.

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 $ι(M)$ so that every point of $\partial ι(M)$ is visible from the infinity?" Under the minNCP hypothesis, we show that if a bounded domain $M$ can be isotoped to $ι(M)$ 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 $ι(M)$, $ι(M)$ can be taken so that $z:ι(M)\to \boldsymbol{R}$ is a Morse height function with minimum number of critical points in the isotopy class of the embedding $M\subset \boldsymbol{R}^3$.

Figures (20)

• Figure 1: The knot exteriors of the trefoil knot (left) and the figure eight knot (right). The left figure has been used in the cover of Sugaku Tsushin published by Mathematical Society of Japan.
• Figure 2: The simple closed curve $c$ on $S$ bounds a compressing disk $D^2$ in $A$. We have a neighborhood $D^2\times [-1,1]\subset A$. $S'=(S\smallsetminus (\partial D^2\times (-1,1))) \cup (D^2\times \{-1,1\})$.
• Figure 3: Non degenerate critical points. The indice are 0 (upper left), 2 (upper right), 1 (lower center), respectively.
• Figure 4: Examples of Morse height functions on the 2-torus
• Figure 5: Two Morse height functions on a solid torus, their Reeb graphs $\mathcal{R}_{F|\partial M}$ ($\mathcal{R}_{F|\partial M"}$), $\mathcal{R}_F$ and the inverse image $f^{-1}(z)$.

Theorems & Definitions (50)

• Definition 1.1: Bounded domains
• Theorem 2.1: Schöflies problem, Alexander theorem (Alexander)
• Remark 2.2
• Theorem 2.3: Solid torus theorem
• Theorem 2.4: The loop theorem
• Theorem 2.5: Dehn's lemma for a boundary simple closed curve
• Proposition 2.6
• Definition 3.1: Handlebodies
• Definition 4.1
• Proposition 4.2