Overview
This research addresses the topological rigidity of contractible 3-manifolds and the interiors of compact handlebodies under specific geometric conditions. The study establishes that a contractible 3-manifold admitting a complete Riemannian metric with nonnegative scalar curvature is diffeomorphic to $\R^3$. Additionally, it demonstrates that if the interior $M_\gamma$ of a compact handlebody of genus $\gamma$ admits such a metric, then the genus $\gamma$ must be less than or equal to 1.
Research Context
The work directly responds to two open questions within the field of geometry and topology. One question was posed by Wang, concerning contractible 3-manifolds. The other, related to handlebody interiors, was raised by Gromov.
The scalar curvature is a fundamental geometric invariant that characterizes the curvature of a Riemannian manifold. Nonnegative scalar curvature imposes constraints on the topology of manifolds. The concept of a complete Riemannian metric ensures that geodesics can be extended indefinitely within the manifold. Diffeomorphism describes a smooth invertible mapping between two smooth manifolds, implying they are topologically and geometrically equivalent from a smooth perspective.
A contractible 3-manifold is a 3-dimensional manifold that can be continuously deformed to a single point. $\R^3$ represents standard Euclidean 3-space. A handlebody of genus $\gamma$ is a 3-dimensional manifold obtained by attaching $\gamma$ handles to a 3-ball. Its interior, $M_\gamma$, is of particular interest in this context.
Findings
- A contractible 3-manifold that possesses a complete Riemannian metric of nonnegative scalar curvature is diffeomorphic to $\R^3$.
- If the interior $M_\gamma$ of a compact handlebody of genus $\gamma$ admits a complete Riemannian metric of nonnegative scalar curvature, then the genus $\gamma$ must satisfy the condition $\gamma \leq 1$.