[Paper Review] The intrinsic geometry of a Jordan domain

Richard L. Bishop|ArXiv.org|Dec 28, 2005

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper establishes that the set of finite-distance points in a Jordan domain equipped with the intrinsic length metric forms a CAT(0) space and is Gromov hyperbolic. It further shows that the space with its boundary at infinity, under the cone topology, is topologically equivalent to the original Jordan domain, linking intrinsic geometric structure to topological type via metric compactification.

ABSTRACT

For a Jordan domain in the plane the length metric space of points connected to an interior point by a curve of finite length is a CAT(0)space and Gromov hyperbolic. With respect to the cone topology, that space plus its boundary at infinity is topologically the same as the original Jordan domain.

Motivation & Objective

• To characterize the intrinsic metric geometry of a Jordan domain as a complete length space.
• To prove that the space of finite-distance points in a Jordan domain is CAT(0).
• To establish that this space is also Gromov hyperbolic.
• To show that the compactification of the space via its boundary at infinity, under the cone topology, is homeomorphic to the original Jordan domain.

Proposed method

• Define the space $X$ as the set of points in a Jordan domain $J$ that are at finite intrinsic distance from an interior point.
• Use local convexity and supporting half-disks to analyze geodesic behavior at boundary points of $X$, particularly on $\partial X$.
• Apply Alexandrov’s criterion: if all geodesic triangles in a simply connected space have angle sum $\leq 2\pi$, the space is CAT(0).
• Show that the geodesic triangle $\bar{p}\bar{q}\bar{r}$ formed by bifurcation points has total curvature $\leq 0$, implying angle sum $\leq 2\pi$, satisfying Alexandrov’s condition.
• Use the cone topology on $X \cup \partial X$ to identify the boundary at infinity with $\partial J$, and prove topological equivalence to $J$ via neighborhood constructions.
• Leverage the convergence of geodesics and convexity of distance functions to relate Euclidean and intrinsic metric convergence.

Experimental results

Research questions

• RQ1 Is the intrinsic length metric space of a Jordan domain CAT(0)?
• RQ2 Is this space Gromov hyperbolic?
• RQ3 Does the compactification of the finite-distance space via its boundary at infinity, under the cone topology, yield a space homeomorphic to the original Jordan domain?
• RQ4 How do geodesics behave asymptotically, and how does their convergence relate to the topology of the boundary?

Key findings

• The space $X$ of finite-distance points in a Jordan domain is a complete CAT(0) space.
• The space $X$ is also Gromov hyperbolic, as a consequence of its CAT(0) structure.
• The boundary at infinity of $X$, equipped with the cone topology, is homeomorphic to the original Jordan domain’s boundary $\partial J$, and the entire compactification $X \cup \partial X$ is homeomorphic to $J$.
• Geodesics in $X$ are locally convex and supported by half-disks at every interior point, with curvature constraints ensuring thin triangles.
• The topology induced by the cone topology on $X \cup \partial X$ coincides with the standard topology on $J$, establishing a topological identification.
• Convergence of geodesics in the intrinsic metric is equivalent to convergence of their endpoints in the Euclidean topology, due to convexity of distance functions and metric majorization.