Gluing $CAT(0)$ domains

TL;DR

The paper investigates when gluing two locally $CAT(0)$ domains along boundary pieces preserves local nonpositive curvature, first in the plane and then on general simply connected Riemannian surfaces with curvature $k\le 0$. It introduces a boundary signed-curvature balance condition: the sum of the boundary curvatures at gluing points must be nonpositive, while one side is nonpositive and the other nonnegative; these conditions ensure the glued space is locally $CAT(0)$. A key technical ingredient is a limit-stability result: limits of locally $CAT(0)$ spaces under natural convergence are themselves $CAT(0)$, enabling a polygonal-approximation approach to transfer curvature bounds through the gluing. The authors extend the Euclidean results to Riemann surfaces using isothermal coordinates and show the same curvature-balance criteria yield locally $CAT(0)$ glued domains, with a discussion of equality points and a counterexample illustrating the necessity of the balance condition.

Abstract

In this work we describe a class of subsets of the Euclidean plane which, with the induced length metric, are locally $CAT(0)$ spaces and we show that the gluing of two such subsets along a piece of their boundary is again a locally $CAT(0)$ space provided that the sum of the signed curvatures at every gluing point is non-positive. A generalization to subsets of smooth Riemannian surfaces of curvature $k\leq 0$ is given.

Figures (9)

• Figure 1: The comparison triangles $\overline{T}$ and $\overline{T_{k}}$ for the geodesic triangles $T\subset A$ and $T_{k} \subset A_{k}$ respectively.
• Figure 2: The geodesic triangle $\bigtriangleup_{k}$ in $A_{k}$ with vertices $x,y,a$ and its comparison triangle $\overline{\bigtriangleup}_{k}$.
• Figure 3: The tangent lines at $z,w$ and the angles involved in the proof of the Claim in Proposition \ref{['ugl']}.
• Figure 4: The case, in the proof of uniqueness in Proposition \ref{['ugl']}, where $x,y\in\Sigma_A$ and the assumed geodesics both intersect $\Sigma_B \setminus \partial \Sigma_B .$
• Figure 5: The case, in the proof of uniqueness in Proposition \ref{['ugl']}, where $x,y\in \Sigma_A$ and only one of the assumed geodesics intersects $\Sigma_B \setminus \partial \Sigma_B .$

Theorems & Definitions (28)

• Proposition 1
• Theorem 2
• Remark 3
• Proposition 4
• proof
• Remark 5
• Proposition 6
• Lemma 7
• proof : Proof of Lemma \ref{['piangle']}
• proof : Proof of Proposition \ref{['ugl']}