Let \(S\) be a surface of finite type. A flat metric \(\phi\) on a surface \(S\) is a non-positively curved Euclidean cone metric whose completion is also a Euclidean cone metric. The holonomy of a flat metric \(\phi\) on \(S\) is the image of the holonomy homomorphism \(\pi_1(S^{\circ}) \rightarrow SO(2)\), where \(S^{\circ}\) is the surface \(S\) with the cone points of \(\phi\) removed. When the order of the holonomy is a finite number \(q\), such a metric is called a \(q\)-flat metric. When a \(q\)-flat metric has no cone points it is said to be fully punctured. A cylinder for a flat metric \(\phi\) is an isometrically immersed open Euclidean cylinder. When this cylinder is embedded, the simple closed curve of the core of the cylinder is called an embedded cylinder curve. This paper studies the set of all embedded cylinder curves, EC\((\phi)\), considered as a subset of the curve complex of \(S\), \(C(S)\), where \(\phi\) is a \(q\)-flat metric on \(S\) with \(2 < q < \infty\).
1. First, the authors give an example of a 4-flat metric on a genus-2 surface which has exactly three embedded cylinder curves. This example generalizes to give a \(2g\)-flat metric on a genus \(g\) surface for which the EC\((\phi)\) contains exactly \(g+1\) curves.
2. Next, the authors use a certain building block surface, which when locally isometrically immersed in a surface \((S, \phi)\) gives cylinder free regions. This is used to produce many examples for which EC\((\phi)\) is empty.
3. Given a pair of filling curves \(\alpha\) and \(\beta\) on a genus-\(g\) surface \(S\), satisfying a few conditions, one can obtain semi-translational structures on \(S^{\circ}\). By deforming \(\beta\) in a certain way, the authors produce fully punctured \(q\)-flat metrics with \(q > 2\) on \(S^{\circ}\) containing the embedded cylinder curves \(\alpha\) and the deformed \(\beta\) which are at arbitrary large distance in \(C(S^{\circ})\).
4. Next, the authors prove that given a point in the Gromov boundary of the curve complex of a surface, there exists a 6-flat metric for which the EC\((\phi)\) accumulates onto that point which then says that EC\((\phi)\) for such a \(\phi\) has infinite diameter.
5. However, in contrast with the above, the authors prove the main theorem of the paper that if \(\phi\) is a fully punctured flat metric on \(S\) with finite order holonomy then EC\((\phi)\) has finite diameter in the curve complex \(C(S)\).
6. The proof of this main theorem is also used to show that if a flat metric \(\phi\) on a surface \(S\) with holonomy of finte order at least 1 is such that EC\((\phi)\) has infinte diameter, then there is a semi-translation structure \(\hat{\phi}\) such that \(\phi\) is obtained from \(\hat{\phi}\) by certain disk-replacements. A criterion for when EC\((\phi)\) accumulates onto the Gromov boundary of \(C(S)\) is also given.