Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces

In this paper we study curves with constant geodesic curvature in rotationally symmetric complete surfaces. Under monotonicity conditions on the Gauss curvature we classify the closed embedded ones in planes, cylinders, spheres and projective planes. We also distinguish the stable ones, i.e., the second order minima of perimeter while keeping constant the area enclosed. We prove existence and nonexistence of isoperimetric domains, and we show the isoperimetric domains when they exist.