The isoperimetric problem in Randers Poincaré disc

Abstract

It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this paper, we show that the circles centered at origin in the Randers Poincaré disc satisfy the isoperimetric equality with respect to different volume forms however, these Randers metrics do not necessarily have constant (negative) flag curvature. In particular, we show that Osserman's result [12] of the Riemannian case cannot be extended to the Finsler geometry as such. © 2024 World Scientific Publishing Company.