Unexpected Behaviour of Flag and S -Curvatures on the Interpolated Poincaré Metric

We endow the disc $$D=\{(x_1,x_2)\in {\mathbb {R}}^2: x_1^2+x_2^2<4\}$$ with a Poincare-type Randers metric $$F_\lambda $$ , $$\lambda \in [0,1]$$ that ’linearly’ interpolates between the usual Riemannian Poincare disc model ( $$\lambda =0$$ , having constant sectional curvature $$-1$$ and zero S-curvature) and the Finsler–Poincare metric ( $$\lambda =1$$ , having constant flag curvature $$-1/4$$ and constant S-curvature with isotropic factor 1/2), respectively. Contrary to our intuition, we show that when $$\lambda \nearrow 1$$ , both the flag and normalized S-curvatures of the metric $$F_\lambda $$ blow up close to $$\partial D$$ for some particular choices of the flagpoles.