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

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Abstract

We endow the disc \(D=\{(x_1,x_2)\in {\mathbb {R}}^2: x_1^2+x_2^2<4\}\) with a Poincaré-type Randers metric \(F_\lambda \), \(\lambda \in [0,1]\) that ’linearly’ interpolates between the usual Riemannian Poincaré disc model (\(\lambda =0\), having constant sectional curvature \(-1\) and zero S-curvature) and the Finsler–Poincaré 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.

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Acknowledgements

Research of A. Kristály is supported by the National Research, Development and Innovation Fund of Hungary, financed under the K\(\_\)18 funding scheme, Project No. 127926.

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Correspondence to Alexandru Kristály.

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Kajántó, S., Kristály, A. Unexpected Behaviour of Flag and S-Curvatures on the Interpolated Poincaré Metric. J Geom Anal 31, 10246–10262 (2021). https://doi.org/10.1007/s12220-021-00644-x

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  • DOI: https://doi.org/10.1007/s12220-021-00644-x

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