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Unexpected Behaviour of Flag and S -Curvatures on the Interpolated Poincaré Metric
The Journal of Geometric Analysis ( IF 1.1 ) Pub Date : 2021-03-17 , DOI: 10.1007/s12220-021-00644-x
Sándor Kajántó , Alexandru Kristály

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.



中文翻译:

插值Poincaré度量上标志和S曲线的意外行为

我们在{\ mathbb {R}} ^ 2:x_1 ^ 2 + x_2 ^ 2 <4 \} \中为光盘\(D = \ {(x_1,x_2)\)赋予Poincaré型Randers度量\(F_ \ lambda \)\(\ lambda \ in [0,1] \)在通常的黎曼Poincaré圆盘模型(\(\ lambda = 0 \)之间,具有恒定的截面曲率\(-1 \)和零S曲率)和Finsler-Poincaré度量(\(\ lambda = 1 \),具有恒定的标志曲率\(-1/4 \)和恒定的S-曲率,各向同性系数1/2)。与我们的直觉相反,我们证明当\(\ lambda \ nearrow 1 \)时,标志和标准化S-curvatures度量的\(F_ \拉姆达\) 炸毁接近\(\局部d \)的旗杆的一些特定的选择。

更新日期:2021-03-18
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