论文标题
鳍磁盘的最小区域，最小化的大地测量学
Minimal area of Finsler disks with minimizing geodesics
论文作者
论文摘要
我们表明，福尔摩斯（Holmes） - 半径$ r $的每个Finsler磁盘的Thompson区域，其内部大地测量学是长度最小化的至少为$ \ frac {6}πr^2 $。此外，我们构建了示例，表明不平等是尖锐的，并且观察到相等案例是通过非旋转对称度量获得的。这与伯格在里曼尼亚案中的猜想形成了鲜明对比，后者断言圆形半球是极端的。为了证明我们的定理，我们使用随机的大地测量学离散鳍度量标准。作为辅助结果，我们表明，Blaschke和Santaló的积分几何公式持有几乎没有被困的大地测量学的Finsler歧管。
We show that the Holmes--Thompson area of every Finsler disk of radius $r$ whose interior geodesics are length-minimizing is at least $\frac{6}π r^2$. Furthermore, we construct examples showing that the inequality is sharp and observe that the equality case is attained by a non-rotationally symmetric metric. This contrasts with Berger's conjecture in the Riemannian case, which asserts that the round hemisphere is extremal. To prove our theorem we discretize the Finsler metric using random geodesics. As an auxiliary result, we show that the integral geometry formulas of Blaschke and Santaló hold on Finsler manifolds with almost no trapped geodesics.
