AI 中文总结
本文将最优传输插值不等式从次黎曼情形扩展到次芬斯勒情形，借助次芬斯勒雅可比场等建立理论，刻画广义畸变系数，推导几何不等式，并证明兰德斯次芬斯勒海森堡群满足测度收缩性质。
AI 中文摘要
在本文中，我们证明了前向理想次芬斯勒流形支持最优传输的插值不等式，将巴里拉里和里齐（arXiv:1705.05380）的结果从次黎曼情形扩展到次芬斯勒情形。次芬斯勒雅可比场的引入以及次芬斯勒流形上最优传输理论的建立起到了关键作用。通过将此传输框架与次芬斯勒雅可比估计相结合，我们刻画了广义畸变系数。作为应用，我们推导了几个基本的几何不等式，包括布伦 - 闵可夫斯基不等式和博雷尔 - 布拉斯坎普 - 利布不等式。最后，对于兰德斯次芬斯勒海森堡群的情形，其度量由受漂移项扰动的次黎曼度量定义，我们明确表明它满足测度收缩性质。
英文摘要
In this paper, we prove that forward ideal sub-Finslerian manifolds support interpolation inequalities for optimal transport, extending the results of Barilari and Rizzi, arXiv:1705.05380, from the sub-Riemannian to the sub-Finslerian setting. A key role is played by the introduction of sub-Finslerian Jacobi fields and the establishment of optimal transport theory on sub-Finslerian manifolds. By combining this transport framework with sub-Finslerian Jacobian estimates, we characterize the generalized distortion coefficients. As an application, we deduce several fundamental geometric inequalities, including the Brunn-Minkowski and Borell-Brascamp-Lieb inequalities. Finally, for the case of the Randers sub-Finslerian Heisenberg group, whose metric is defined by a sub-Riemannian metric perturbed by a drift term, we explicitly show that it satisfies the measure contraction property.
