等距叶上的测度收缩性质与单调纤维
Measure contraction property on isometric leaves and monotone fibres
- Beijing Institute of Mathematical Sciences and Applications（北京数学科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明非扩张映射的等距叶上测度收缩性质不变，并推广到单调纤维，同时通过反例否证Klartag猜想，揭示余维数一情形下的继承条件。
AI中文摘要：
对于凸欧几里得支撑上具有正密度的有限测度，我们证明$MCP(\kappa,N)$性质以不变的参数传递给任意非扩张映射的几乎每个等距叶。证明基于几何条件密度的尖锐收缩不等式，其指数等于叶的余维数。继承的维数参数是最优的。关于预解图的全变差极限将结果推广到极大单调关系的逆纤维，包括凸梯度纤维。我们还通过三维中的强非扩张例子和四维中的梯度例子否证了Klartag的曲率维数继承猜想。在余维数为一的情形下，几何密度的仿射性导出曲率维数继承。第一个例子也给出了单调纤维上的失败情形。两种构造都允许任意大的曲率损失，包括对于正商测度叶族上的固定高斯环境测度。
英文摘要：
For finite measures with positive densities on convex Euclidean supports, we prove that $MCP(κ,N)$ passes with unchanged parameters to almost every isometric leaf of an arbitrary nonexpansive map. The proof rests on a sharp contraction inequality for geometric conditional densities, with exponent equal to the leaf codimension. The inherited dimension parameter is optimal. A total-variation limit on resolvent graphs extends the result to inverse fibres of maximal monotone relations, including convex gradient fibres. We also disprove Klartag's curvature-dimension inheritance conjecture by a firmly nonexpansive example in dimension three and a gradient example in dimension four. In codimension one, affinity of the geometric density yields curvature-dimension inheritance. The first example also gives failure on monotone fibres. Both constructions admit arbitrarily large curvature loss, including for a fixed Gaussian ambient measure on families of leaves of positive quotient measure.
