一种不具有测度压缩性质的二步理想次黎曼结构
A two-step ideal sub-Riemannian structure with no measure contraction properties
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中文总结 AI 辅助
该研究构造了步长为2的紧致等正则理想次黎曼流形，证明其对所有光滑正测度均不满足MCP性质，从而表明测度压缩定理中的实解析性假设不可被光滑性替代。
中文摘要 AI 辅助
我们构造了一个紧致、等正则、理想的步长为2的次黎曼流形，使得对于每个光滑正测度，$\nmathrm{MCP}(K,N)$ 对所有 $K\in\mathbb{R}$ 和 $N \in (1,\infty)$ 均失效。这表明Badreddine和Rifford在arXiv:1712.09900v2中的测度压缩定理中的实解析性假设不能被光滑性所替代。此外，我们的结构是理想的，即它不允许非平凡的异常极小测地线。尽管最近在更高步长设置中获得了理想结构的测度压缩性质的失效，但我们的构造表明该现象已经可以发生在步长为2的紧致、等正则、理想结构上。证明利用了测度压缩性质的一个微分推论，即平方距离的次拉普拉斯算子在其基点附近的一致上界，并构造了一个该量无界的结构。
英文摘要
We construct a compact, equiregular, ideal sub-Riemannian manifold of step $2$ such that, for every smooth positive measure, the $\mathrm{MCP}(K,N)$ fails for all $K\in\mathbb{R}$ and $N \in (1,\infty)$. This shows that the real-analyticity assumption in the measure contraction theorem of Badreddine and Rifford in arXiv:1712.09900v2 cannot be replaced by smoothness. Moreover, our structure is ideal, i.e. it admits no non-trivial abnormal minimizing geodesics. Although failures of the measure contraction property for ideal structures were recently obtained in the higher-step setting, our construction shows that the phenomenon can already occur on compact, equiregular, ideal structures of step $2$. The proof exploits a differential consequence of the measure contraction property, namely a uniform upper bound for the sub-Laplacian of the squared distance near its base point, and constructs a structure for which this quantity is unbounded.
发表机构
- SISSA（国际高等研究学院）
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