MCP度量测度空间上胶合定理的一个反例
A counterexample to gluing theorems for MCP metric measure spaces
摘要：
佩雷尔曼的加倍定理断言，通过沿着边界粘合两个具有曲率$\geq κ$的亚历山德罗夫空间的副本而获得的度量空间是具有相同维数并满足相同曲率下界的亚历山德罗夫空间。我们证明了这个结果不能推广到满足$\mathrm{MCP}$意义下的合成里奇曲率界的度量测度空间。反例由Grushin半平面给出，它满足$\mathrm{MCP}(0,N)$当且仅当$N\geq 4$，而它的双平面满足$\mathrm{MCP}(0,N)$当且仅当$N\geq 5$。
英文摘要：
Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature $\geq κ$ is an Alexandrov space with the same dimension and satisfying the same curvature lower bound. We show that this result cannot be extended to metric measure spaces satisfying synthetic Ricci curvature bounds in the $\mathrm{MCP}$ sense. The counterexample is given by the Grushin half-plane, which satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 4$, while its double satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 5$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
