This note is dedicated to the study of the asymptotic behaviour of sets of finite perimeter over \({{\,\mathrm{RCD}\,}}(K,N)\) metric measure spaces. Our main result asserts existence of a Euclidean tangent half-space almost everywhere with respect to the perimeter measure and it can be improved to an existence and uniqueness statement when the ambient is non collapsed. As an intermediate tool, we provide a complete characterization of the class of \({{\,\mathrm{RCD}\,}}(0,N)\) spaces for which there exists a nontrivial function satisfying the equality in the 1-Bakry–Émery inequality. This result is of independent interest and it is new, up to our knowledge, even in the smooth framework.

中文翻译：

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RCD空间中1-Bakry-Émery不等式的刚性和有限周集

此注释致力于研究度量（K，N）\（{{\\\ mathrm {RCD} \，}}（K，N）\）上的有限周长集合的渐近行为。我们的主要结果断言，相对于周长度量，欧几里德正切半空间的存在几乎无处不在，并且当周围环境不塌陷时，可以将其改进为存在和唯一性陈述。作为一种中间工具，我们提供\（{{\，\ mathrm {RCD} \，}}（0，N）\）类空间的完整刻画，对于这些空间，存在一个满足1中相等的非平凡函数。 -Bakry–Émery不平等。该结果具有独立的利益，就我们所知，即使是在平稳的框架中，也是新的。