论文标题
通过直接方法在不稳定的度量度量空间中通过直接方法具有较低RICCI边界的等值问题
The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds
论文作者
论文摘要
我们建立了一个结构定理，以最大程度地减少在非2级$ \ mathsf {rcd}（k，k，n）$ spaces $（x，x，\ mathsf {d}，\ mathcal {h}^n）$的序列中的序列。在鞋底（必要的）假设下，单位球的度量均匀地远离零，我们证明，这种序列的极限是通过在点序序列的尖头gromov-hausdorff限制的有限集合的Gromov-Hausdorff限制中所包含的。根据最小化序列的度量，此类区域的数量是线性界定的。 结果来自新的广义紧凑定理，该定理识别了$ e_i \ subset x_i $的限制，其x_i $具有均匀边界的度量和周围，其中$（x_i，\ sathsf {d} _i，\ natcal {h}^n）$是$ $ sect的$ sect n $ sect $ sect $ sect $ sect（k） 还讨论了一个最小化序列的抽象标准，而无需在无穷大的情况下将质量融合到等速度集合。后一个标准也是新的，用于光滑的riemannian空间。
We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact $\mathsf{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathcal{H}^N)$. Under the sole (necessary) assumption that the measure of unit balls is uniformly bounded away from zero, we prove that the limit of such a sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov--Hausdorff limits of the ambient space $X$ along diverging sequences of points. The number of such regions is bounded linearly in terms of the measure of the minimizing sequence. The result follows from a new generalized compactness theorem, which identifies the limit of a sequence of sets $E_i\subset X_i$ with uniformly bounded measure and perimeter, where $(X_i,\mathsf{d}_i,\mathcal{H}^N)$ is an arbitrary sequence of $\mathsf{RCD}(K,N)$ spaces. An abstract criterion for a minimizing sequence to converge without losing mass at infinity to an isoperimetric set is also discussed. The latter criterion is new also for smooth Riemannian spaces.
