论文标题
边界规律性和稳定性的RICCI界限
Boundary regularity and stability for spaces with Ricci bounded below
论文作者
论文摘要
本文研究了非汇总$ \ text {rcd}（k，n）$ spaces中边界的结构和稳定性，即公制 - 量化空间$（x，x，\ mathsf {d}，\ mathscr {h}^n）$，较低的ricci curvature curvature bouncation bounded bounded。我们的主要结构结果是，边界$ \ partial x $是远离一组Codimension 2的多种形态，并且是$ n-1 $可整流的。一路上，我们显示了边界及其管状邻居的有效度量界限。这些结果即使对于gromov-hausdorff限制$（m_i^n，\ mathsf {d} _ {g_i}，p_i），p_i）\ rightarrow（x，x，\ mathsf {d}，p），与边界平稳形式相比，以及需要新技术的典型结构，以供宣布为类似的结构，并需要进行类似的结构。 $ \ partial x $。 本地的关键结果是$ε$ - 型定理，它告诉我们，如果一个球$ b_ {2}（p）（p）\ subset x $足够接近半空间$ b_ {2}（0）\ subset \ subset \ subbb {r}^n _+in in gromov-hausdorff sense，ant $ b_1 $ \ mathbb {r}^n _+$。特别是，$ \ partial x $本身是同型对$ b_1（0^{n-1}）$附近$ b_1（p）$。此外，边界$ \ partial x $是$ n-1 $ rectifififiable，边界度量$ \ mathscr {h}^{n-1} _ {\ partial x} $是$ b_1（p）$的ahlfors常规，其体积接近欧几里得体积。 我们的第二个结果集合涉及边界相对于非汇合的MGH收敛$ x_i \ to x $的稳定性。具体来说，我们显示了一个边界量的收敛，该量告诉我们，$ n-1 $ hausdorff的量度对$ \ mathscr {h}^{n-1} _ {\ partial x_i} \ to \ mathscr {h}^{我们将看到这一点的结果是，如果$ x_i $是无边界的，那么$ x $。
This paper studies the structure and stability of boundaries in noncollapsed $\text{RCD}(K,N)$ spaces, that is, metric-measure spaces $(X,\mathsf{d},\mathscr{H}^N)$ with lower Ricci curvature bounded below. Our main structural result is that the boundary $\partial X$ is homeomorphic to a manifold away from a set of codimension 2, and is $N-1$ rectifiable. Along the way we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov-Hausdorff limits $(M_i^N,\mathsf{d}_{g_i},p_i) \rightarrow (X,\mathsf{d},p)$ of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $\partial X$. The key local result is an $ε$-regularity theorem, which tells us that if a ball $B_{2}(p)\subset X$ is sufficiently close to a half space $B_{2}(0)\subset \mathbb{R}^N_+$ in the Gromov-Hausdorff sense, then $B_1(p)$ is biHölder to an open set of $\mathbb{R}^N_+$. In particular, $\partial X$ is itself homeomorphic to $B_1(0^{N-1})$ near $B_1(p)$. Further, the boundary $\partial X$ is $N-1$ rectifiable and the boundary measure $\mathscr{H}^{N-1}_{\partial X}$ is Ahlfors regular on $B_1(p)$ with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $X_i\to X$. Specifically, we show a boundary volume convergence which tells us that the $N-1$ Hausdorff measures on the boundaries converge $\mathscr{H}^{N-1}_{\partial X_i}\to \mathscr{H}^{N-1}_{\partial X}$ to the limit Hausdorff measure on $\partial X$. We will see that a consequence of this is that if the $X_i$ are boundary free then so is $X$.
