English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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. Ourmain structural result is that the boundary $\partial X$ is homeomorphic to amanifold away from a set of codimension 2, and is $N-1$ rectifiable. Along theway we show effective measure bounds on the boundary and its tubularneighborhoods. 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 manifoldswith boundary, and require new techniques beyond those needed to prove theanalogous statements for the regular set, in particular when it comes to themanifold structure of the boundary $\partial X$.  The key local result is an $\epsilon$-regularity theorem, which tells us thatif 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 biH\"older 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 volumeclose to the Euclidean volume.  Our second collection of results involve the stability of the boundary withrespect to noncollapsed mGH convergence $X_i\to X$. Specifically, we show aboundary volume convergence which tells us that the $N-1$ Hausdorff measures onthe 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 freethen so is $X$.

Boundary regularity and stability for spaces with Ricci bounded below