论文标题
RCD（K，N）空间中的切线锥的Hölder连续性以及非分支的应用
Hölder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching
论文作者
论文摘要
在本文中，我们证明了满足有限的riemannian曲率差异条件$ {\ sf rcd}（k，n）$满足有限的riemannian曲率差异条件的公制量$（x，d，m）$是非分支的，并且来自同一reccallings reccalence in everior n everior n everior n everior n everior n everior n everior n everior codedence in $ x $ x $ x $ x $。更确切地说，我们表明，小半径的球几何形状以内部的内部为中心，最多最多是Hölder的连续方式，沿着尖锐的Gromov-Hausdorff距离。这通过冷 - 纳伯（Colding-Naber）改善了RICCI限制设置的结果，其中显示了至少一个地理位置，并显示了任何两个点之间具有此类特性的结果。与RICCI限制案例一样，这意味着$ {\ sf rcd}（k，n）$ space的常规集具有$ m $ -a.e。恒定尺寸，这是Bruè-Semola已经建立的结果，是$ M $ -A.E凸。这也意味着最高维度的常规集是弱凸，因此连接。在证明主要定理时，我们在$ {\ sf rcd}（k，n）$中开发了沿某个矢量场的常规拉格朗格流动的距离函数的预期二阶插值公式，该公式使用其协证衍生物。
In this paper we prove that a metric measure space $(X,d,m)$ satisfying the finite Riemannian curvature-dimension condition ${\sf RCD}(K,N)$ is non-branching and that tangent cones from the same sequence of rescalings are Hölder continuous along the interior of every geodesic in $X$. More precisely, we show that the geometry of balls of small radius centred in the interior of any geodesic changes in at most a Hölder continuous way along the geodesic in pointed Gromov-Hausdorff distance. This improves a result in the Ricci limit setting by Colding-Naber where the existence of at least one geodesic with such properties between any two points is shown. As in the Ricci limit case, this implies that the regular set of an ${\sf RCD}(K,N)$ space has $m$-a.e. constant dimension, a result already established by Bruè-Semola, and is $m$-a.e convex. It also implies that the top dimension regular set is weakly convex and, therefore, connected. In proving the main theorems, we develop in the ${\sf RCD}(K,N)$ setting the expected second order interpolation formula for the distance function along the Regular Lagrangian flow of some vector field using its covariant derivative.
