Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications

Abstract

Consider a limit space $(M_{\alpha },g_{\alpha },p_{\alpha })\stackrel{GH}{\to }(Y,d_Y,p)$, where the $M_{\alpha }^n$ have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of $Y$ at a point $p\in Y$ are known to be metric cones $C(X)$, however they need not be unique. Let $\barΩ_{Y,p}\subseteq\cM_{GH}$ be the closed subset of compact metric spaces $X$ which arise as cross sections for the tangents cones of $Y$ at $p$. In this paper we study the properties of ${\bar{\Omega }}_{Y,p}$. In particular, we give necessary and sufficient conditions for an open smooth family $\Omega \equiv (X_s,g_s)$ of closed manifolds to satisfy $\bar{\Omega }={\bar{\Omega }}_{Y,p}$ for {\it some} limit $Y$ and point $p\in Y$ as above, where $\bar{\Omega }$ is the closure of $\Omega$ in the set of metric spaces equipped with the Gromov-Hausdorff topology. We use this characterization to construct examples which exhibit fundamentally new behaviors. The first application is to construct limit spaces $(Y^n,d_Y,p)$ with $n\ge 3$ such that at $p$ there exists for every $0\le k\le n-2$ a tangent cone at $p$ of the form $\RR^{k}\times C(X^{n-k-1})$, where $X^{n-k-1}$ is a smooth manifold not isometric to the standard sphere. In particular, this is the first example which shows that a stratification of a limit space $Y$ based on the Euclidean behavior of tangent cones is not possible or even well defined. It is also the first example of a three dimensional limit space with nonunique tangent cones. The second application is to construct a limit space $(Y^5,d_Y,p)$, such that at $p$ the tangent cones are not only not unique, but not homeomorphic. Specifically, some tangent cones are homeomorphic to cones over $\CC P^2\sharp\bar{\CC P}^2$ while others are homeomorphic to cones over $\Sn^4$.

References in corpus (1)

• Lower Ricci Curvature, Branching, and Bi-Lipschitz Structure of Uniform Reifenberg Spaces

Cited by in corpus (2)

• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
• Lower Ricci Curvature, Branching, and Bi-Lipschitz Structure of Uniform Reifenberg Spaces