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

TL;DR

This paper studies the structure of tangent cones in limit spaces with lower Ricci bounds, revealing nonuniqueness and complex stratification behaviors, and constructs examples with novel tangent cone properties.

Contribution

It provides necessary and sufficient conditions for tangent cone cross sections and constructs examples showing nonuniqueness and non-homeomorphic tangent cones in limit spaces.

Findings

Existence of limit spaces with tangent cones of varying Euclidean factors.

First example of a 3D limit space with nonunique tangent cones.

Construction of a 5D limit space with non-homeomorphic tangent cones.

Abstract

Consider a limit space $(M_\alpha,g_\alpha,p_\alpha)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, 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 $\bar\Omega_{Y,p}\subseteq\cM_{GH}$ 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 $n\geq 3$ such that at $p$ there exists for every $0\leq k\leq n-2$ 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 $\Sn^4$.