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

Consider a limit space (M_αg_αp_α →^GH (Y,d_Y,p), where the M_αⁿ have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of Y at a point p ∈ Y are known to be metric cones C(X), however they need not be unique. Let Ω_Y,p ⊆ M_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 Ω_Y,p. In particular, we give necessary and sufficient conditions for an open smooth family Ω≡ (X,g_s) of closed manifolds to satisfy Ω =Ω_Y,p for some limit Y and point Ω is the closure of Ω 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 (Yⁿ, d_Y, p) with n ≥ 3 such that at p there exists for every 0 ≤ k ≤ n-2 a tangent cone at p of the form ℝ^k × 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 (Y⁵, 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 ℂP²#ℂP̄² while others are homeomorphic to cones over S⁴.