We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map f : X = X → Y between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of X. We furthermore characterize the metric structure on Y with respect to X when f is also onto. This implies the converse of Petrunin's Gluing Theorem: if a gluing of two Alexandrov spaces via a bijection between their boundaries produces an Alexandrov space, then the bijection must be an isometry.
Let X ∈ Alex n (−1) be an n-dimensional Alexandrov space with curvature ≥ −1. Let the r-scale (k,)-singular set S k , r (X) be the collection of x ∈ X so that B r (x) is not rclose to a ball in any splitting space ℝ k+1 × Z. We show that there exists C(n,) > 0 and (n,) > 0 , independent of the volume, so that for any disjoint collection B r i (x i) ∶ x i ∈ S k , r i (X) ∩ B 1 , r i ≤ 1 , the packing estimate ∑ r k i ≤ C holds. Consequently, we obtain the Hausdorff measure estimates H k (S k (X) ∩ B 1) ≤ C and H n B r (S k , r (X)) ∩ B 1 (p) ≤ C r n−k. This answers an open question in Kapovitch et al. (Metric-measure boundary and geodesic flow on Alexandrov spaces. arXiv : 1705.04767 (2017)). We also show that the k-singular set S k (X) = ⋃ >0 � ⋂ r>0 S k , r � is k-rectifiable and construct examples to show that such a structure is sharp. For instance, in the k = 1 case we can build for any closed set T ⊆ 1 and > 0 a space Y ∈ Alex 3 (0) with S 1 (Y) = (T) , where ∶ 1 → Y is a bi-Lipschitz embedding. Taking T to be a Cantor set it gives rise to an example where the singular set is a 1-rectifiable, 1-Cantor set with positive 1-Hausdorff measure.
Abstract. We prove a Lipschitz-Volume rigidity theorem for the non-collapsed GromovHausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.
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