Lower bounds for codimension-1 measure in metric manifolds

Abstract

We establish Euclidean-type lower bounds for the codimension-1 Hausdorff measure of sets that separate points in doubling and linearly locally contractible metric manifolds. This gives a quantitative topological isoperimetric inequality in the setting of metric manifolds, in the sense that lower bounds for the codimension-1 measure of a set depend not on some notion of filling or volume but rather on in-radii of complementary components. As a consequence, we show that balls in a closed, connected, doubling, and linearly locally contractible metric n-manifold (M, d) with radius 0 < r ≤ diam(M ) have n-dimensional Hausdorff measure at least c · r n , where c > 0 depends only on n and on the doubling and linear local contractibility constants.

Non-removability of the Sierpiński gasket

“…This is a folklore statement, and it is discussed in [Ra17, Section 16]; see also [Ki18,Corollary 1.4]. We give a quick proof for the sake of completeness.…”

Geometric and analytic structures on metric spaces homeomorphic to a manifold

“…Proof. This follows directly from a slight adaptation of the proof of a result of Kinneberg [28,Corollary 1.4].…”

“…The proof of Theorem 1.1 uses a recent important result of Bate [5] about the Hausdorff measure of Lipschitz images of purely unrectifiable sets, and assumption (1.1) is only needed to apply Bate's result. It is for example satisfied if X with H n (X) < ∞ is a linearly locally contractible metric space homeomorphic to a closed, oriented, smooth n-manifold and X is doubling or, more generally, has finite Nagata dimension (see Lemma 2.5 and also [28]). Actually, using a recently announced deep result of Csörnyei and Jones, it can be shown that (1.1) is never needed; see [5,Remark 6.7].…”

Thin Loewner Carpets and Their Quasisymmetric Embeddings in S2

“…The tools to prove the quasisymmetric embeddability of general (inexplicit) Loewner carpets were inspired by discussions with Bonk, and his description of partial results with Kleiner. Bonk also pointed out the relevance of the reference [47] and encouraged us to pursue the use of substitution rules.…”