Weakly non-collapsed RCD spaces are strongly non-collapsed

Brena, Camillo; Gigli, Nicola; Honda, Shinya; Zhu, Xingyu

doi:10.1515/crelle-2022-0071

Cited by 8 publications

(3 citation statements)

References 52 publications

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“…Then Theorem 2.11 implies that X splits a line, but, since X is assumed to have one end, this contradicts Lemma 2. 33.…”

Section: Theorem 222 (Asymptotic Mass Decompositionmentioning

confidence: 99%

“…If m = H 2 , then by [79] we know that (X, d) is a 2-dimensional CBB(0) metric space and we are done by Theorem 5.5. If (X, d, m) is not noncollapsed according to the definition of [52], by using the results of [52,38,33] we get that its essential dimension is either 0, or 1. The case in which the dimension is 0 is trivial, and so we can reduce to the case where the essential dimension is 1.…”

Section: Isoperimetric Problemmentioning

confidence: 99%

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Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

Antonelli¹,

Pozzetta²

2023

Preprint

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In this paper we consider nonnegatively curved finite dimensional Alexandrov spaces such that unit balls have volumes uniformly bounded from below away from zero. We study the relation between the isoperimetric profile, the existence of isoperimetric sets, and the asymptotic structure at infinity of such spaces.In the previous setting, we prove that the following conditions are equivalent: the space has linear volume growth; it is Gromov-Hausdorff asymptotic to one cylinder at infinity; it has uniformly bounded isoperimetric profile; the entire space is a tubular neighborhood of either a line or a ray.Moreover, on a space satisfying any of the previous conditions, we prove existence of isoperimetric sets for sufficiently large volumes, and, in such generality, we characterize the geometric rigidity at the level of the isoperimetric profile.Specializing our study to the 2-dimensional case, we prove that unit balls have always volumes uniformly bounded from below away from zero, and we prove existence of isoperimetric sets for every volume, characterizing also their topology when the space has no boundary.The proofs exploit an approach by direct method and the results in particular apply to Riemannian manifolds with nonnegative sectional curvature and to Euclidean convex bodies. Up to the authors' knowledge, most of the results are new even in these smooth cases.

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“…Then Theorem 2.11 implies that X splits a line, but, since X is assumed to have one end, this contradicts Lemma 2. 33.…”

Section: Theorem 222 (Asymptotic Mass Decompositionmentioning

confidence: 99%

Section: Isoperimetric Problemmentioning

confidence: 99%

Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

Antonelli¹,

Pozzetta²

2023

Preprint

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“…Applying Lemma 4.14 below with A := Hess(b p ) and v := ∇bp |∇bp| , and recalling that the Laplacian is the trace of the Hessian on noncollapsed RCD spaces (see for instance [20]), we get…”

Section: The Uniform Convergence Bmentioning

confidence: 99%

Boundary regularity and stability for spaces with Ricci bounded below

2022

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This paper studies the structure and stability of boundaries in noncollapsed $${{\,\mathrm{RCD}\,}}(K,N)$$ RCD ( K , N ) spaces, that is, metric-measure spaces $$(X,{\mathsf {d}},{\mathscr {H}}^N)$$ ( X , d , H N ) with Ricci curvature bounded below. Our main structural result is that the boundary $$\partial X$$ ∂ X is homeomorphic to a manifold away from a set of codimension 2, and is $$N-1$$ N - 1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits $$(M_i^N,{\mathsf {d}}_{g_i},p_i) \rightarrow (X,{\mathsf {d}},p)$$ ( M i N , d g i , p i ) → ( X , d , p ) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $$\partial X$$ ∂ X . The key local result is an $$\varepsilon $$ ε -regularity theorem, which tells us that if a ball $$B_{2}(p)\subset X$$ B 2 ( p ) ⊂ X is sufficiently close to a half space $$B_{2}(0)\subset {\mathbb {R}}^N_+$$ B 2 ( 0 ) ⊂ R + N in the Gromov–Hausdorff sense, then $$B_1(p)$$ B 1 ( p ) is biHölder to an open set of $${\mathbb {R}}^N_+$$ R + N . In particular, $$\partial X$$ ∂ X is itself homeomorphic to $$B_1(0^{N-1})$$ B 1 ( 0 N - 1 ) near $$B_1(p)$$ B 1 ( p ) . Further, the boundary $$\partial X$$ ∂ X is $$N-1$$ N - 1 rectifiable and the boundary measure "Equation missing" is Ahlfors regular on $$B_1(p)$$ B 1 ( p ) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $$X_i\rightarrow X$$ X i → X . Specifically, we show a boundary volume convergence which tells us that the $$N-1$$ N - 1 Hausdorff measures on the boundaries converge "Equation missing" to the limit Hausdorff measure on $$\partial X$$ ∂ X . We will see that a consequence of this is that if the $$X_i$$ X i are boundary free then so is X.

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The metric measure boundary of spaces with Ricci curvature bounded below

2023

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We solve a conjecture raised by Kapovitch, Lytchak and Petrunin in [KLP21] by showing that the metric measure boundary is vanishing on any $${{\,\textrm{RCD}\,}}(K,N)$$ RCD ( K , N ) space $$(X,{\textsf{d}},{\mathscr {H}}^N)$$ ( X , d , H N ) without boundary. Our result, combined with [KLP21], settles an open question about the existence of infinite geodesics on Alexandrov spaces without boundary raised by Perelman and Petrunin in 1996.

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Weakly non-collapsed RCD spaces are strongly non-collapsed

Cited by 8 publications

References 52 publications

Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

Boundary regularity and stability for spaces with Ricci bounded below

The metric measure boundary of spaces with Ricci curvature bounded below

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