On one-dimensionality of metric measure spaces

Schultz, Timo

doi:10.1090/proc/15162

Cited by 5 publications

(5 citation statements)

References 20 publications

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“…Moreover, at the other points the tangent is still unique and isometric to a half line pointed at the extreme (otherwise there would be a full line in the tangent and we would be in the previous case). Arguing as in the proof of [84,Theorem 3.1] we conclude that each point in Z has a neighborhood isometric either to (−ε, ε) or to [0, ε). Hence the metric conclusion follows from the characterization of one dimensional Riemannian manifolds.…”

Section: Remark 36mentioning

confidence: 61%

“…Adapting the arguments of [63] (see also [84] for a recent generalization with simplified arguments relying on optimal transport tools), it is possible to prove that at points of Z where there is a line in the tangent there is a small ball isometric to the Euclidean one. Moreover, at the other points the tangent is still unique and isometric to a half line pointed at the extreme (otherwise there would be a full line in the tangent and we would be in the previous case).…”

Section: Remark 36mentioning

confidence: 99%

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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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Section: Remark 36mentioning

confidence: 61%

Section: Remark 36mentioning

confidence: 99%

Boundary regularity and stability for spaces with Ricci bounded below

2022

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“…With such a complex definition it is not completely obvious that the map M satisfies condition (12), but this can be easily proven.…”

Section: Definition Of the Midpointmentioning

confidence: 99%

Example of a Highly Branching CD Space

Mattia

2022

J Geom Anal

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In Ketterer and Rajala (Potential Anal 42:645–655, 2014) showed an example of metric measure space, satisfying the measure contraction property $$\mathsf {MCP}(0,3)$$ MCP ( 0 , 3 ) , that has different topological dimensions at different regions of the space. In this article I propose a refinement of that example, which satisfies the $$\mathsf {CD}(0,\infty )$$ CD ( 0 , ∞ ) condition, proving the non-constancy of topological dimension for CD spaces. This example also shows that the weak curvature dimension bound, in the sense of Lott–Sturm–Villani, is not sufficient to deduce any reasonable non-branching condition. Moreover, it allows to answer to some open question proposed by Schultz in (Calc Var Partial Differ Equ 57:1–11, 2018), about strict curvature dimension bounds and their stability with respect to the measured Gromov–Hausdorff convergence.

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“…• the very strict CD condition studied by Schultz ([7], [8], [9]) is strictly stronger than the weak one,…”

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confidence: 99%

Example of an Highly Branching CD Space

Mattia¹

2021

Preprint

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In [3] Ketterer and Rajala showed an example of metric measure space, satisfying the measure contraction property MCP(0, 3), that has different topological dimensions at different regions of the space. In this article I propose a refinement of that example, which satisfies the CD(0, ∞) condition, proving the non-constancy of topological dimension for CD spaces. This example also shows that the weak curvature dimension bound, in the sense of Lott-Sturm-Villani, is not sufficient to deduce any reasonable non-branching condition. Moreover, it allows to answer to some open question proposed by Schultz in [7], about strict curvature dimension bounds and their stability with respect to the measured Gromov Hausdorff convergence.

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On one-dimensionality of metric measure spaces

Cited by 5 publications

References 20 publications

Boundary regularity and stability for spaces with Ricci bounded below

Boundary regularity and stability for spaces with Ricci bounded below

Example of a Highly Branching CD Space

Example of an Highly Branching CD Space

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