Volume Bounds for the Quantitative Singular Strata of Non Collapsed RCD Metric Measure Spaces

Antonelli, Gioacchino; Bruè, Elia; Semola, Daniele

doi:10.1515/agms-2019-0008

Cited by 22 publications

(25 citation statements)

References 31 publications

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“…The rectifiability of the top dimensional singular stratum was conjectured both in [59,Conjecture 4.10] and in [37], together with the local finiteness of the H N −1 -measure. Moreover, with (1.15) we sharpen the volume bound for the tubular neighbourhood of the top dimensional singular set obtained in [14,Corollary 2.7] by adapting the techniques developed in [30] to the synthetic framework. The topological regularity part of Theorem 1.4 improves upon [59,Theorem 4.11], including the boundary in the statements.…”

Section: Structure Of Boundaries and Of Spaces With Boundarymentioning

confidence: 93%

“…Let us start by restating a quantitative version of the cone splitting lemma [30,Lemma 4.1] tailored for RCD(K , N ) spaces (see [14] for the present version).…”

Section: Cone Splitting Via Contentmentioning

confidence: 99%

“…By the functional splitting theorem (cf. [14,Lemma 1.20]), Y splits off a factor R N −1 . Moreover, it is a metric cone and it is not isometric to R N , thanks to Definition 4.3 (ii).…”

Section: Yielding (414)mentioning

confidence: 99%

“…In the same work some properties valid for noncollapsed Ricci limits have been generalized to the synthetic framework, such as the volume convergence and the stratification of the singular set. More recent contributions dealt with topological regularity [59], volume bounds for the singular strata [14] and differential characterizations [53].…”

Section: Introductionmentioning

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: Structure Of Boundaries and Of Spaces With Boundarymentioning

confidence: 93%

“…Let us start by restating a quantitative version of the cone splitting lemma [30,Lemma 4.1] tailored for RCD(K , N ) spaces (see [14] for the present version).…”

Section: Cone Splitting Via Contentmentioning

confidence: 99%

“…By the functional splitting theorem (cf. [14,Lemma 1.20]), Y splits off a factor R N −1 . Moreover, it is a metric cone and it is not isometric to R N , thanks to Definition 4.3 (ii).…”

Section: Yielding (414)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Boundary regularity and stability for spaces with Ricci bounded below

2022

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“…The case of arbitrary volume measures m appears to be more involved and related to a better understanding of the properties of the density of m with respect to the Hausdorff measure of the essential dimension of the space. We stress that the class of N-dimensional RCD(K, N) spaces, that has been recently introduced and studied in the works [44,35,14,25], is the non-smooth generalization of the class of non collapsed Ricci limit spaces [30], in which a volume convergence theorem holds. The Riemannian assumption is necessary here to exploit the convergence and stability results of [6].…”

Section: Introductionmentioning

confidence: 99%

The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds

Antonelli,

Nardulli,

Pozzetta

2022

Preprint

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We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact RCD(K, N ) spaces (X, d, H N ). Under the sole (necessary) assumption that the measure of unit balls is uniformly bounded away from zero, we prove that the limit of such a sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov-Hausdorff limits of the ambient space X along diverging sequences of points. The number of such regions is bounded linearly in terms of the measure of the minimizing sequence.The result follows from a new generalized compactness theorem, which identifies the limit of a sequence of sets E i ⊂ X i with uniformly bounded measure and perimeter, where (X i , d i , H N ) is an arbitrary sequence of RCD(K, N ) spaces.An abstract criterion for a minimizing sequence to converge without losing mass at infinity to an isoperimetric set is also discussed. The latter criterion is new also for smooth Riemannian spaces.

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On Well-posed Variational Problems Involving Multidimensional Integral Functionals

Treanţă

2022

Forum for Interdisciplinary Mathematics

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Volume Bounds for the Quantitative Singular Strata of Non Collapsed RCD Metric Measure Spaces

Cited by 22 publications

References 31 publications

Boundary regularity and stability for spaces with Ricci bounded below

Boundary regularity and stability for spaces with Ricci bounded below

The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds

On Well-posed Variational Problems Involving Multidimensional Integral Functionals

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