Ricci curvature in dimension 2

Lytchak, Alexander; Stadler, Stephan

doi:10.4171/jems/1196

Cited by 10 publications

(18 citation statements)

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“…In particular, (X, d, H 2 ) is an RCD(0, 2)-space. Thanks to Theorem 1.1 of [23], X is a topological surface, which implies π 1 (X) ≃ π 1 (X). However, up to homeomorphism, there are only three compact topological surfaces with a finite fundamental group, namely S 2 , RP 2 and D. This concludes the proof.…”

Section: Essential Dimension and Topological Obstructionsmentioning

confidence: 99%

“…Therefore, applying Proposition 4.1, there exists a > 0 such that m = aH 2 . In particular, thanks to Theorem 1.1 of [23] , (X, d) is an Alexandrov space with nonnegative curvature. Therefore, proceeding exactly as in the last part of the proof of Proposition 5.1, we obtain the following result.…”

Section: 2mentioning

confidence: 99%

“…We can easily take care of the case k(X) ∈ {1, 2} using the first part of the proof. The final case k(X) = 0 implies that (X, d, m) is a simply connected compact topological surface with boundary (using the results of [23]). It will then be easy to conclude the proof.…”

Section: Introductionmentioning

confidence: 99%

“…Applying the same ideas, we can show that M 0,2 (M 2 ) is homeomorphic to R 3 (see Proposition 7.3) Now, observe that the remaining cases are S 2 , RP 2 , and D; in particular, they are all compact topological surfaces (possibly with boundary). Moreover, given a compact topological surface X, we can apply results from [18], [14], and [23] to show that M 0,2 (X) is homeomorphic to R × M curv≥0 (X), where M curv≥0 (X) is the moduli space of metrics on X that are nonnegatively curved in the Alexandrov sense (see Lemma 5.1). In particular, to conclude, we only need to describe M curv≥0 (S 2 ), M curv≥0 (RP 2 ), and M curv≥0 (D).…”

Section: Introductionmentioning

confidence: 99%

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Contractibility of moduli spaces of RCD(0,2)-structures

Navarro¹

2022

Preprint

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This paper focuses on RCD(0, 2)-spaces, which can be thought of as possibly non-smooth metric measure spaces with non-negative Ricci curvature and dimension less than 2. First, we establish a list of the compact topological spaces admitting an RCD(0, 2)-structure. Then, we describe the associated moduli space of RCD(0, 2)-structures for each of them. In particular, we show that all these moduli spaces are contractible. CONTENTS1. Introduction 1.1. Basic definitions 1.2. Main results 1.3. Sketch of the proofs Organization of the paper Acknowledgements Preliminaries 2. Equivariance 3. Albanese variety and soul 4. Essential dimension and topological obstructions Computations of Moduli spaces 5. Helpful results 5.1. The case of compact flat manifolds 5.2. The case of surfaces 6. The 1-dimensional case 6.1. The circle S 1 6.2. The closed unit interval I 7. The 2-dimensional case 7.1. The 2-Torus T 2 and the Klein bottle K 2 7.2. The Möbius band M 2 and the cylinder S 1 × I 7.3. The 2-sphere S 2 , the projective plane RP 2 , and the closed disc D References

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Section: Essential Dimension and Topological Obstructionsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Contractibility of moduli spaces of RCD(0,2)-structures

Navarro¹

2022

Preprint

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“…Note that the essential dimension is not necessary to coincide with the Hausdorff dimension by [PW21] and that the question (Q1) has a positive answer if N = 1, 2 or if N = 3 and X i is smooth, see [LS22,ST21]. It is well-known that in general the question (Q1) has a negative answer even if each (X i , d i ) is isometric to a Ricci flat manifold and the essential dimensions are equal to N = 4 (see for example [KT87,P78]).…”

Section: Introduction 1topological Stability Theoremmentioning

confidence: 99%

A note on the topological stability theorem from RCD spaces to Riemannian manifolds

Honda¹,

Peng²

2022

Preprint

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Inspired by a recent work of , in this note we prove that for a fixed n-dimensional closed Riemannian manifold (M n , g), if an RCD(K, n) space (X, d, m) is Gromov-Hausdorff close to M n , then there exists a regular homeomorphism F from X to M n such that F is Lipschitz continuous and that F −1 is Hölder continuous, where the Lipschitz constant of F , the Hölder exponent and the Hölder constant of F −1 can be chosen arbitrary close to 1. This is sharp in the sense that in general such a map cannot be improved to being bi-Lipschitz. Moreover if X is smooth, then such a homeomorphism can be chosen as a diffeomorphism, which improves the intrinsic Reifenberg theorem established by Cheeger-Colding. The Nash embedding theorem plays a key role in the proof.

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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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Ricci curvature in dimension 2

Cited by 10 publications

References 0 publications

Contractibility of moduli spaces of RCD(0,2)-structures

Contractibility of moduli spaces of RCD(0,2)-structures

A note on the topological stability theorem from RCD spaces to Riemannian manifolds

Boundary regularity and stability for spaces with Ricci bounded below

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