Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces

Li, Chao

doi:10.3842/sigma.2020.099

Cited by 8 publications

(9 citation statements)

References 31 publications

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“…Gromov also made the conjecture for parabolic cubes in [Gro14], and it was naturally important in the study of scalar curvature with a negative lower bound. Li [Li20b] confirmed Gromov dihedral rigidity conjecture for parabolic cubes in dimensions up to seven.…”

Section: Introductionsupporting

confidence: 55%

“…We also use the notation F i F j . Motivated by the result regarding the hyperbolic mass in [JM21], Appendix B, and the dihedral rigidity conjectures [Gro14], [Li20c], [Li20a] and [Li20b], we conjecture the following.…”

Section: Introductionmentioning

confidence: 99%

“…Sometimes we need to study P with the Euclidean metric, to emphasize the Euclidean background, we call P a flat reference. We have mentioned the parabolic cube in the work of [Gro14,Li20b], and a parabolic cube is a cube with two base faces lying on the horospheres.…”

Section: Introductionmentioning

confidence: 99%

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Dihedral rigidity in hyperbolic 3-space

Chai¹,

Wang²

2022

Preprint

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Section: Introductionsupporting

confidence: 55%

Section: Introductionmentioning

confidence: 99%

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Dihedral rigidity in hyperbolic 3-space

Chai¹,

Wang²

2022

Preprint

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“…He applies Jean Taylor's proof of the existence of capillary surfaces as integral currents [142] and then improves the regularity of the surfaces so that he can apply the technique Bray-Brendle-Neves used in [29] to prove the Scalar Cover Splitting Theorem. He has also proven a hyperbolic version of this theorem assuming Scal ≥ −1 in [90].…”

Section: The Geometry Of Scalar Curvaturementioning

confidence: 98%

“…If so, we would suggest trying to prove the asymptotically hyperbolic version of this conjecture in that setting as well. See also Chao Li's paper [90].…”

Section: Geometric Stability Of Zero Mass Rigidity [Lee-sormani]mentioning

confidence: 99%

Conjectures on Convergence and Scalar Curvature

Sormani¹,

Curvature²,

Convergence³

2021

Preprint

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Here we survey the compactness and geometric stability conjectures formulated by the participants at the 2018 IAS Emerging Topics Workshop on Scalar Curvature and Convergence. We have tried to survey all the progress towards these conjectures as well as related examples, although it is impossible to cover everything. We focus primarily on sequences of compact Riemannian manifolds with nonnegative scalar curvature and their limit spaces. Christina Sormani is grateful to have had the opportunity to write up our ideas and has done her best to credit everyone involved within the paper even though she is the only author listed above. In truth we are a team of over thirty people working together and apart on these deep questions and we welcome everyone who is interested in these conjectures to join us.

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Pointwise lower scalar curvature bounds for $$C^0$$ metrics via regularizing Ricci flow

Burkhardt-Guim

2019

Geom. Funct. Anal.

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In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C 0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C 0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C 0 initial data.Date: September 12, 2019. Theorem 1.5 implies:Corollary 1.6. If (M, g) is a closed Riemannian manifold with C 0 metric g, and if there exists β ∈ (0, 1/2) such that, at every point in M , g has scalar curvature bounded below by κ in the β-weak sense, then there exists a sequence of C 2 metrics on M with scalar curvature bounded below by κ that converges uniformly to g.Theorem 1.5 also implies:Theorem 1.7. Let g be a C 0 metric on a closed manifold M which admits a uniform approximation by C 0 metrics g i such that, for some β ∈ (0, 1/2), g i has scalar curvature bounded below by κ i in the β-weak sense everywhere on M , where κ i is some sequence of numbers such that κ i − −− → i→∞ κ for some number κ. Then g has scalar curvature bounded below by κ in the β-weak sense. In particular, any regularizing Ricci flow (g(t)) t∈(0,T ] for g satisfies R(g(t)) ≥ κ for all t ∈ (0, T ], so g admits a uniform approximation by smooth metrics with scalar curvature bounded below by κ.As a corollary of Theorem 1.7, we may answer the following question, posed by Gromov in [8]:Question 1 ([8, Page 1119]). Let g be a continuous Riemannian metric on a closed manifold M which admits a C 0 -approximation by smooth Riemannian metrics g i with R(g i ) ≥ −ε i − −− → i→∞ 0. Does M admit a smooth metric of nonnegative scalar curvature?By setting κ i = −ε i in Theorem 1.7, we obtain the following response:Corollary 1.8. If (M, g) is as in Question 1, then any regularizing Ricci flow (g t ) t∈(0,T ] for g satisfies R(g t ) ≥ 0 for all t ∈ (0, T ]. In particular, M admits a smooth metric of nonnegative scalar curvature, and moreover, g admits a uniform approximation by smooth metrics with nonnegative scalar curvature.We use similar methods to show a torus rigidity result, motivated by the scalar torus rigidity theorem, which was first proven by Schoen and Yau [14] for dimensions ≤ 7, and later proven by Gromov and Lawson [7] for all dimensions, and which says that any Riemannian manifold with nonnegative scalar curvature that is diffeomorphic to the torus must be isometric to the flat torus. We show: Corollary 1.9. Suppose g is a C 0 metric on the torus T, and that there is some β ∈ (0, 1/2) such that g has nonnegative scalar curvature in the β-weak sense everywhere. Then (T, g) is isometric as a metric space to the standard flat metric on T.Remark 1.10. Corollary 1.9 is in fact the optimal result, i.e. it is not possible to show that there is a Riemannian isometry between g and the standard flat metric. In the case where g 1 and g 2 are smooth metrics, a metric space...

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Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces

Cited by 8 publications

References 31 publications

Dihedral rigidity in hyperbolic 3-space

Dihedral rigidity in hyperbolic 3-space

Conjectures on Convergence and Scalar Curvature

Pointwise lower scalar curvature bounds for $$C^0$$ metrics via regularizing Ricci flow

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