Scalar Curvature via Local Extent

Veronelli, Giona

doi:10.1515/agms-2018-0008

Cited by 7 publications

(6 citation statements)

References 21 publications

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“…What about a synthetic version of the non negative scalar curvature? An attempt to give a non differential definition of scalar curvature was made in [67], which could lead to a notion of scalar curvature for metric spaces. Some recent works by M. Gromov seem to pave the way towards such a notion; the interested reader is referred to [32,33] and the preprints posted here.…”

Section: Discussionmentioning

confidence: 99%

On Scalar and Ricci Curvatures

Besson¹,

Gallot²

2021

SIGMA

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The purpose of this report is to acknowledge the influence of M. Gromov's vision of geometry on our own works. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds carry a complete Riemannian metric of positive or non negative scalar curvature? In the second part we look for weak forms of the notion of ''lower bounds of the Ricci curvature'' on non necessarily smooth metric measure spaces. We describe recent results some of which are already posted in [arXiv:1712.08386] where we proposed to use the volume entropy. We also attempt to give a new synthetic version of Ricci curvature bounded below using Bishop-Gromov's inequality.

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Section: Discussionmentioning

confidence: 99%

On Scalar and Ricci Curvatures

Besson¹,

Gallot²

2021

SIGMA

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“…What about a synthetic version of the non negative scalar curvature ? An attempt to give a non differential definition of scalar curvature was made in [Ver18], which could lead to a notion of scalar curvature for metric spaces. Some recent works by M. Gromov seem to pave the way towards such a notion; the interested reader is referred to [Gro18b,Gro18a] and the preprints posted here.…”

Section: Discussionmentioning

confidence: 99%

On Scalar and Ricci Curvatures

Besson,

Gallot

2020

Preprint

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The purpose of this report is to acknowledge the influence of M. Gromov's vision of Geometry on our own work. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds carry a complete Riemannian metric of positive or non negative scalar curvature ? In the second part we look for weak forms of the notion of "lower bounds of the Ricci curvature" on non necessarily smooth metric measured spaces. We describe recent results some of which are already posted in [BCGS20b] where we proposed to use the volume entropy. We also attempt to give a new synthetic version of Ricci curvature bounded below using Bishop-Gromov's Inequality.

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“…The enlargeable length-structure may be used to deal with positive scalar curvature in the metric geometry setting. For instance, the definition of scalar curvature for length metrics was given in [30].…”

Section: Remark 22mentioning

confidence: 99%

Enlargeable length-structure and scalar curvatures

Deng

2021

Ann Glob Anal Geom

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We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive MV-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

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Scalar Curvature via Local Extent

Cited by 7 publications

References 21 publications

On Scalar and Ricci Curvatures

On Scalar and Ricci Curvatures

On Scalar and Ricci Curvatures

Enlargeable length-structure and scalar curvatures

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