Desingularizing positive scalar curvature 4-manifolds

Kazaras, Demetre

doi:10.1007/s00208-024-02829-5

Cited by 1 publication

(2 citation statements)

References 33 publications

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“…We note that the proof of Theorem C follows the same lines as in the works of Kazaras [23] and Li-Mantoulidis [28]. Our contribution is identifying the correct KO-theoretic obstruction to complete psc metrics which already appeared in work of Cecchini [6], see also §3.2].…”

Section: Theorem Bmentioning

confidence: 61%

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Scalar Curvature and Generalized Callias Operators

Cecchini

Zeidler

2023

Perspectives in Scalar Curvature

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We show that in every dimension n ≥ 8, there exists a smooth closed manifold M n which does not admit a smooth positive scalar curvature ("psc") metric, but M admits an L ∞ -metric which is smooth and has psc outside a singular set of codimension ≥ 8. This provides counterexamples to a conjecture of Schoen. In fact, there are such examples of arbitrarily high dimension with only single point singularities. In addition, we provide examples of L ∞ -metrics on R n for certain n ≥ 8 which are smooth and have psc outside the origin, but cannot be smoothly approximated away from the origin by everywhere smooth Riemannian metrics of non-negative scalar curvature. This stands in precise contrast to established smoothing results via Ricci-DeTurck flow for singular metrics with stronger regularity assumptions. Finally, as a positive result, we describe a KO-theoretic condition which obstructs the existence of L ∞ -metrics that are smooth and of psc outside a finite subset. This shows that closed enlargeable spin manifolds do not carry such metrics.

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Section: Theorem Bmentioning

confidence: 61%

“…For instance, if dim(M ) = 3, it holds in general by Li-Mantoulidis [28,Corollary 1.6]. In the 4-dimensional case, far-reaching results have been established by Kazaras [23]. In a closely related direction, a version of Llarull's theorem for metrics with L ∞ point singularities was recently proved by Chu-Lee-Zhu [12].…”

Section: Introductionmentioning

confidence: 99%

Scalar Curvature and Generalized Callias Operators

Cecchini

Zeidler

2023

Perspectives in Scalar Curvature

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Desingularizing positive scalar curvature 4-manifolds

Cited by 1 publication

References 33 publications

Scalar Curvature and Generalized Callias Operators

Scalar Curvature and Generalized Callias Operators

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