Positive scalar curvature with skeleton singularities

Abstract

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean (L ∞ ) metrics that consolidate Gromov's scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles ≤ 2π along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1-skeletons, exhibiting edge singularities (angles ≤ 2π) and arbitrary L ∞ isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities. arXiv:1708.08211v3 [math.DG] 18 Sep 2018 Edge singularitiesThe starting point of our discussion is the classical example of isolated conical singularities on two-dimensional Riemannian manifolds.

Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces

“…Then (M, g) is isometric to a flat polyhedron in R n . Conjecture 1.2 and related problems have been studied and extended in recent years (see, e.g., [16,17,18,22,23,24,25]), leading to a range of interesting new discoveries and questions on manifolds with nonnegative scalar curvature. In this paper, we investigate the analogous polyhedral comparison principle, together with the rigidity phenomenon, for metrics with negative scalar curvature lower bound.…”

Nonexistence of the NNSC-cobordism of Bartnik data

“…Then consider the closed manifold (Ω n ,g) obtained by the doubling (Ω, g) along the minimal boundary (Σ n−1 1 , γ 1 ). Similar to the proof of [13,Theorem 3] (see also [10,18]),Ω n admits a metricg ′ with positive scalar curvature. However,Ω n = T n #K#T n , where K is an n-dimensional closed orientable manifold obtained by the doubling of Ω n 2 ∪ Ω n 1 , and thenΩ n admits no metric with positive scalar curvature by [15,Corollary 2], which leads to a contradiction.…”

Weak scalar curvature lower bounds along Ricci flow

“…In this section, let us discuss some problems related to singular metrics. To study the scalar curvature of low-regularity metrics, Schoen proposed such a conjecture (see [LM19]): Conjecture 6.1 (Schoen). Let g be a C 0 metric on M which is smooth away from a submanifold Σ ⊂ M with codim(Σ ⊂ M) ≥ 3, σ(M) ≤ 0 and R g ≥ 0 on M \ Σ, then g smoothly extends to M and Ric g ≡ 0.…”

“…He proposed a question that if the Yamabe invariant σ(M) is nonpositive, the metric admits singularity in a subset and the scalar curvature is at least σ(M) away from the singular set, then whether we can prove that the metric is smooth and Ricci flat provided that the singular set is small, see [LM19]. Li and Mantoulidis [LM19] gave an answer for his skeleton metrics and for 3-manifolds with metrics admitting point singularities. For more related results, see the survey of Sormani [So21].…”