Główne pojęcia
Closed oriented 4-manifolds with tolerable metric singularities along points and embedded circles can be desingularized to obtain smooth closed oriented manifolds with positive scalar curvature.
Streszczenie
The paper presents a desingularization process for 4-manifolds with positive scalar curvature (psc) metrics that have metric singularities along points and embedded circles.
Key highlights:
The authors show that the bordism group of closed 3-manifolds with psc metrics is trivial, using scalar-flat Kähler ALE surfaces discovered by Lock-Viaclovsky.
They develop a desingularization technique that allows them to prove a non-existence result for singular psc metrics on enlargeable 4-manifolds with uniformly Euclidean geometry.
As a consequence, the authors obtain a positive mass theorem for asymptotically flat 4-manifolds with non-negative scalar curvature and low regularity.
The desingularization process generally changes the underlying topology of the manifold, increasing the second Betti number, but a degree-1 map back to the original manifold is constructed.
The key technical tool is the triviality of the 3-dimensional oriented psc-bordism group ΩSO,+3 (S) where S is a finite 1-complex, proven in Theorem B.
Statystyki
There are no key metrics or figures used to support the author's main arguments.
Cytaty
There are no striking quotes supporting the author's key logics.