Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundaryopen access
- Authors
- Kim, Seunghyeok; Musso, Monica; Wei, Juncheng
- Issue Date
- Nov-2021
- Publisher
- ELSEVIER
- Keywords
- Boundary Yamabe problem; Compactness; Blow-up analysis; Positive mass theorem
- Citation
- ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, v.38, no.6, pp.1763 - 1793
- Indexed
- SCIE
SCOPUS
- Journal Title
- ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
- Volume
- 38
- Number
- 6
- Start Page
- 1763
- End Page
- 1793
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/140602
- DOI
- 10.1016/j.anihpc.2021.01.005
- ISSN
- 0294-1449
- Abstract
- We concern C2-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are 4, 5 or 6. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions. Applying this result and the positive mass theorem, we deduce the C2-compactness for all 4-manifolds (which may be non-umbilic). For the 5-dimensional case, we also establish that a sum of the second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points. We essentially use this fact to obtain the C2-compactness for all 5-manifolds. Finally, we show that the C2-compactness on 6-manifolds is true if the trace-free second fundamental form on the boundary never vanishes.
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