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Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundaryopen access

Authors
Kim, SeunghyeokMusso, MonicaWei, 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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COLLEGE OF NATURAL SCIENCES (DEPARTMENT OF MATHEMATICS)
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