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Abstract
We concern C^{2}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 tracefree second fundamental form must vanish at possible blowup points of a sequence of blowingup solutions. Applying this result and the positive mass theorem, we deduce the C^{2}compactness for all 4manifolds (which may be nonumbilic). For the 5dimensional case, we also establish that a sum of the secondorder derivatives of the tracefree second fundamental form is nonnegative at possible blowup points. We essentially use this fact to obtain the C^{2}compactness for all 5manifolds. Finally, we show that the C^{2}compactness on 6manifolds is true if the tracefree second fundamental form on the boundary never vanishes.
Original language  English 

Pages (fromto)  17631793 
Number of pages  31 
Journal  Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire 
Volume  38 
Issue number  6 
Early online date  19 Feb 2021 
DOIs  
Publication status  Published  19 Oct 2021 
Bibliographical note
Funding Information:S. Kim is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2017R1C1B5076384, NRF2020R1C1C1A01010133, NRF2020R1A4A3079066), and the associate member problem of Korea institute for advanced study (KIAS). M. Musso has been supported by EPSRC research Grant EP/T008458/1. The research of J. Wei is partially supported by NSERC of Canada.
Keywords
 Blowup analysis
 Boundary Yamabe problem
 Compactness
 Positive mass theorem
ASJC Scopus subject areas
 Analysis
 Mathematical Physics
 Applied Mathematics
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Concentration phenomena in nonlinear analysis
Engineering and Physical Sciences Research Council
27/04/20 → 31/07/24
Project: Research council