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

Seunghyeok Kim, Monica Musso, Juncheng Wei

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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.

Original languageEnglish
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Early online date19 Feb 2021
Publication statusE-pub ahead of print - 19 Feb 2021


  • Blow-up analysis
  • Boundary Yamabe problem
  • Compactness
  • Positive mass theorem

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

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