## 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 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 C^{2}-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 C^{2}-compactness for all 5-manifolds. Finally, we show that the C^{2}-compactness on 6-manifolds is true if the trace-free second fundamental form on the boundary never vanishes.

Original language | English |
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Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Early online date | 19 Feb 2021 |

DOIs | |

Publication status | E-pub ahead of print - 19 Feb 2021 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Mathematical Physics
- Applied Mathematics