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

### Monica Musso

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom### Juncheng Wei

Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada### Seunghyeok Kim

Department of Mathematics and Research Institute for Natural Sciences, College of Natural Sciences, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Republic of Korea

## 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.

## Cite this article

Monica Musso, Juncheng Wei, Seunghyeok Kim, Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 6, pp. 1763–1793

DOI 10.1016/J.ANIHPC.2021.01.005