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

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.

29 pages, This version treats general 5-manifolds as well, Comments are welcome

References in corpus (4)

• Barycenter technique and the Riemann mapping problem of Escobar
• Compactness of conformal metrics with constant $Q$-curvature. I
• A compactness theorem of the fractional Yamabe problem, Part I: The non-umbilic conformal infinity
• A compactness result for scalar-flat metrics on manifolds with umbilic boundary