2022/11/15 by Cruz-Blázquez, Sergio, Pistoia, Angela, Vaira, Giusi · 1 citation
#35B44 #58J32 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2211.08219
We consider the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a n-dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if n=3 all the blow-up points are isolated and simple. In this work we prove that, for a linear perturbation, this is not true anymore in low dimensions 4≤ n≤ 7. In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and is a local minimizer of the norm of the trace-free second fundamental form of the boundary.