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.
CLUSTERING PHENOMENA IN LOW DIMENSIONS FOR A BOUNDARY YAMABE PROBLEM
GIUSI VAIRA
In corso di stampa
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
