Department of Mathematics, Aula Seminari
In this talk we consider a boundary case of the classical Kazdan-Warner problem in dimension greater or equal than four, corresponding to the prescription of scalar and boundary mean curvatures under conformal transformations of
the metric. We deal in particular with the case of negative scalar curvature and boundary mean curvature of arbitrary sign, which has been less investigated.
In [C.-Malchiodi-Ruiz] existence results are obtained using Variational Methods on manifolds of dimension three and nonnegative Escobar class, which are crucially supported by the fact that the blow-up is isolated and simple in that case, but there is little information available for higher dimensions. Motivated by this fact, we construct clustering solutions for a linear perturbation of the problem around nondegenerate critical points of the norm of the second fundamental form on manifolds with positive Escobar class, and show that the blow-up is not isolated for dimensions 4,5,6 and 7.
This is joint work with Angela Pistoia (University of Rome ”La Sapienza”, Italy) and Giusi Vaira (University of Bari, Italy).