Ricerc@Sapienza

Let M be a connected compact C∞ manifold of dimension n≥4 without boundary. Let Mk be the set of all Ck Riemannian metrics on M. Any g∈Mk determines the Weyl tensor Wg:M→R4n,Wg(ξ):=(Wgijkl(ξ))i,j,k,l=1,…,n. We prove that the set A:=-g∈Mk:|Wg(ξ)|+|DWg(ξ)|+|D2Wg(ξ)|>0 for any ξ∈M} is an open dense subset of M k .

Let (M,g) be a non-locally conformally flat compact Riemannian manifold with dimension N≥7. We are interested in finding positive solutions to the linear perturbation of the Yamabe problem
−Lgu+ϵu=uN+2N−2 in (M,g)

We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n≥11.

We build blowing-up solutions for linear perturbation of the Yamabe problem
on manifolds with boundary, provided the dimension of the manifold is n ≥ 7
and the trace-free part of the second fundamental form is non-zero everywhere on the
boundary

Given (M, g), a smooth compact Riemannian n-manifold, we prove that for a generic Riemannian
metric g the critical points of the norm of the Weyl's tensor are non-degenerate.