Fredholm properties of the jacobi Operator of minimal conical hypersurfaces

In this paper we study non-degeneracy properties of $Σ$ via the Jacobi operator J_Σ:=Δ_Σ+|A_Σ|² of a given minimal hypersurface $Σ$ asymptotic to a cone C⊂ RN+1 of co-dimension one. Here Δ_Σ is the Laplace Beltrami operator of $Σ$ and |A_Σ| is the norm of the second fundamental form of $Σ$. We also construct a right inverse of J_Σ, that is, we prove that the Jacobi equation J_Σϕ=f is solvable in $Σ$, at least under some suitable non-degeneracy assumptions about $Σ$ and about the asymptotic behavior of f at infinity. We also discuss some examples where our results can be applied.