Symmetry properties of the Jacobi Operator of minimal conical hypersurfaces

In this paper, we study nondegeneracy and symmetry properties of minimal hypersurfaces Σ, asymptotic to a nontrivial minimal cone C ⊂ RN+1 of co-dimension one. Our analysis is based on a careful study of the symmetries and asymptotic behavior of the elements of the kernel of the associated Jacobi operator JΣ := ∆Σ + |AΣ|2. Here, ∆Σ is the Laplace Beltrami operator of Σ, and |AΣ| is the norm of the second fundamental form of Σ. Also, under some suitable nondegeneracy assumptions about Σ, we construct a symmetry-preserving right inverse of JΣ. In other words, if f : Σ → R enjoys some symmetry properties and has suitable asymptotic behavior at infinity, we construct a solution ϕ : Σ → R to the Jacobi equation JΣϕ = f in Σ, enjoying the same symmetry properties as the given f . We also discuss some examples where our results can be applied.