Fredholm properties of the jacobi Operator of minimal conical hypersurfaces

Abstract

In this paper we study non-degeneracy properties of $\Sigma$ via the Jacobi operator $J_\Sigma:=\Delta_\Sigma+|A_\Sigma|^2$ of a given minimal hypersurface $\Sigma$ asymptotic to a cone $C\subset \mathbb{R}^{N+1}$ of co-dimension one. Here $\Delta_{\Sigma}$ is the Laplace Beltrami operator of $\Sigma$ and $|A_{\Sigma}|$ is the norm of the second fundamental form of $\Sigma$. We also construct a right inverse of $J_{\Sigma}$, that is, we prove that the Jacobi equation $J_\Sigma\phi=f$ is solvable in $\Sigma$, at least under some suitable non-degeneracy assumptions about $\Sigma$ and about the asymptotic behavior of $f$ at infinity. We also discuss some examples where our results can be applied.