Fredholm properties of the jacobi Operator of minimal conical hypersurfaces

TL;DR

This paper investigates the non-degeneracy and solvability of the Jacobi operator on minimal hypersurfaces asymptotic to cones, providing conditions for invertibility and applications to specific examples.

Contribution

It establishes the Fredholm properties of the Jacobi operator for minimal conical hypersurfaces and constructs a right inverse under certain non-degeneracy conditions.

Findings

The Jacobi operator is shown to be Fredholm under specific conditions.

A right inverse for the Jacobi operator is constructed, ensuring solvability of the Jacobi equation.

Applications to particular minimal hypersurface examples are discussed.

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.