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 $C\subset \mathbb{R}^{N+1}$ 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.

💡 Research Summary

The paper develops a comprehensive Fredholm theory for the Jacobi operator of minimal hypersurfaces that are asymptotic to a non‑trivial cone. Let Σ⊂ℝ^{N+1} be a smooth minimal hypersurface (co‑dimension one) which, outside a compact set, can be written as a normal graph over a cone C={rθ : r>0, θ∈Γ} with a small function w(r,θ). The cone C is assumed to be S‑invariant for a subgroup S⊂O(N+1) and Σ inherits the same symmetry. The authors impose four structural hypotheses (H1‑H4): C is non‑trivial and S‑invariant, Σ is S‑invariant, Σ is asymptotic to C in the precise C^{2} sense, and S is maximal with respect to (H1).

The Jacobi operator is J_Σ=Δ_Σ+|A_Σ|². Its geometric kernel consists of Jacobi fields generated by ambient translations, dilations and rotations. These are denoted ζ_j (translations), ζ_0 (dilation) and ζ_M (rotations). The decay rate of ζ_0 at infinity is measured by \