Fredholm properties of the jacobi Operator of minimal conical hypersurfaces

2025/11/11 by Rico, Oscar Ivan Agudelo, Rizzi, Matteo
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.11804

Abstract

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⊂ 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.

Citations

• The Jacobi operator of some special minimal hypersurfaces
• k-ended O(m) times O(n) invariant solutions to the Allen-Cahn equation with infinite Morse index
• Symmetry properties of some solutions to some semilinear elliptic equations
• Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones
• Entire Solutions of the Allen-Cahn equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in \R³

Related

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• Maximizing the first eigenvalue of the Jacobi operator
• An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere
• Spectrum of the Laplacian and the Jacobi operator on Generalized rotational minimal hypersurfaces of spheres
• Index and topology of minimal hypersurfaces in Rⁿ