Generic nondegeneracy of constant mean curvature radial graphs
Abstract
Given a smooth domain D⊂𝕊N−1, N≥ 3, and a positive constant H, we study radial graphs over D with constant mean curvature H which meet orthogonally the boundary of the cone spanned by D. The corresponding problem is given by a quasilinear elliptic equation on D with homogeneous Neumann boundary condition. We investigate the structure of the solution set under smooth perturbations of the domain which are parametrized by an open set of C3,α-diffeomorphisms of the sphere. After pulling the equation back to a fixed reference domain, we apply a classical transversality theorem and prove that, in the class of domains for which N−1 does not belong to the Neumann spectrum of the Laplace-Beltrami operator, there is a residual set for which every solution is nondegenerate. A consequence of our result is that, for every such domain and every R>0, the set of solutions whose C2,α-norm is bounded by R is finite.
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BibTeX
@misc{iacopetti2026generic,
title = {{Generic nondegeneracy of constant mean curvature radial graphs}},
author = {Alessandro Iacopetti and Filomena Pacella},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14755},
eprint = {2610.14755},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14755},
}
RIS
TY - UNPB
TI - Generic nondegeneracy of constant mean curvature radial graphs
AU - Alessandro Iacopetti
AU - Filomena Pacella
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14755
UR - https://arxiv.org/abs/2610.14755
AB - Given a smooth domain D⊂𝕊^N−1, N≥ 3, and a positive constant H, we study radial graphs over D with constant mean curvature H which meet orthogonally the boundary of the cone spanned by D. The corresponding problem is given by a quasilinear elliptic equation on D with homogeneous Neumann boundary condition. We investigate the structure of the solution set under smooth perturbations of the domain which are parametrized by an open set of C^3,α-diffeomorphisms of the sphere. After pulling the equation back to a fixed reference domain, we apply a classical transversality theorem and prove that, in the class of domains for which N−1 does not belong to the Neumann spectrum of the Laplace-Beltrami operator, there is a residual set for which every solution is nondegenerate. A consequence of our result is that, for every such domain and every R>0, the set of solutions whose C^2,α-norm is bounded by R is finite.
ER -