Nondegeneracy of low-energy nodal solutions to the Yamabe equation
Abstract
We study the linearized Yamabe equation in Rn, or equivalently on sn, at a low-energy sign-changing solution with two nodal domains. We consider odd dimensions n=2k−1 and a solution invariant under Γ=O(k)× O(k). Its regular level sets are isoparametric hypersurfaces, which foliate the sphere away from the two focal submanifolds. This symmetry reduces the equation for the solution to a singular ODE. In the balanced case, we choose a single-node profile that is odd under reflection across the midpoint. We prove nondegeneracy in the Γ-invariant subspace for every k≥2, and in the full space H1(sn) for sufficiently large k: every kernel element is generated by a conformal invariance of the equation. The proof uses separation of variables, Wronskian identities and comparison functions adapted to the singular equations for the angular modes.
Related
- Classification of finite energy positive solutions to Yamabe-type equation on the fifteen dimensional octonionic Heisenberg group
- Uniqueness and radial nondegeneracy of nodal states for a mixed-power scalar field equation
- The nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations
- Symmetry, uniqueness and nondegeneracy of positive solutions of the Choquard equation
- Isolated singularities of solutions to the Yamabe equation in higher dimensions I
BibTeX
@misc{neves2026nondegeneracy,
title = {{Nondegeneracy of low-energy nodal solutions to the Yamabe equation}},
author = {Sérgio L. N. Neves and Angela Pistoia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40960},
eprint = {2610.40960},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40960},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nondegeneracy of low-energy nodal solutions to the Yamabe equation
AU - Sérgio L. N. Neves
AU - Angela Pistoia
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40960
UR - https://arxiv.org/abs/2610.40960
AB - We study the linearized Yamabe equation in R^n, or equivalently on s^n, at a low-energy sign-changing solution with two nodal domains. We consider odd dimensions n=2k−1 and a solution invariant under Γ=O(k)× O(k). Its regular level sets are isoparametric hypersurfaces, which foliate the sphere away from the two focal submanifolds. This symmetry reduces the equation for the solution to a singular ODE. In the balanced case, we choose a single-node profile that is odd under reflection across the midpoint. We prove nondegeneracy in the Γ-invariant subspace for every k≥2, and in the full space H^1(s^n) for sufficiently large k: every kernel element is generated by a conformal invariance of the equation. The proof uses separation of variables, Wronskian identities and comparison functions adapted to the singular equations for the angular modes.
ER -