Nondegeneracy of low-energy nodal solutions to the Yamabe equation

Sérgio L. N. Neves; Angela Pistoia

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arXiv · 2610.40960

Nondegeneracy of low-energy nodal solutions to the Yamabe equation

Sérgio L. N. Neves·Angela Pistoia

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Abstract

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 $Γ=O(k)\times O(k)$. 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 $k\ge2$, 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.

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Sérgio L. N. Neves, Angela Pistoia. 2026-10-29. Nondegeneracy of low-energy nodal solutions to the Yamabe equation. https://arxiv.org/abs/2610.40960

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