Nondegeneracy of low-energy nodal solutions to the Yamabe equation

Abstract

We study the linearized Yamabe equation in ${\mathbb{R}}^n$, 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 $\Gamma =O(k)\times O(k)$. 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 $\Gamma$-invariant subspace for every $k\ge 2$, 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.