Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in \(\mathbb{R}^3\) (Q680705)

The paper under review makes an interesting contribution to our understanding of nodal sets, that is, sets of zeros, of Laplacian eigenfunctions. As the author notes, the geometry of the nodal set of a single eigenfunction can be quite complicated and we still have a rather poor understanding of them; here, however, following a paper of \textit{J. Bourgain} and \textit{Z. Rudnick} [Invent. Math. 185, No. 1, 199--237 (2011; Zbl 1223.58025)] and building on ideas of the author and \textit{E. T. Quinto} [J. Funct. Anal. 139, No. 2, 383--414 (1996; Zbl 0860.44002); Contemp. Math. 405, 1--10 (2006; Zbl 1107.35307)], the idea is to show that sets on which a ``large number'' of eigenfunctions all vanish must have a special structure. \par More precisely, the author considers a Paley-Wiener family of Laplacian eigenfunctions, that is, a family of solutions $\varphi_\lambda$ of the Helmholtz equation \[ \Delta \varphi_\lambda = -\lambda^2\varphi_\lambda \] on $\mathbb{R}^3$, which can be extended in the complex plane $\lambda \in \mathbb{C}$ to an even, nonzero entire function satisfying the growth bound \[|\varphi_\lambda (x)| \leq C(1+|\lambda|)^N e^{(R+|x|)|\mathrm{Im}\lambda|} \] for constants $C,R>0$ and $N \in \mathbb{N}$. The sets on which these functions are to vanish are taken as surfaces $S \subset \mathbb{R}^3$ which are irreducible real analytically ruled surfaces: roughly speaking, there exists a closed, locally analytic curve $\gamma \in \mathbb{R}^3$ (satisfying certain hypotheses) such that $S$ is the union of straight lines passing through $\gamma$. \par The first main result then states that, under a mild genericity condition, such a surface $S$ is the common nodal set of a Paley-Wiener family of eigenfunctions, then it must be an harmonic cone, a subset of the zero set (up to a shift) of an harmonic homogeneous polynomial; the second is for surfaces consisting of finite unions of the type described above. Further similar results are presented under stronger assumptions on $S$ or the eigenfunctions; for example, if $S$ is an immersed $C^1$ manifold, then $S$ is a so-called Coxeter system of planes. \par These results have an equivalent formulation in terms of ruled injectivity sets for the spherical mean transform, following the lines of the work of the author and Quinto [loc. cit.], and in particular answer a conjecture from that work, in the case of ruled surfaces in $\mathbb{R}^3$.