Topological aspects of \(\mathbb{Z} / 2 \mathbb{Z}\) eigenfunctions for the Laplacian on \(S^2\) (Q6593666)

This paper discusses how the eigenfunctions and eigenvalues of the spherical Laplacian acting on the space of sections of a real line bundle defined on the complement of an even numbers of points in \(S^2\) can behave if they are viewed as functions on the configuration space of each point.\N\NThe explanation in this paper is very detailed, and the total number of pages is \(84\), which is a bit long. Nevertheless, what is strange is that there are no results that the authors call theorems (though there are lemmas, propositions, and corollaries).\N\NInstead, in chapter 7, the authors give the following two interesting examples \((\mathrm{i})\) and \((\mathrm{ii})\), whereas it is usually the case in mathematical physics and differential equations that the multiplicity of the smallest eigenvalue of the Laplacian is one:\N\N\((\mathrm{i})\) The dimension of the eigenspace of the smallest eigenvalue on \(\mathcal{C}_2\) is two when the two points of the configuration are at antipodal points, \((\mathrm{ii})\) there are configurations in any \(n>1\) version of \(\mathcal{C}_{2n}\) where the multiplicity of the minimum eigenvalue of the spherical Laplacian is at least four. \N\NFurthermore, the authors mention as a future task to determine whether these critical points all lie in some fixed \(\mathcal{C}_{2j}\) strata with \(j\) having an \(n\) and \(m\) (\(n\) is natural number, \(m\) is non-negative integer) independent upper bound. In the above, \(\mathcal{C}_{2n}\) denote the space of unordered \(2n\)-tuples of points in \(S^2\) and \(\overline{\mathcal{C}}_{2n}\) denotes its compactification.\N\NFor \(\mathfrak{p}\in \mathcal{C}_{2n}\), let \(\mathcal{I}_{\mathfrak{p}}\) denote the real line bundle over \(S^2-\mathfrak{p}\) with monodromy \(-1\) on any embedded circle in \(S^2-\mathfrak{p}\). Let \(f\) be a cross section on \(\mathcal{I}_{\mathfrak{p}}\) such that the integral of \(f^2\) over \(S^2\) is \(1\). Let \(\mathcal{E}\) be a function defined for such \(f\) by \(E(\mathfrak{p},f)=\int_{S^2-\mathfrak{p}}|df|^2\), where \(df\) denote the corresponding covariant derivative.\N\NThen, by Lemma 4.2, for \(\mathbb{R}\mathbb{P}\) with the structure of a fiber bundle on \(\mathcal{C}_{2n}\) and its extension \(\overline{\mathbb{R}\mathbb{P}}\) to \(\overline{\mathcal{C}}_{2n}\) (see equation (5.1)), \(\overline{\mathbb{R}\mathbb{P}}\) has non-zero \(\mathbb{Z}/2\mathbb{Z}\) homology with degree \(4n+2m+1\), but in particular as Proposition 6.3, this class gives critical points on \(\mathcal{E}\) on \(\overline{\mathbb{R}\mathbb{P}}\), and as Lemma 6.2 it is shown that these critical points are unbounded.\N\NNext, the following is a mechanical enumeration of the titles of the chapters of this paper: \N1. Introduction (Proposition 1.1.); 2. Basic technology; 3. The lowest eigenvalue as a function on \(\mathcal{C}_{2n}\) (Lemma 3.1.); 4. The \(\mathbb{R}\mathbb{P}^{\infty}\) bundle; 5. The extension of \(\mathbb{R}\mathbb{P}\) and \(\mathcal{E}\) to \(\overline{\mathcal{C}}_{2n}\). 6. Min-max for \(\mathcal{E}\) on \(\overline{\mathbb{R}\mathbb{P}}\); 7. The case of \(\mathcal{C}_2\); Appendix A. The cohomology of a weak \(\overline{\mathbb{R}\mathbb{P}}^{\infty}\) bundle (Proposition A.1.). \N\NAt the end of the text, the authors close the paper with the following words: ``One can imagine that there is a topological explanation for the interesting spectral flow over \(\mathcal{C}_2\) via an analysis of the effect on \(\overline{\mathbb{R}\mathbb{P}}\) of the rotation group's action on \(S^2\).''