Compact minimal submanifolds of the Riemannian symmetric spaces \(\mathbf{S}U(n)/\mathbf{SO}(n)\), \(\mathbf{S}p(n)/\mathbf{U}(n)\), \(\mathbf{SO}(2n)/\mathbf{U}(n)\), \(\mathbf{S}U(2n)/\mathbf{S}p(n)\) via complex-valued eigenfunctions (Q6622297)

The authors construct a large number of explicit minimal submanifolds of codimension \(2\) in the classical Riemannian symmetric spaces \(\mathrm{SU}(n)/\mathrm{SO}(n)\), \(\mathrm{Sp}(n)/\mathrm{U}(n)\), \(\mathrm{SO}(2n)/\mathrm{U}(n)\), and \(\mathrm{SU}(2n)/\mathrm{Sp}(n)\) (Theorem 2.1, Theorem 2.2, Theorem 2.3, and Theorem 2.4). The main technical tool is a result (reproduced in Theorem 4.1), obtained originally in [\textit{S. Gudmundsson} and \textit{T. J. Munn}, J. Geom. Anal. 34, No. 6, Paper No. 190, 22 p. (2024; Zbl 1542.53063)] about so-called complex-valued eigenfunctions on the Riemannian ambient space, that is, functions which are eigen both with respect to the classical Laplace-Beltrami and the so-called conformality operator: Let \(\phi:(M,g) \rightarrow \mathbb{C}\) be a complex-valued eigenfunction on a Riemannian manifold, such that \(0\in \phi(M)\) is a regular value for \(\phi\), then the fiber \(\phi^{-1}({0})\) is a minimal submanifold of \(M\) of codimension two.\N\NBased on this idea, the authors construct families of suitable complex-valued eigenfunctions for each of the indicated symmetric spaces, which leads to the proof of the wanted results.