Classification of minimal immersions of conformally flat 3-tori and 4-tori into spheres by the first eigenfunctions (Q6624789)

The paper under review explores intimate relationships between the spectral geometry and the theory of minimal submanifolds in spheres dealing with the notion of \(\lambda_1\)-minimality.\N\NBy definition, an isometric immersion \(x:(M^m,g)\to \mathbb S^n\) of a Riemannian manifold \((M^m,g)\) into the sphere \(\mathbb S^n\) is called \(\lambda_1\)-minimal, if the coordinate functions of \(x\) are eigenfunctions corresponding to the first eigenvalue \(\lambda_1\) of the Laplacian on \((M^m,g)\), c.f. [\textit{T. Takahashi}, J. Math. Soc. Japan 18, 380--385 (1966; Zbl 0145.18601); \textit{A. El Soufi} and \textit{S. Ilias}, Pac. J. Math. 195, No. 1, 91--99 (2000; Zbl 1030.53043)].\N\NThe main attention in the paper is paid to \(\lambda_1\)-minimal immersions of \(m\)-dimensional tori \(\mathbb T^m\).\N\NThe only examples of two-dimensional \(\lambda_1\)-minimal tori are the Clifford torus in \(\mathbb S^3\) and the equilateral torus in \(\mathbb S^5\). The authors are interested in searching for \(\lambda_1\)-minimal tori in higher dimensions \(m>2\). Notice that properly rescaled products of \(\lambda_1\)-minimal tori generate \(\lambda_1\)-minimal tori referred to as reducible, and hence the question is to find \(\lambda_1\)-minimal tori which are irreducible.\N\NThe authors construct explicit examples of \(\lambda_1\)-minimal tori in the case of dimensions \(m=3, 4\). For \(m=3\) the list includes one example of \(\lambda_1\)-minimal torus in \(\mathbb S^n\) for each of \(n=5, 9, 11\), and two non-congruent examples in \(\mathbb S^7\). For \(m=4\) the list includes one example of \(\lambda_1\)-minimal torus in \(\mathbb S^{19}\), two non-congruent examples in \(\mathbb S^n\) for each of \(n=7, 17\), three non-congruent examples in \(\mathbb S^n\) for each of \(n=9, 13\), four non-congruent examples in \(\mathbb S^{11}\), a one-parametric family and two exceptional non-congruent examples in \(\mathbb S^{15}\), and a two-parametric family of non-congruent examples in \(\mathbb S^{23}\). The lists include irreducible \(\lambda_1\)-minimal tori as well as reducible ones.\N\NAs the main result, it is shown that the constructed examples provide an exhaustive classification of \(\lambda_1\)-minimal conformally flat tori of dimensions \(m=3, 4\).\N\NMoreover, motivated by the Berger problem, the authors consider the functional \(L_1\) which to any Riemannian metric \(g\) on the topological \(m\)-torus associates the dilatation-invariant quantity \(L(g) = \lambda_1(g)\cdot Vol^{\frac{2}{m}}(g)\). The problem of maximization of \(L_1\) is known to be related to \(\lambda_1\)-minimal tori in spheres. The authors compute the explicit values of \(L_1\) for all the examples they constructed above and then apply the results to prove the following statements concerning the Berger problem, c.f. [\textit{A. El Soufi} and \textit{S. Ilias}, Proc. Am. Math. Soc. 131, No. 5, 1611--1618 (2003; Zbl 1027.58010)].\N\NTheorem.\N\begin{itemize}\N\item[1.] For all flat metrics on the 3-dimensional topological torus one has \(L_1\leq 4(2)^{\frac{1}{3}}\pi^2\). \N\item[2.] For all flat metrics on the 4-dimensional topological torus one has \(L_1\leq 4(2)^{\frac{1}{2}}\pi^2\).\N\end{itemize}\N\NBoth estimates are optimal, the equalities are attained by particular \(\lambda_1\)-minimal tori discussed above.\N\NTheorem. Suppose \(g\) is a smooth Riemannian metric on the topological \(m\)-torus. If \(g\) is conformal equivalent to a flat metric whose first eigenspace is of dimension no less than \(2n\), then \(L_1(g)\leq 4(2)^{\frac{1}{3}}\pi^2\) if \(m=3\) and \(L_1(g)\leq 4(2)^{\frac{1}{2}}\pi^2\) if \(m=4\).